On the Orthogonality of q-Classical Polynomials of the Hahn Class

The central idea behind this review article is to discuss in a unified sense the orthogonality of all possible polynomial solutions of the q-hypergeometric difference equation on a q-linear lattice by means of a qualitative analysis of the q-Pearson equation. To be more specific, a geometrical appro...

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Опубліковано в: :	Symmetry, Integrability and Geometry: Methods and Applications
Дата:	2012
Автори:	Álvarez-Nodarse, R., Adıgüzel, R.S., Taşeli, H.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2012
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/148454
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Цитувати:	On the Orthogonality of q-Classical Polynomials of the Hahn Class / R. Álvarez-Nodarse, R.S. Adıgüzel, H. Taşeli // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 33 назв. — англ.

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spelling	Álvarez-Nodarse, R. Adıgüzel, R.S. Taşeli, H. 2019-02-18T12:56:54Z 2019-02-18T12:56:54Z 2012 On the Orthogonality of q-Classical Polynomials of the Hahn Class / R. Álvarez-Nodarse, R.S. Adıgüzel, H. Taşeli // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 33 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D45; 42C05 DOI: http://dx.doi.org/10.3842/SIGMA.2012.042 https://nasplib.isofts.kiev.ua/handle/123456789/148454 The central idea behind this review article is to discuss in a unified sense the orthogonality of all possible polynomial solutions of the q-hypergeometric difference equation on a q-linear lattice by means of a qualitative analysis of the q-Pearson equation. To be more specific, a geometrical approach has been used by taking into account every possible rational form of the polynomial coefficients in the q-Pearson equation, together with various relative positions of their zeros, to describe a desired q-weight function supported on a suitable set of points. Therefore, our method differs from the standard ones which are based on the Favard theorem, the three-term recurrence relation and the difference equation of hypergeometric type. Our approach enables us to extend the orthogonality relations for some well-known q-polynomials of the Hahn class to a larger set of their parameters. We thank the referees for their careful reading of the manuscript and the suggestions that helped us to improve the paper. This work was partially supported by MTM2009-12740-C03-02 (Ministerio de Econom´ıa y Competitividad), FQM-262, FQM-4643, FQM-7276 (Junta de Andaluc´ıa), Feder Funds (European Union), and METU OYP program (RSA). The second author (RSA) thanks the Departamento de An´alisis Matem´atico and IMUS for their kind hospitality during her stay in Sevilla. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Orthogonality of q-Classical Polynomials of the Hahn Class Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	On the Orthogonality of q-Classical Polynomials of the Hahn Class
spellingShingle	On the Orthogonality of q-Classical Polynomials of the Hahn Class Álvarez-Nodarse, R. Adıgüzel, R.S. Taşeli, H.
title_short	On the Orthogonality of q-Classical Polynomials of the Hahn Class
title_full	On the Orthogonality of q-Classical Polynomials of the Hahn Class
title_fullStr	On the Orthogonality of q-Classical Polynomials of the Hahn Class
title_full_unstemmed	On the Orthogonality of q-Classical Polynomials of the Hahn Class
title_sort	on the orthogonality of q-classical polynomials of the hahn class
author	Álvarez-Nodarse, R. Adıgüzel, R.S. Taşeli, H.
author_facet	Álvarez-Nodarse, R. Adıgüzel, R.S. Taşeli, H.
publishDate	2012
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	The central idea behind this review article is to discuss in a unified sense the orthogonality of all possible polynomial solutions of the q-hypergeometric difference equation on a q-linear lattice by means of a qualitative analysis of the q-Pearson equation. To be more specific, a geometrical approach has been used by taking into account every possible rational form of the polynomial coefficients in the q-Pearson equation, together with various relative positions of their zeros, to describe a desired q-weight function supported on a suitable set of points. Therefore, our method differs from the standard ones which are based on the Favard theorem, the three-term recurrence relation and the difference equation of hypergeometric type. Our approach enables us to extend the orthogonality relations for some well-known q-polynomials of the Hahn class to a larger set of their parameters.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/148454
citation_txt	On the Orthogonality of q-Classical Polynomials of the Hahn Class / R. Álvarez-Nodarse, R.S. Adıgüzel, H. Taşeli // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 33 назв. — англ.
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first_indexed	2025-11-25T07:14:46Z
last_indexed	2025-11-25T07:14:46Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 042, 30 pages On the Orthogonality of q-Classical Polynomials of the Hahn Class Renato ÁLVAREZ-NODARSE †, Rezan SEVİNİK ADIGÜZEL ‡ and Hasan TAŞELİ ‡ † IMUS & Departamento de Análisis Matemático, Universidad de Sevilla, Apdo. 1160, E-41080 Sevilla, Spain E-mail: ran@us.es URL: http://euler.us.es/~renato/ ‡ Department of Mathematics, Middle East Technical University (METU), 06531, Ankara, Turkey E-mail: sevinikrezan@gmail.com, taseli@metu.edu.tr Received July 29, 2011, in final form July 02, 2012; Published online July 11, 2012 http://dx.doi.org/10.3842/SIGMA.2012.042 Abstract. The central idea behind this review article is to discuss in a unified sense the orthogonality of all possible polynomial solutions of the q-hypergeometric difference equation on a q-linear lattice by means of a qualitative analysis of the q-Pearson equation. To be more specific, a geometrical approach has been used by taking into account every possible rational form of the polynomial coefficients in the q-Pearson equation, together with various relative positions of their zeros, to describe a desired q-weight function supported on a suitable set of points. Therefore, our method differs from the standard ones which are based on the Favard theorem, the three-term recurrence relation and the difference equation of hypergeometric type. Our approach enables us to extend the orthogonality relations for some well-known q-polynomials of the Hahn class to a larger set of their parameters. Key words: q-polynomials; orthogonal polynomials on q-linear lattices; q-Hahn class 2010 Mathematics Subject Classification: 33D45; 42C05 1 Introduction The so-called q-polynomials are of great interest inside the class of special functions since they play an important role in the treatment of several problems such as Eulerian series and continued fractions [8, 15], q-algebras and quantum groups [23, 24, 33] and q-oscillators [4, 10, 18], and references therein, among others. A q-analog of the Chebychev’s discrete orthogonal polynomials is due to Markov in 1884 [7, p. 43], which can be regarded as the first example of a q-polynomial family. In 1949, Hahn introduced the q-Hahn class [19] including the big q-Jacobi polynomials, on the exponential lattice although he did not use this terminology. In fact, he did not give the orthogonality relations of the big q-Jacobi polynomials in [19] which was done by Andrew and Askey [7]. During the last decades the q-polynomials have been studied by many authors from different points of view. There are two most recognized approaches. The first approach, initiated by the work of Askey and Wilson [9] (see also Andrews and Askey [7]) is based on the basic hypergeometric series [8, 17]. The second approach is due to Nikiforov and Uvarov [29, 31] and uses the analysis of difference equations on non-uniform lattices. The readers are also referred to the surveys [11, 28, 30, 32]. These approaches are associated with the so-called q- Askey scheme [21] and the Nikiforov–Uvarov scheme [30], respectively. Another approach was published in [27] where the authors proved several characterizations of the q-polynomials starting mailto:ran@us.es http://euler.us.es/~renato/ mailto:sevinikrezan@gmail.com mailto:taseli@metu.edu.tr http://dx.doi.org/10.3842/SIGMA.2012.042 2 R. Álvarez-Nodarse, R. Sevinik Adıgüzel and H. Taşeli from the so-called distributional q-Pearson equation (for the non q-case see, e.g., [16, 26] and references therein). In particular, in [27] a classification of all possible families of orthogonal polynomials on the exponential lattice was established, and latter on in [5] the comparison with the q-Askey and Nikiforov–Uvarov schemes was done, resulting in two new families of orthogonal polynomials. Furthermore, an important contribution to the theory of (orthogonal) q-polynomials, and in particular, to the theory of orthogonal q-polynomials on the linear exponential lattice, appeared in the recent book [21]. The corresponding table is generally called the q-Hahn tableau (see, e.g., Koornwinder [23]). The q-polynomials belonging to this class are the solutions of the q-difference equation of hypergeometric type (q-EHT) [19] σ1(x; q)Dq−1Dqy(x, q) + τ(x, q)Dqy(x, q) + λ(q)y(x, q) = 0. (1.1) One way of deriving the q-EHT (1.1) whose bounded solutions are the q-polynomials of the Hahn class, is to discretize the classical differential equation of hypergeometric type (EHT) σ(x)y′′ + τ(x)y′ + λy = 0, (1.2) where σ(x) and τ(x) are polynomials of at most second and first degree, respectively, and λ is a constant [2, 11, 26, 28, 31]. To this end, we can use the approximations (see e.g. [31, § 13, p. 142]) y′(x) ∼ 1 1 + q [Dqy(x) + qDq−1y(x)] and y′′(x) ∼ 2q 1 + q DqDq−1y(x) as q → 1 for the derivatives in (1.2), where we use the standard notation for the q and q−1-Jackson derivatives of y(x) [17, 20], i.e., Dζy(x) = y(x)− y(ζx) (1− ζ)x , ζ ∈ C \ {0,±1} for x 6= 0 and Dζy(0) = y′(0), provided that y′(0) exists. This leads to the q-EHT (1.1) where σ1(x; q) := 2 1 + q [ σ(x)− 1 2 (q − 1)xτ(x) ] , τ(x, q) := τ(x), λ(q) := λ, y(x, q) := y(x). Notice here the relations Dq = Dq−1 + (q − 1)xDqDq−1 and DqDq−1 = q−1Dq−1Dq so that (1.1) can be rewritten in the equivalent form σ2(x; q)DqDq−1y(x, q) + τ(x, q)Dq−1y(x, q) + λ(q)y(x, q) = 0, (1.3) where σ2(x, q) := q [ σ1(x, q) + ( 1− q−1 ) xτ(x, q) ] . (1.4) It should be noted that the q-EHT (1.1) and (1.3) correspond to the second order linear difference equations of hypergeometric type on the linear exponential lattices x(s) = c1q s + c2 and x(s) = c1q −s + c2, respectively [2, 11, 28]. Notice also that (1.1) (or (1.3)) can be written in a very convenient form [5, 21, 22] σ2(x, q)Dqy(x, q)− qσ1(x, q)Dq−1y(x, q) + (q − 1)xλ(q)y(x, q) = 0, where the coefficients σ1(x; q) and σ2(x; q) are polynomials of at most 2nd degree and τ(x, q) is a 1st degree polynomial in x. On the Orthogonality of q-Classical Polynomials of the Hahn Class 3 Notice that the q-EHT (1.1) can be written in the self-adjoint form Dq [ σ1(x, q)ρ(x, q)Dq−1y(x) ] + q−1λ(q)ρ(x, q)y(x) = 0, where ρ is a function satisfying the so-called q-Pearson equation Dq [σ1(x, q)ρ] = q−1τ(x, q)ρ that can be written as ρ(qx, q) ρ(x, q) = σ1(x, q) + (1− q−1)xτ(x, q) σ1(qx, q) = q−1σ2(x, q) σ1(qx, q) , (1.5) or, equivalently, σ2(x, q)ρ(x, q) = qσ1(qx, q)ρ(qx, q). (1.6) In this paper we study, without loss of generality, the q-EHT (1.1), assume 0 < q < 1 and take λ(q) as λ(q) := λn(q) = −[n]q [ τ ′(0, q) + 1 2 [n− 1]q−1σ′′1(0, q) ] , n ∈ N0 = N ∪ {0} since we are interested only in the polynomial solutions [2, 11, 28]. For more details on the q-polynomials of the q-Hahn tableau we refer the readers to the works [1, 2, 3, 5, 11, 13, 21, 23, 25, 28, 29, 30, 31, 32], and references therein. In this paper, we deal with the orthogonality properties of the q-polynomials of the q-Hahn tableau from a different viewpoint than the one used in [21]. In [21], the authors presented a unified study of the orthogonality of q-polynomials based on the Favard theorem. Here, the main idea is to provide a relatively simple geometrical analysis of the q-Pearson equation by taking into account every possible rational form of the polynomial coefficients of the q-difference equation. Roughly, our qualitative analysis is concerned with the examination of the behavior of the graphs of the ratio ρ(qx, q)/ρ(x, q) by means of the definite right hand side (r.h.s.) of (1.5) in order to find out a suitable q-weight function. Such a qualitative analysis implies all possible orthogonality relations among the polynomial solutions of the q-difference equation in question. Moreover, it allows us to extend the orthogonality relations for some well-known q-polynomials of the Hahn class to a larger set of their parameters (see Sections 4.1 and 4.2). A first attempt of using a geometrical approach for studying the orthogonality of q-polynomials of the q-Hahn class was presented in [13]. However, the study is far from being complete and only some partial results were obtained. We will fill this gap in this review paper. Our main goal is to study each orthogonal polynomial system or sequence (OPS), which is orthogonal with respect to (w.r.t.) a q-weight function ρ(x, q) > 0 satisfying the q-Pearson equation as well as certain boundary conditions (BCs) to be introduced in Section 2. For each family of polynomial solutions of (1.1) we search for the ones that are orthogonal in a suitable intervals depending on the range of the parameters coming from the coefficients of (1.1) and the corresponding q-Pearson equation. Hence, in Section 2, we present the candidate intervals by inspecting the BCs as well as some preliminary results. Theorems which help to calculate q-weight functions are given in Section 3. Section 4 deals with the qualitative analysis including the theorems stating the main results of our article. The last section concludes the paper with some final remarks. 2 The orthogonality and preliminary results We first introduce the so-called q-Jackson integrals and afterward a well known theorem for the orthogonality of polynomial solutions of (1.1) in order to make the article self-contained [2, 12, 28]. 4 R. Álvarez-Nodarse, R. Sevinik Adıgüzel and H. Taşeli The q-Jackson integrals for q ∈ (0, 1) [17, 20] are defined by∫ a 0 f(x)dqx = (1− q)a ∞∑ j=0 qjf(qja) and ∫ 0 a f(x)dqx = (1− q)(−a) ∞∑ j=0 qjf(qja) (2.1) if a > 0 and a < 0, respectively. Therefore, we have∫ b a f(x)dqx := ∫ b 0 f(x)dqx− ∫ a 0 f(x)dqx and∫ b a f(x)dqx := ∫ 0 a f(x)dqx+ ∫ b 0 f(x)dqx, (2.2) when 0 < a < b and a < 0 < b, respectively. Furthermore, we make use of the improper q-Jackson integrals∫ ∞ 0 f(x)dqx = (1− q) ∞∑ j=−∞ qjf(qj) and ∫ ∞ −∞ f(x)dqx = (1− q) ∞∑ j=−∞ qj [f(qj) + f(−qj)], (2.3) where the second one is sometimes called the bilateral q-integral. The q−1-Jackson integrals are defined similarly. For instance, the improper q−1-Jackson integral on (a,∞) is given by∫ ∞ a f(x)dq−1x = ( q−1 − 1 ) a ∞∑ j=0 q−jf ( q−ja ) , a > 0 (2.4) provided that limj→∞ q −jf(q−ja) = 0 and the series is convergent. Theorem 2.1. Let ρ be a function satisfying the q-Pearson equation (1.5) in such a way that the BCs σ1(x, q)ρ(x, q)xk ∣∣ x=a,b = σ2 ( q−1x, q ) ρ ( q−1x, q ) xk ∣∣ x=a,b = 0, k ∈ N0 (2.5) also hold. Then the sequence {Pn(x, q)} of polynomial solutions of (1.1) are orthogonal on (a, b) w.r.t. ρ(x, q) in the sense that∫ b a Pn(x, q)Pm(x, q)ρ(x, q)dqx = d2n(q)δmn, (2.6) where dn(q) and δmn denote the norm of the polynomials Pn and the Kronecker delta, respectively. Analogously, if the conditions σ2(x, q)ρ(x, q)xk ∣∣ x=a,b = σ1(qx, q)ρ(qx, q)xk ∣∣ x=a,b = 0, k ∈ N0 (2.7) are fulfilled, the q-polynomials then satisfy the relation∫ b a Pn(x, q)Pm(x, q)ρ(x, q)dq−1x = d2n(q)δmn. (2.8) On the Orthogonality of q-Classical Polynomials of the Hahn Class 5 Remark 2.2. The relation (2.6) means that the polynomials Pn(x, q) are orthogonal with respect to a measure supported on the set of points {qka}k∈N0 and {qkb}k∈N0 . Since we are interested in the positive definite cases, i.e., when ρ(x, q) > 0, then, • when a = 0, the measure is supported on the set of points {qkb}k∈N0 in (0, b]; • when a > 0, the measure should be supported on the finite set of points {qkb}Nk=0 being a = qN+1b; • when a < 0, the measure is supported on the set {qka}k∈N0 ⋃ {qkb}k∈N0 in [a, 0) ⋃ (0, b]. A similar analysis can be done for the relation (2.8). According to Theorem 2.1, we have to determine an interval (a, b) in which ρ is q-integrable and ρ > 0 on the lattice points of the types αq±k and βq±k for k ∈ N0 and α, β ∈ R. Such a weight function will be a solution of the q-Pearson equation (1.5). To this end, a qualitative analysis of the q-Pearson equation is presented by a detailed inspection of the r.h.s. of (1.5). Note that the r.h.s. of (1.5) consists of the polynomial coefficients σ1 and σ2 of the q-EHT which can be made definite for possible forms of the coefficients. As a result, the possible behavior of ρ on the left hand side (l.h.s.) of (1.5) and the candidate intervals can be obtained accordingly. OPSs on finite (a, b) intervals First assume that (a, b) denotes a finite interval. We list the following possibilities for finding ρ which obeys the BCs in (2.5) or in (2.7). PI. This is the simplest case where σ1 vanishes at both x = a and b, i.e., σ1(a, q) = σ1(b, q) = 0. Using (1.5) rewritten of the form ρ ( q−1x, q ) = qσ1(x, q) σ2(q−1x, q) ρ(x, q) (2.9) we see that the function ρ(x, q) becomes zero at the points q−ka and q−kb for k ∈ N. However, we have to take into consideration three different situations. (i) Let a < 0 < b. Since the points q−ka and q−kb lie outside the interval (a, b) and BCs are fulfilled at x = a and b, there could be an OPS w.r.t. a measure supported on the union of the set of points {aqk}k∈N0 and {bqk}k∈N0 in [a, 0) ∪ (0, b], if ρ is positive. (ii) Let 0 < a < b. In this case ρ(x, q) vanishes at the points q−ka in (a, b) and q−kb out of (a, b). Then, the only possibility to have an OPS on (a, b] depends on the existence of N such that qN+1b = a. This condition, however, implies that bqk = aq−(N−k) and that ρ vanishes at bqk for k = 0, 1, . . . , N , and, therefore, it must be rejected. The similar statement is true when a < b < 0, which can be obtained by a simple linear scaling transformation so that it does not represent an independent case. (iii) Let a = 0 < b (or, a < b = 0). This case is much more involved. First of all, if a = 0 is a zero of σ1(x, q) then it is a zero of σ2(x, q) as well, both containing a factor x. Therefore, the r.h.s. of q-Pearson equation (1.5) can be simplified and PI(i)-(ii) are not valid anymore. In fact, in this case an OPS w.r.t. a measure supported on the set of points {bqk}k∈N0 in (0, b] can be defined. PII. The relation in (1.6) suggest an alternative possibility to define an OPS on (a, b). Namely, if q−1a and q−1b are both zeros of σ2(x, q), by using (1.5) rewritten of the form ρ(qx, q) = q−1σ2(x, q) σ1(qx, q) ρ(x, q), (2.10) 6 R. Álvarez-Nodarse, R. Sevinik Adıgüzel and H. Taşeli it follows that ρ(x, q) vanishes at the points qka and qkb for k ∈ N0. Then two different situations appear depending on whether a < 0 < b or 0 < a < b. In the first case, ρ(x, q) = 0 at the points qka and qkb for k ∈ N0 in [a, b], which is not interesting. In the second case, the qkb are in [a, b] whereas the qka remain out of [a, b], so that we could have an OPS if there exists N such that q−N−1a = q−1b. However, since q−ka = qN−kb, ρ vanishes at the q−ka which are in [a, b] as well. PIII. Let q−1a and b be the roots of σ2 and σ1, respectively. Then we see, from (2.9) and (2.10), that ρ = 0 at x = q−kb for k ∈ N and at x = qka for k ∈ N0. That is, if a < 0 < b, ρ = 0 on x ∈ (a, 0) and, therefore, an OPS can not be constructed on (a, b) unless a → 0−. In this limiting case of x ∈ (0, b], it can be possible to introduce a desired weight function supported on the set {bqk}k∈N0 . If 0 < a < b, on the other hand, ρ vanishes for x < a and x > b. Thus there could be an OPS w.r.t. a measure supported on the finite set of points {qkb}Nk=0 provided that qN+1b = a for some finite N integer. Alternatively, we can define an equivalent OPS w.r.t. a measure supported on the equivalent finite set of points {q−ka}Nk=0 provided now that q−N−1a = b, where N is a finite integer. Note that in the limiting case of a → 0+ the set of points {qkb}k∈N0 becomes infinity. PIV. Assume that a and q−1b are the roots of σ1(x, q) and σ2(x, q), respectively. Then, from (2.9) and (2.10), it follows that ρ(x, q) vanishes at the points q−ka, k ∈ N and qkb, k ∈ N0. So, if a < 0 < b, it is not possible to find a weight function satisfying the BCs. Nevertheless, as in PIII, in the limiting case of b → 0+ an OPS w.r.t. a measure supported at the points qka, k ∈ N0 in [a, 0) can be constructed. If 0 < a < b, there is no possibility to introduce an OPS. Note that when a = 0 < b, an OPS also does not exist. OPSs on infinite intervals Assume now that (a, b) is an infinite interval. Without any loss of generality, let a be a finite number and b→∞. In fact, the system on the infinite interval (−∞, b) is not independent which may be transformed into (a,∞) on replacing x by −x. Obviously one BC in (2.5) reads as lim b→∞ σ1(b, q)ρ(b, q)bk = 0 or lim b→∞ σ2(b, q)ρ(b, q)bk = 0, k ∈ N0, and there are additional cases for x = a. PV. If x = a 6= 0 is root of σ1(x, q) then, from (2.9), ρ(x, q) vanishes at the points q−ka for k ∈ N which are interior points of (a,∞) when a > 0. Therefore there is no OPS on (a,∞) for a > 0. If a < 0 we can find a q-weight function in [a, 0) ∪ (0,∞) supported on the union of the sets {qka}k∈N0 and {q±kα}k∈N0 for arbitrary α > 0 where α can be taken as unity. If a = 0, on the other hand, then a weight function in (0,∞) can be defined at the points q±kα for arbitrary α > 0 and k ∈ N0. PVI. If x = q−1a is a root of σ2(x, q), as we have already discussed, ρ is zero at qka for k ∈ N0. Therefore, for a > 0 a q-weight function can exist in (q−1a,∞) supported on the set of points {q−ka}k∈N. An OPS does not exist if a < 0. Finally, if a = 0 it is possible to find a ρ on (0,∞) supported at the points q±kα for arbitrary α > 0 and k ∈ N0. PVII. Finally, we consider the possibility of satisfying the BC lim a→−∞ σ1(a, q)ρ(a, q)ak = 0 in the limiting case as a → −∞. If this condition holds a weight function and, hence, an OPS w.r.t. a measure supported on the set of points {±q±kα}k∈N0 , for arbitrary α > 0, can be defined. The aforementioned considerations are expressible as a theorem. On the Orthogonality of q-Classical Polynomials of the Hahn Class 7 Theorem 2.3. Let a1(q), b1(q) and a2(q), b2(q) denote the zeros of σ1(x, q) and σ2(x, q), respec- tively. Let ρ be a bounded and non-negative function satisfying the q-Pearson equation (1.5) as well as the BCs (2.5) or (2.7). Then ρ is a desired weight function for the polynomial solutions Pn(x, q) of (1.1) only in the following cases: 1. Let a < 0 < b, where a = a1(q) and b = b1(q). Then ρ is supported on {qka}k∈N0 ⋃ {qkb}k∈N0 and∫ b1(q) a1(q) Pn(x, q)Pm(x, q)ρ(x, q)dqx = d2n(q)δmn, (2.11) where the q-Jackson integral is of type (2.2). 2. Let a = 0 < b, where b = a1(q). Then ρ is supported on the set of points {qkb}k∈N0 in (0, b] and∫ a1(q) 0 Pn(x, q)Pm(x, q)ρ(x, q)dqx = d2n(q)δmn, (2.12) where the q-Jackson integral is of type (2.1). 3. Let 0 < a < b, where a = a2(q) and b = q−1a1(q). Then ρ is supported on the finite set of points {q−ka}Nk=0 when q−N−1a = b and∫ q−1a1(q)=q−N−1a2(q) a2(q) Pn(x, q)Pm(x, q)ρ(x, q)dq−1x = d2n(q)δmn, (2.13) which is the finite sum of the form∫ q−N−1a2(q) a2(q) [·]dq−1x = ( 1− q−1 ) a2(q) N∑ k=0 Pn(q−ka2(q), q)Pm(q−ka2(q), q)ρ(q−ka2(q), q). 4. Let a = a1(q) < 0 and b→∞. Then ρ is supported on the set {qka}k∈N0 ⋃ {q∓kα}k∈N0 for arbitrary α > 0 and∫ ∞ a1(q) Pn(x, q)Pm(x, q)ρ(x, q)dqx := ∫ 0 a1(q) [·]dqx+ ∫ ∞ 0 [·]dqx = d2n(q)δmn, (2.14) where the first q-Jackson integral is of type (2.1) and the second one is of type (2.3), respectively. 5. Let a = a2(q) > 0 and b → ∞. Then ρ is supported on the set of points {q−ka}k∈N0 in [a,∞) and∫ ∞ a2(q) Pn(x, q)Pm(x, q)ρ(x, q)dq−1x = d2n(q)δmn, (2.15) where the q−1-Jackson integral is of type (2.4). 6. Let a = 0 and b→∞. Then ρ is supported on the set of points {q±kα}k∈N0 for arbitrary α > 0 and∫ ∞ 0 Pn(x, q)Pm(x, q)ρ(x, q)dqx = d2n(q)δmn, (2.16) where the q-Jackson integral is of type (2.3). 8 R. Álvarez-Nodarse, R. Sevinik Adıgüzel and H. Taşeli 7. Let a → −∞ and b → ∞. Then ρ is supported on the set of points {∓q±kα}k∈N0 for arbitrary α > 0 and∫ ∞ −∞ Pn(x, q)Pm(x, q)ρ(x, q)dqx = d2n(q)δmn, (2.17) where the bilateral q-Jackson integral is of type (2.3). Before starting our analysis, let us mention that in accordance with [5, 27, 30] the q- polynomials can be classified by means of the degrees of the polynomial coefficients σ1 and σ2 and the fact that either σ1(0, q)σ2(0, q) 6= 0 or σ1(0, q) = σ2(0, q) = 0. Therefore, we can define two classes, namely, the non-zero (∅) and zero (0) classes corresponding to the cases where σ1(0, q)σ2(0, q) 6= 0 and σ1(0, q) = σ2(0, q) = 0, respectively (this is a consequence of the fact that σ2(0, q) = qσ1(0, q), i.e., σ1 and σ2 both have the same constant terms). In each class we consider all possible degrees of the polynomial coefficients σ1(x, q) and σ2(x, q) as shown in [27, p. 182]. We follow the notation introduced in [5, 27], i.e., the statement ∅-Laguerre/Jacobi im- plies that deg σ2 = 1, deg σ1 = 2, and σ1(0, q)σ2(0, q) 6= 0 and the statement 0-Jacobi/Laguerre indicates that deg σ2 = 2, deg σ1 = 1 and σ1(0, q) = σ2(0, q) = 0. In the following we use the Taylor polynomial expansion for the coefficients τ(x, q) = τ ′(0, q)x+ τ(0, q), τ ′(0, q) 6= 0, σ1(x, q) = 1 2 σ′′1(0, q)x2 + σ′1(0, q)x+ σ1(0, q) = 1 2 σ′′1(0, q)[x− a1(q)][x− b1(q)], (2.18) σ2(x, q) = 1 2 σ′′2(0, q)x2 + σ′2(0, q)x+ σ2(0, q) = 1 2 σ′′2(0, q)[x− a2(q)][x− b2(q)]. Theorem 2.4 (classification of the OPS of the q-Hahn class [5, 27]). All orthogonal polynomial solutions of the q-difference equations (1.1) and (1.3) can be classified as follows: 1. ∅-Jacobi/Jacobi polynomials where σ′′2(0, q) 6= 0 and σ′′1(0, q) 6= 0 with σ1(0, q)σ2(0, q) 6= 0. 2. ∅-Jacobi/Laguerre polynomials where σ′′2(0, q) 6= 0 and σ′′1(0, q) = 0, σ′1(0, q) 6= 0 with σ1(0, q)σ2(0, q) 6= 0. 3. ∅-Jacobi/Hermite polynomials where σ′′2(0, q) 6= 0, σ′′1(0, q) = 0, σ′1(0, q) = 0 and σ1(0, q) 6= 0 with σ1(0, q)σ2(0, q) 6= 0. 4. ∅-Laguerre/Jacobi polynomials where σ′′2(0, q) = 0, σ′2(0, q) 6= 0 and σ′′1(0, q) 6= 0 with σ1(0, q)σ2(0, q) 6= 0. 5. ∅-Hermite/Jacobi polynomials where σ′′2(0, q) = 0, σ′2(0, q) = 0, σ2(0, q) 6= 0 and σ′′1(0, q) 6= 0 with σ1(0, q)σ2(0, q) 6= 0. 6. 0-Jacobi/Jacobi polynomials where σ′′2(0, q) 6= 0, σ′2(0, q) 6= 0 and σ′′1(0, q) 6= 0, σ′1(0, q) 6= 0 with σ2(0, q) = σ1(0, q) = 0. 7. 0-Jacobi/Laguerre polynomials where σ′′2(0, q) 6= 0, σ′2(0, q) 6= 0 and σ′′1(0, q) = 0, σ′1(0, q) 6= 0 with σ2(0, q) = σ1(0, q) = 0. 8. 0-Bessel/Jacobi polynomials where σ′′2(0, q) 6= 0, σ′2(0, q) = 0 and σ′′1(0, q) 6= 0, σ′1(0, q) 6= 0 with σ2(0, q) = σ1(0, q) = 0. 9. 0-Bessel/Laguerre polynomials where σ′′2(0, q) 6= 0, σ′2(0, q) = 0 and σ′′1(0, q) = 0, σ′1(0, q) 6= 0 with σ2(0, q) = σ1(0, q) = 0. 10. 0-Laguerre/Jacobi polynomials where σ′′2(0, q) = 0, σ′2(0, q) 6= 0 and σ′′1(0, q) 6= 0, σ′1(0, q) 6= 0 with σ2(0, q) = σ1(0, q) = 0. On the Orthogonality of q-Classical Polynomials of the Hahn Class 9 3 The q-weight function In the following sections we will discuss the solutions of the q-Pearson equation (1.5) defined on the q-linear lattices enumerated in Remark 2.2. Since it is a linear difference equation on a given lattice, its solution can be uniquely determined by the equation (1.5) and the boundary conditions (for more details on the general theory of linear difference equations, see e.g. [14, § 1.2]). In fact, the explicit form of a q-weight function can be deduced by means of Theo- rem 3.1. Theorem 3.1. Let f satisfy the difference equation f(qx; q) f(x; q) = a(x; q) b(x; q) , (3.1) such that the limits lim x→0 f(x; q) = f(0, q) and lim x→∞ f(x; q) = f(∞, q) exist, where a(x; q) and b(x; q) are definite functions. Then f(x; q) admits the two q-integral representations f(x, q) = f(0, q) exp [∫ x 0 1 (q − 1)t ln [ a(t, q) b(t, q) ] dqt ] (3.2) and f(x, q) = f(∞, q) exp [∫ ∞ x 1 (1− q−1)t ln [ a(t, q) b(t, q) ] dq−1t ] (3.3) provided that the integrals are convergent. Proof. Taking the logarithms of both sides of (3.1), multiplying by 1/(q − 1)t and then inte- grating from 0 to x, we have∫ x 0 1 (q − 1)t ln [ f(qt, q) f(t, q) ] dqt = ∫ x 0 1 (q − 1)t ln [ a(t, q) b(t, q) ] dqt. The l.h.s. is expressible as∫ x 0 1 (q − 1)t ln [ f(qt, q) f(t, q) ] dqt = lim n→∞ n∑ j=0 [ ln ( f(qjx, q) ) − ln ( f(qj+1x, q) )] = ln [f(x, q)]− ln [f(0, q)] , which completes the proof, on using the fact that f(qn+1x, q)→ f(0, q) as n→∞ for 0 < q < 1. The second representation (3.3) can be proven in a similar way. � This theorem can be used to derive the q-weight functions for every σ1 and σ2. However, here we take into account the quadratic coefficients leading to ∅-Jacobi/Jacobi and 0-Jacobi/Jacobi cases. The results, some of which can be found in [5], are stated by the next theorem. Theorem 3.2. In the ∅-Jacobi/Jacobi case, let σ1(x, q) and σ2(x, q) be of forms (2.18) in which σ′′1(0, q)a1(q)b1(q) 6= 0 and σ′′2(0, q)a2(q)b2(q) 6= 0. And let, in 0-Jacobi/Jacobi case, b1(q) = b2(q) = 0, σ′′1(0, q) 6= 0 and σ′′2(0, q) 6= 0. Then a solution ρ(x, q) of q-Pearson equation (1.5) is expressible in the equivalent forms shown in Table 1 where H(1)(x) is given by [5] H(1)(x) = √ xlogq x−1 = \|x\|− 1 2 q 1 2 (logq x) 2 . (3.4) 10 R. Álvarez-Nodarse, R. Sevinik Adıgüzel and H. Taşeli Table 1. Expressions for the q-weight function ρ(x, q) in the Jacobi/Jacobi cases. ∅-Jacobi/Jacobi case 1. (a−11 qx, b−11 qx; q)∞ (a−12 x, b−12 x; q)∞ 2. \|x\|α (b−11 qx, a2q/x; q)∞ (a1/x, b −1 2 x; q)∞ where qα = q−2σ′′2 (0, q)b2 σ′′1 (0, q)b1 0-Jacobi/Jacobi case 3. \|x\|α (a−11 qx; q)∞ (a−12 x; q)∞ where qα = q−2σ′′2 (0, q)a2 σ′′1 (0, q)a1 4. \|x\|αH(1)(x)(qa−11 x, qa2/x; q)∞ where qα = q−2σ′′2 (0, q) σ′′1 (0, q)a1 Proof. We start proving the first expression in Table 1. Keeping in mind that q−1σ2(0, q) = σ1(0, q) and that q−1σ′′2(0, q)a2(q)b2(q) = σ′′1(0, q)a1(q)b1(q) we have from (1.5) ρ(qx, q) ρ(x, q) = q−1σ′′2(0, q)[x− a2(q)][x− b2(q)] σ′′1(0, q)[qx− a1(q)][qx− b1(q)] = [1− a−12 (q)x][1− b−12 (q)x] [1− a−11 (q)qx][1− b−11 (q)qx] (3.5) which gives ρ(x, q) = ρ(0, q) exp {∫ x 0 1 (q − 1)t [ ln ( 1− a−12 t )( 1− b−12 t ) − ln ( 1− a−11 qt )( 1− b−11 qt )] dqt } on using (3.2). By definition (2.1) of the q-integral, we first obtain ρ(x, q) = ρ(0, q) exp { ln ∞∏ k=0 ( 1− a−11 qk+1x )( 1− b−11 qk+1x ) − ln ∞∏ k=0 ( 1− a−12 qkx )( 1− b−12 qkx )} and, therefore, ρ(x, q) = ρ(0, q) (a−11 qx, b−11 qx; q)∞ (a−12 x, b−12 x; q)∞ , ρ(0, q) 6= 0. (3.6) This implies that a1(q)q −1−k and b1(q)q −1−k for k ≥ 0 are zeros of ρ. Furthermore, a2(q)q −j and b2(q)q −j for j ≥ 0 stand for the simple poles of ρ. Note here that ρ(0, q) can be made unity, and a1(q), b1(q), a2(q) and b2(q) are non-zero constants. Therefore the solution in (3.6) is continuous everywhere except at the simple poles. To show 4, we rewrite the q-Pearson equation in the form ρ(qx, q) ρ(x, q) = ax[1− a2(q)/x] [1− a−11 (q)qx] , a = q−2σ′′2(0, q) σ′′1(0, q)a1(q) (3.7) and assume that ρ is a product of three functions ρ(x, q) = f(x, q)g(x, q)h(x, q). Hence, if f , g and h are solutions of f(qx, q) f(x, q) = ax, g(qx, q) g(x, q) = 1 [1− a−11 (q)qx] and h(qx, q) h(x, q) = [ 1− a2(q) x ] (3.8) On the Orthogonality of q-Classical Polynomials of the Hahn Class 11 respectively, then ρ = fgh is a solution of (3.7). A solution of (3.8) for f(x, q) is of the form f(x, q) = \|x\|αH(1)(x), where H(1)(x) is given by (3.4) and α 6= 0 is such that qα = a, which can be verified by direct substitution. The equation in (3.8) for g(x, q) can be solved in a way similar to that of (3.5). So we find that g(x, q) = g(0, q)(a−11 qx; q)∞, where g(0, q) = 1. The expres- sion (3.2) is not suitable in finding h(x, q) which gives a divergent infinite product. Instead, we employ (3.3) so that the equation for h(x, q) becomes h(q−1x, q) h(x, q) = 1 [1− qa2(q)/x] whose solution is of the form h(x, q) = h(∞, q)(qa2/x; q)∞, where h(∞, q) can be taken again as unity without loss of generality. Clearly h(x, q) is uniformly convergent in any compact subset of the complex plane that does not contain the point at the origin. Moreover, the product converges to an arbitrary constant c, which has been set to unity, as x→∞. Thus, ρ(x, q) = f(x, q)g(x, q)h(x, q) = \|x\|αH(1)(x) ( qa−11 x, qa2/x; q ) ∞. In order to obtain the expressions 2 and 3 in Table 1 for the weight function we use the same procedure as before, but starting from the q-Pearson equation written in the forms ρ(qx, q) ρ(x, q) = a [1− a2(q)/x][1− b−12 (q)x] [1− a1(q)q−1/x][1− b−11 (q)qx] , a = q−2σ′′2(0, q)b2(q) σ′′1(0, q)b1(q) (3.9) and ρ(qx, q) ρ(x, q) = a [1− a−12 (q)x] [1− a−11 (q)qx] , a = q−2σ′′2(0, q)a2(q) σ′′1(0, q)a1(q) , (3.10) respectively. This completes the proof. � Remark 3.3. Notice that for getting the expressions of the weight function we have used the q- Pearson equation rewritten in different forms, namely (3.5), (3.7), (3.9) and (3.10), and different solution procedure in each case, therefore, it is not surprising that ρ has several equivalent representations displayed in Table 1. However, they all satisfy the same linear equation and, therefore, they differ only by a multiplicative constant. For the sake of the completeness, let us obtain the analytic representations of q-weight func- tions satisfying (1.5) for the other cases. Theorem 3.4. Let σ1 and σ2 be polynomials of at most 2nd degree in x as the form (2.18). Then a solution ρ(x, q) of q-Pearson equation (1.5) for each ∅-Jacobi/Laguerre, ∅-Jacobi/Hermite, ∅- Laguerre/Jacobi, ∅-Hermite/Jacobi, 0-Jacobi/Laguerre, 0-Bessel/Jacobi, 0-Bessel/Laguerre and 0-Laguerre/Jacobi case is expressible in the equivalent forms shown in Table 2 where H(1)(x) is given by (3.4). Proof. The proof is similar to the previous one. That is, to obtain the second formula for the ∅-Jacobi/Laguerre family we rewrite the q-Pearson equation (1.5) in the form ρ(qx, q) ρ(x, q) = ax [1− a2/x][1− b2/x] [1− a1q−1/x] , a = q−2 12σ ′′ 2(0, q) σ′1(0, q) and then apply the same procedure described in the proof of the previous theorem. � 12 R. Álvarez-Nodarse, R. Sevinik Adıgüzel and H. Taşeli Table 2. Expressions for the q-weight function ρ(x, q) for the other cases. ∅-Jacobi/Laguerre 1. (a−11 qx; q)∞ (a−12 x, b−12 x; q)∞ 2. \|x\|αH(1)(x) (a2q/x, b2q/x; q)∞ (a1/x; q)∞ , qα = 1 2σ ′′ 2 (0, q)q−2 σ′1(0, q) 3. \|x\|α+1 [H(1)(x)]2(qa−11 x, qa2/x, qb2/x; q)∞, qα = − q−2 1 2σ ′′ 2 (0, q) σ′1(0, q)a1 ∅-Jacobi/Hermite 1. 1 (a−12 x, b−12 x; q)∞ 2. \|x\|α [H(1)(x)]2(a2q/x, b2q/x; q)∞, qα = 1 2σ ′′ 2 (0, q)q−1 σ1(0, q) ∅-Laguerre/Jacobi 1. (a−11 qx, b−11 qx; q)∞ (a−12 x; q)∞ 2. \|x\|α (qa2/x, qb −1 1 x; q)∞ (a1/x; q)∞ , qα = − q −2σ′2(0, q) 1 2σ ′′ 1 (0, q)b1 ∅-Hermite/Jacobi (a−11 qx, b−11 qx; q)∞ 0-Jacobi/Laguerre 1. \|x\|α 1 (a−12 x; q)∞ , qα = − q−2 1 2σ ′′ 2 (0, q)a2 σ′1(0, q) 2. \|x\|αH(1)(x)(qa2/x; q)∞, qα = q−2 1 2σ ′′ 2 (0, q) σ′1(0, q) 0-Bessel/Jacobi \|x\|αH(1)(x)(a−11 qx; q)∞, qα = − q−2 1 2σ ′′ 2 (0, q) 1 2σ ′′ 1 (0, q)a1 0-Bessel/Laguerre \|x\|αH(1)(x), qα = q−2 1 2σ ′′ 2 (0, q) σ′1(0, q) 0-Laguerre/Jacobi \|x\|α (a−11 qx; q)∞, qα = − q −2σ′2(0, q) 1 2σ ′′ 1 (0, q)a1 4 The orthogonality of q-polynomials In this section we study the orthogonality of all families of q-polynomials of the Hahn class by taking into account the rational function on the r.h.s. of the q-Pearson equation (1.5). Since it is the ratio of two polynomials σ1 and σ2 of at most second degree, we deal with a definite rational function having at most two zeros and two poles. In the analysis of the unknown quantity ρ(qx, q)/ρ(x, q) on the l.h.s. of (1.5), we sketch roughly its graph by using every possible form of the definite rational function in question. In particular, we split the x-interval into subintervals according to whether ρ(qx, q)/ρ(x, q) < 1 or ρ(qx, q)/ρ(x, q) > 1, which yields valuable information about the monotonicity of ρ(x, q). Other significant properties of ρ are provided by the asymptotes, if there exist any, of ρ(qx, q)/ρ(x, q). A full analysis along these lines is sufficient for a complete characterization of the orthogonal q-polynomials. Since ∅-Jacobi/Jacobi and ∅-Jacobi/Laguerre families include new results, we present the detailed analysis of these two cases in Sections 4.1 and 4.2, respectively. For the other families we only include a list leading to a positive definite OPS and compare them with those obtained in [21]. For a complete study see [6]. 4.1 q-classical ∅-Jacobi/Jacobi polynomials Let the coefficients σ2 and σ1 be quadratic polynomials in x such that σ1(0, q)σ2(0, q) 6= 0. If σ1 is written in terms of its roots, i.e., σ1(x, q) = 1 2σ ′′ 1(0, q)[x− a1(q)][x− b1(q)] then, from (1.4), σ2(x, q) = [ σ′′1(0, q) 2 + ( 1− q−1 ) τ ′(0, q) ] qx2 − [ σ′′1(0, q) 2 (a1 + b1)− ( 1− q−1 ) τ(0, q) ] qx+ σ′′1(0, q) 2 a1b1, On the Orthogonality of q-Classical Polynomials of the Hahn Class 13 where 1 2σ ′′ 1(0, q) + (1− q−1)τ ′(0, q) 6= 0 by hypothesis. Then q-Pearson equation (1.5) takes the form f(x, q) := ρ(qx, q) ρ(x, q) = q−1σ2(x, q) σ1(qx, q) = [ 1 + (1− q−1)τ ′(0, q) 1 2σ ′′ 1(0, q) ] [x− a2(q)][x− b2(q)] [qx− a1(q)][qx− b1(q)] (4.1) provided that the discriminant, denoted by ∆q, ∆q := [ a1(q) + b1(q)− (1− q−1)τ(0, q) 1 2σ ′′ 1(0, q) ]2 − 4a1(q)b1(q) [ 1 + (1− q−1)τ ′(0, q) 1 2σ ′′ 1(0, q) ] , of the quadratic polynomial in the nominator of f(x, q) in (4.1) is non-zero. Here x = a2 and x = b2 denote the zeros of f , and they are constant multiples of the roots of σ2(x, q). We see that the lines x = q−1a1 and x = q−1b1 stand for the vertical asymptotes of f(x, q) and the point y = 1 is always its y-intercept since σ2(0, q) = qσ1(0, q). Moreover, the locations of the zeros of f are determined by the straightforward lemma. Lemma 4.1. Define the parameter Λq = 1 q2 [ 1 + (1− q−1)τ ′(0, q) 1 2σ ′′ 1(0, q) ] 6= 0 (4.2) so that the line y = Λq denotes the horizontal asymptote of f(x, q). Then we encounter the following cases for the roots of the equation f(x, q) = 0. Case 1. If Λq > 0 and a1(q) < 0 < b1(q), f has two real and distinct roots with opposite signs. Case 2. If Λq > 0 and 0 < a1(q) < b1(q), there exist three possibilities (a) if ∆q > 0, f has two real roots with the same signs, (b) if ∆q = 0, f has a double root, (c) if ∆q < 0, f has a pair of complex conjugate roots. Case 3. If Λq < 0 and a1(q) < 0 < b1(q), there exist three possibilities (a) if ∆q > 0, f has two real roots with the same signs, (b) if ∆q = 0, f has a double root, (c) if ∆q < 0, f has a pair of complex conjugate roots. Case 4. If Λq < 0 and 0 < a1(q) < b1(q), f has two real distinct roots with opposite signs. From (4.1) it is clear that we need to consider the cases Λq > 1 and 0 < Λq < 1 separately. Now, our strategy consists of sketching first the graphs of f(x, q) depending on all possible relative positions of the zeros of σ1 and σ2. To obtain the behaviors of q-weight functions ρ from the graphs of f(x, q) = ρ(qx, q)/ρ(x, q), we divide the real line into subintervals in which ρ is either monotonic decreasing or increasing. We take into consideration only the subintervals where ρ > 0. Note that if ρ is initially positive then we have ρ > 0 everywhere in an interval where ρ(qx, q)/ρ(x, q) > 0. Then we find suitable intervals in cooperation with Theorem 2.3. In Fig. 1A, the intervals (q−1a1, a2) and (b2, q −1b1) are rejected immediately since f is nega- tive. The subinterval (a2, b2) should also be rejected in which ρ = 0 by PII. For the same reason (q−1b1,∞) and (−∞, q−1a1), by symmetry, are not suitable by PV. Therefore, an OPS fails to exist. Let us analyze the problem presented in Fig. 1B. The positivity of ρ implies that the intervals (q−1a1, a2) and (q−1b1, b2) should be eliminated. With the transformation x = −t, we eliminate 14 R. Álvarez-Nodarse, R. Sevinik Adıgüzel and H. Taşeli 0 0 a 2q −1 a 1 1 A b 2 y x q −1 b 1 0 0 1 B x y q −1 a 1 q −1 b 1 b 2 a 2 Figure 1. The graph of f(x, q) in Case 1 with Λq > 1. In A, the zeros are in order q−1a1 < a2 < 0 < b2 < q−1b1, and in B, q−1a1 < a2 < 0 < q−1b1 < b2. also (−∞, q−1a1) by PV. The interval (a2, q −1b1) is not suitable too, by PIII. So it remains only (b2,∞) to examine which coincides with the 5th case in Theorem 2.3. Since ρ(qx, q)/ρ(x, q) = 1 at x0 = −τ(0, q)/τ ′(0, q) > b2(q), then ρ is increasing on (b2, x0) and decreasing on (x0,∞). As is shown from the figure f has a finite limit as x→ +∞ so that we could have the case ρ→ 0 as x→∞. Even if ρ→ 0 as x→∞, we must show also that σ1(x, q)ρ(x, q)xk → 0 as x→∞ to satisfy the BC. In fact, instead of the usual q-Pearson equation we have to consider the equation g(x, q) := σ1(qx, q)ρ(qx, q)(qx)k σ1(x, q)ρ(x, q)xk = qk σ1(x, q) + (1− q−1)xτ(x, q) σ1(x, q) = qk q−1σ2(x, q) σ1(x, q) (4.3) in case of an infinite interval, what we call it here the extended q-Pearson equation to determine the behavior of the quantity σ1(x, q)ρ(x, q)xk as x→∞, which has been easily derived from (1.5). It is obvious that the extended q-Pearson equation is the difference equation not for the weight function ρ(x, q) but for σ1(x, q)ρ(x, q)xk. 0 0 q k a 1 a 2 b 1 b 2 1 y x Figure 2. The graph of g(x, q) corresponding to Fig. 1B. In Fig. 2 we draw the graph of a typical g for some 0 < q < 1, where k is large enough. From this figure we see that g < 1 for x > b2 so that σ1(x, q)ρ(x, q)xk does not vanish at ∞ since it is increasing as x increases. Thus we cannot find a weight function ρ on (b2,∞). From Fig. 3A, we first eliminate the intervals (a2, q −1a1) and (q−1b1, b2) because of the positivity of ρ. The interval (b2,∞) coincides again with the 5th case in Theorem 2.3. However, f(x, q) < 1 on this interval so that ρ is increasing on (b2,∞) which implies that ρ can not vanish as x → ∞. Thus σ1(x, q)ρ(x, q)xk is never zero as x → ∞ for some k ∈ N0. The same is On the Orthogonality of q-Classical Polynomials of the Hahn Class 15 0 0 1 A a 2 y x q −1 a 1 q −1 b 1 b 2 0 0 1 B a 2 q −1 a 1 b 2 q −1 b 1 y x Figure 3. The graph of f(x, q) in Case 1 with 0 < Λq < 1. In A, the zeros are in order a2 < q−1a1 < 0 < q−1b1 < b2 and in B, a2 < q−1a1 < 0 < b2 < q−1b1. true for (−∞, a2) by symmetry. For the last subinterval (q−1a1, q −1b1), we face the 1th case in Theorem 2.3. Since ρ(qx, q)/ρ(x, q) = 1 at q−1a1 < x0 = −τ(0, q)/τ ′(0, q) < q−1b1, then ρ is increasing on (q−1a1, x0) and decreasing on (x0, q −1b1). Furthermore, ρ(qx, q)/ρ(x, q)→∞, and hence ρ→ 0, as x→ q−1a+1 and x→ q−1b−1 . As a result, the typical shape of ρ is shown in Fig. 4 assuming a positive initial value of ρ in each subinterval. Then, an OPS with such a weight function in Fig. 4 supported on the union of set of points {qka1(q)}k∈N0 and {qkb1(q)}k∈N0 exists (see Theorem 2.3-1). This OPS can be stated in the Theorem 4.2. 0 0 q −1 a 1 q −1 b 1 b 2 a 2 y x Figure 4. The graph of ρ(x, q) associated with the case in Fig. 3A. Theorem 4.2. Consider the case where a2 < a1 < 0 < b1 < b2 and 0 < q2Λq < 1. Let a = a1(q) and b = b1(q) be zeros of σ1(x, q). Then there exists a sequence of polynomials {Pn} for n ∈ N0 orthogonal w.r.t. the weight function (see equation 1 in Table 1) ρ(x, q) = (qa−1x, qb−1x; q)∞ (a−12 x, b−12 x; q)∞ , (4.4) supported on {qka}k∈N0 ⋃ {qkb}k∈N0, (see (2.11) of Theorem 2.3-1). The OPS in Theorem 4.2 coincides with the case VIIa1 in Chapter 10 of [21, p. 292 and p. 318]. In fact, a typical example of this family is the big q-Jacobi polynomials Pn(x; a, b, c; q) satisfying the q-EHT with the coefficients σ1(x, q) = q−2(x− a1)(x− b1), σ2(x, q) = abq(x− a2)(x− b2), 16 R. Álvarez-Nodarse, R. Sevinik Adıgüzel and H. Taşeli τ(x, q) = 1− abq2 (1− q)q x+ a(bq − 1) + c(aq − 1) 1− q and λn(q) = q−n[n]q 1− abqn+1 q − 1 , (4.5) where a1 = cq, b1 = aq, a2 = b−1c and b2 = 1. The conditions 0 < q2Λq < 1 and a2 < a1 < 0 < b1 < b2 give the known constrains c < 0, 0 < b < q−1 and 0 < a < q−1 on the parameters of Pn(x; a, b, c; q) with orthogonality on {cq, cq2, cq3, . . . } ⋃ {. . . , aq3, aq2, aq} in the sense (2.11) where d2n = (a− c)q(1− q)(q, abq2, a−1cq, ac−1q; q)∞ (aq, bq, cq, abc−1q; q)∞ (q, abq; q)n (abq, abq2; q)2n × ( aq, bq, cq, abc−1q; q ) n (−ac)nq n(n+3) 2 . It should be noted that the difference between these conditions and those of Fig. 3 comes from the fact that we have considered not only the conditions on ρ but also on σ1ρ in Theorem 4.2. Finally, the analysis of the case in Fig. 3B does not yield an OPS. 0 0 1 A q −1 a 1 a 2 x y q −1 b 1 b 2 0 0 1 y q −1 a 1 q −1 b 1 x Figure 5. The graph of f(x, q) in Case 3 with Λq < 0. In A, we have Case 3(a) with q−1a1 < 0 < q−1b1 < a2 < b2, and In B, we have Case 3(c) with q−1a1 < 0 < q−1b1 and a2, b2 ∈ C. In Fig. 5A, the only suitable interval is (q−1a1, q −1b1) which coincides with the 1st case in Theorem 2.3. In fact, ρ(qx, q)/ρ(x, q) = 1 at q−1a1 < x0 = −τ(0, q)/τ ′(0, q) < q−1b1, then it follows that ρ is increasing on (q−1a1, x0) and decreasing on (x0, q −1b1) with ρ→ 0 as x→ q−1a+1 and x→ q−1b−1 since ρ(qx, q)/ρ(x, q)→∞. Notice that the BCs (2.5) hold at x = a1 and x = b1. Then, there exists an OPS w.r.t. a ρ supported on the set of points {qka1}k∈N0 ⋃ {qkb1}k∈N0 . Thus, we have the following result. Theorem 4.3. Consider the case a1 < 0 < b1 < a2 ≤ b2, and q2Λq < 0. Let a = a1(q) and b = b1(q) be zeros of σ1(x, q). Then there exists a sequence of polynomials {Pn} for n ∈ N0 orthogonal w.r.t. the weight function (4.4) supported on {qka}k∈N0 ⋃ {qkb}k∈N0 (see (2.11) of Theorem 2.3-1). An example of this family is again the big q-Jacobi polynomials which are orthogonal on the set {cq, cq2, cq3, . . . } ⋃ {. . . , aq3, aq2, aq}. They satisfy the q-EHT with the coefficients in (4.5) where a1 = cq, b1 = aq, a2 = b−1c and b2 = 1. This case corresponds to the case VIIa1 in Chapter 10 of [21, p. 292 and p. 318]. However, notice that the conditions, a1 < 0 < b1 < a2 ≤ b2 and q2Λq < 0, lead to the new constrains c < 0, b < 0, abc−1q ≤ 1 and 0 < a < q−1, which give a larger set of parameters for the orthogonality of the big q-Jacobi polynomials than the one reported in [21, p. 319]. In Fig. 5B, the interval (q−1a1, q −1b1) coincides with the 1st case in Theorem 2.3. Notice that ρ(qx, q)/ρ(x, q) = 1 at q−1a1 < x0 = −τ(0, q)/τ ′(0, q) < q−1b1, then ρ is increasing on (q−1a1, x0) and decreasing on (x0, q −1b1) with ρ(x, q) → 0 as x → q−1a+1 and x → q−1b−1 since On the Orthogonality of q-Classical Polynomials of the Hahn Class 17 ρ(qx, q)/ρ(x, q)→∞. Then there is a suitable ρ supported on the set {qka1}k∈N0 ⋃ {qkb1}k∈N0 . Therefore, the following theorem holds: Theorem 4.4. Consider the case a1 < 0 < b1, a2, b2 ∈ C and q2Λq < 0. Let a = a1(q) and b = b1(q) be zeros of σ1(x, q). Then there exists a sequence of polynomials {Pn} for n ∈ N0 orthogonal w.r.t. the weight function (4.4), supported on {qka}k∈N0 ⋃ {qkb}k∈N0, (see (2.11) of Theorem 2.3-1) with d2n = (b1 − a1) (1− q)qn(n−1)/2 (−a1b1)n (q, q−1a−12 b−12 a1b1; q)n (q−1a−12 b−12 a1b1, a −1 2 b−12 a1b1; q)2n × ( a−12 a1, a −1 2 b1, b −1 2 a1, b −1 2 b1; q ) n (q, qb1a −1 1 , qa1b −1 1 , a−12 b−12 a1b1; q)∞ (a−12 a1, a −1 2 b1, b −1 2 a1, b −1 2 b1; q)∞ , where q2Λq = a1b1a −1 2 b−12 , a2 = iα, b2 = a2 = −iα, α ∈ R. This case is included in the case VIIa1 in Chapter 10 of [21, p. 292 and p. 318] (with γ2 = γ1) but it is not mentioned there. In fact, this case is similar to the big q-Jacobi polynomials studied in (Cases 1 in Fig. 3A and Case 3(a) in Fig. 5A). The difference is that the roots a2(q) and b2(q) are complex numbers. 0 0 1 q −1 a 1 a 2 b 2 q −1 b 1 y A x 0 0 1 B y a 2 b 2 x q −1 a 1 q −1 b 1 Figure 6. The graph of f(x, q) with Λq < 0. In A, we have Case 3(a) with q−1a1 < 0 < a2 < b2 < q−1b1, and in B, we have Case 4 with a2 < 0 < q−1a1 < b2 < q−1b1. In Fig. 6A, the only possible interval is [b2, q −1b1) which corresponds to the 3th case in Theorem 2.3. In this case ρ(qx, q)/ρ(x, q) = 1 at b2 < x0 = −τ(0, q)/τ ′(0, q) < q−1b1, then it follows that ρ is increasing on [b2, x0) and decreasing on (x0, q −1b1). Furthermore, ρ(qb2, q) = 0 and ρ(x, q) → 0 as x → q−1b−1 since ρ(qb2, q)/ρ(b2, q) = 0 and ρ(qx, q)/ρ(x, q) → ∞ as x → q−1b−1 . Thus there is a suitable ρ supported on the set of points {q−kb2}Nk=0 where q−N−1b2 = q−1b1 (see Theorem 2.3-3). Hence we state the following theorem. Theorem 4.5. Consider the case where a1 < 0 < a2 ≤ b2 < b1 and q2Λq < 0. Let a = b2(q) and b = q−1b1(q) be zeros of σ2(x, q) and σ1(qx, q), respectively. Then there exists a finite family of polynomials {Pn} orthogonal w.r.t. the weight function (see equation 2 in Table 1) ρ(x, q) = \|x\|ι ( qax , b −1x; q)∞ (a1x , a −1 2 x; q)∞ , qι = q−3σ′′2(0, q)a2 σ′′1(0, q)b (4.6) supported on the set of points {q−ka}Nk=0 where q−N−1a = b (see (2.13) of Theorem 2.3-3). A typical example of this family is the q-Hahn polynomials satisfying (1.1) and (1.3) with σ1(x, q) = q−2(x− a1)(x− b1), σ2(x, q) = αβq(x− a2)(x− b2), 18 R. Álvarez-Nodarse, R. Sevinik Adıgüzel and H. Taşeli τ(x, q) = 1− αβq2 (1− q)q x+ αq−N + αβq − α− q−N−1 1− q and λn(q) = −q−n[n]q 1− αβqn+1 1− q , (4.7) where a1 = αq, b1 = q−N , a2 = β−1q−N−1 and b2 = 1. They are orthogonal on the set of points {1, q−1, q−2, . . . , q−N} in the sense (2.13) where d2n = (q, qN+1, β−1, α−1β−1q−N−1; q)∞ (αq, βqN+1, β−1q−N , α−1β−1q−1; q)∞ × (q, αβq, αq, q−N , βq, αβqN+2; q)n (αβq, αβq2; q)2n(q−1 − 1)−1 (−αq−N )nq n(n+1) 2 . The conditions a1 < 0 < a2 ≤ b2 < b1 and q2Λq < 0 lead to the orthogonality relation for the q-Hahn polynomials that is valid in a larger set of the parameters, α < 0 and β ≥ q−N−1. This new parameter set is not mentioned in [21]. In Fig. 6B, the only possible interval is [b2, q −1b1) which corresponds to the one described in Theorem 2.3-3. A similar analysis shows that there exists a q-weight function defined on the interval [b2, q −1b1) supported at the points q−kb2 for k = 0, 1, . . . , N where q−N−1b2 = q−1b1 which lead to the following theorem: Theorem 4.6. Consider the case where a2 < 0 < a1 < b2 < b1 and q2Λq < 0. Let a = b2(q) and b = q−1b1(q) be zeros of σ2(x, q) and σ1(qx, q), respectively. Then there exists a finite family of polynomials {Pn} orthogonal w.r.t. the weight function (4.6) supported on the set of points {q−ka}Nk=0 where q−N−1a = b (see (2.13) of Theorem 2.3-3). An example of this family is again the q-Hahn polynomials orthogonal on the finite set of points {1, q−1, q−2, . . . , q−N}. They satisfy (1.1) and (1.3) with the coefficients (4.7) where a1 = αq, b1 = q−N , a2 = β−1q−N−1 and b2 = 1. The conditions a2 < 0 < a1 < b2 < b1 and q2Λq < 0 lead to another new constrains 0 < α < q−1 and β < 0 on the parameters of the q-Hahn polynomials which extend the orthogonality relation for the q-Hahn polynomials and it has been not reported in [21]. For the sake of the shortness, in the following theorem, we summarize the other cases coming from the analysis of the ∅-Jacobi/Jacobi family. Theorem 4.7. The other positive definite orthogonal polynomial solutions of the q-EHT (1.1) and (1.3) for the ∅-Jacobi/Jacobi case can be divided into two cases. Case 2(a). 0 < a1 < a2 < b1 < b2 and 0 < q2Λq < 1. The q-Hahn polynomials satisfying the q-EHT with the coefficients (4.7) where a1 = αq, b1 = q−N , a2 = 1 and b2 = β−1q−N−1. They are orthogonal w.r.t. the weight function (see equation 2 in Table 1) on the set of points {1, q−1, q−2, . . . , q−N} with 0 < α < q−1 and 0 < β < q−1 (see (2.13)). In the literature, this relation is usually written as a finite sum [21, p. 367]. This case corresponds to the case IIIb9 in Chapter 11 of [21, p. 366]. Case 2(a). 0 < a2 ≤ b2 < a1 ≤ b1 and q2Λq > 1. The q-Hahn polynomials satisfying the q-EHT with the coefficients (4.7) where a1 = q−N , b1 = αq, a2 = β−1q−N−1 and b2 = 1. They are orthogonal w.r.t. the weight function (see equation 2 in Table 1) on the set of points {1, q−1, q−2, . . . , q−N} with α ≥ q−N−1 and β ≥ q−N−1. The authors did not mention this different set of the parameters for the q-Hahn polynomials in [21]. However it is given in [22, p. 76]. On the Orthogonality of q-Classical Polynomials of the Hahn Class 19 4.2 q-classical ∅-Jacobi/Laguerre polynomials Let the coefficients σ2 and σ1 be quadratic and linear polynomials in x, respectively, such that σ1(0, q)σ2(0, q) 6= 0. If σ1 is written in terms of its root, i.e., σ1(x, q) = σ′1(0, q)[x − a1(q)], a1(q) = −σ1(0,q) σ′1(0,q) then from (1.4) σ2(x, q) = (q − 1)τ ′(0, q)x2 + [ qσ′1(0, q) + (q − 1)τ(0, q) ] x− qσ′1(0, q)a1(q), where τ ′(0, q) 6= 0 by hypothesis. Then the q-Pearson equation (1.5) takes the form f(x, q) := ρ(qx, q) ρ(x, q) = q−1σ2(x, q) σ1(qx, q) = (1− q−1) τ ′(0,q) σ′1(0,q) [x− a2(q)][x− b2(q)] qx− a1(q) (4.8) provided that the discriminant denoted by ∆q, ∆q := [ 1 + (1− q−1)τ(0, q) σ′1(0, q) ]2 + 4a1(q) ( 1− q−1 ) τ ′(0, q) σ′1(0, q) of the quadratic polynomial in the nominator of f in (4.8) is non-zero. Note that here x = a2 and x = b2 are roots of f which are constant multiplies of the roots of σ2. Moreover, x = q−1a1 is the vertical asymptote of f and y = 1 is its y-intercept since σ2(0, q) = qσ1(0, q). On the other hand, the locations of the zeros of f are introduced by the following straightforward lemma. Lemma 4.8. Let Λq = τ ′(0,q) σ′1(0,q) 6= 0. Then, we have the following cases for the roots of the equation f(x, q) = 0. Case 1. If Λq and a1(q) have opposite signs, then there are two real distinct roots with opposite signs. Case 2. If Λq and a1(q) have same signs, then there exist three possibilities (a) if ∆q > 0, f has two real roots with same signs, (b) if ∆q = 0, f has a double root, (c) if ∆q < 0, f has a pair of complex conjugate roots. 0 0 1 A a 2 q −1 a 1 b 2 x y 0 0 1 q −1 a 1 a 2 b 2 B y x Figure 7. The graph of f(x, q) with Λq < 0. In A, we have Case 1 with a2 < 0 < q−1a1 < b2, and in B, we have Case 2(a) q−1a1 < 0 < a2 < b2. In Fig. 7A, we first start with the positivity condition of q-weight function which allows us to exclude the intervals (−∞, a2) and (q−1a1, b2). Moreover, due to PIII (a2, q −1a1) can not be used. On the other hand, the interval (b2,∞) coincides with the 5th case of Theorem 2.3. 20 R. Álvarez-Nodarse, R. Sevinik Adıgüzel and H. Taşeli Notice that since ρ(qx, q)/ρ(x, q) = 1 at x0 = −τ(0, q)/τ ′(0, q) > b2, ρ is decreasing on (x0,∞). Moreover, since ρ(qx, q)/ρ(x, q) has an infinite limit as x → +∞, we have ρ → 0 as x → ∞. However, since it is infinite interval, we should check that σ1(x, q)ρ(x, q)xk → 0 as x → ∞ by using extended q-Pearson equation (4.3). The graph of the function g defined in (4.3) looks like the one for f . Then the analysis of the extended q-Pearson equation leads to σ1(x, q)ρ(x, q)xk → 0 as x → ∞. Therefore, there exists a suitable ρ supported on the set of points {q−kb2}k∈N0 . Thus, we have the following theorem. Theorem 4.9. Let a2 < 0 < a1 < b2 and Λq < 0. Let a = b2(q) be the zero of σ2(x, q) and b → ∞. Then, there exists a sequence of polynomials (Pn)n for n ∈ N0 orthogonal w.r.t. the weight function (see the 2nd expression of the ∅-Jacobi/Laguerre case in Table 2) ρ(x, q) = \|x\|αH(1)(x) (qa2/x, qa/x; q)∞ (a1/x; q)∞ , qα = q−2 12σ ′′ 2(0, q) σ′1(0, q) (4.9) supported on {q−ka}k∈N0 (see (2.15) of Theorem 2.3-5). Here H(1)(x) is defined by (3.4). The OPS in Theorem 4.9 coincides with the case IIa2 in Chapter 11 of [21, p. 337 and p. 358]. An example of this family is the q-Meixner polynomials Mn(x; b, c; q) satisfying the q-EHT with σ1(x, q) = cq−2(x− a1), σ2(x, q) = (x− a2)(x− b2), τ(x, q) = − 1 1− q x+ cq−1 − bc+ 1 1− q and λn(q) = [n]q 1− q , (4.10) where a1 = bq, a2 = −bc and b2 = 1. The conditions Λq < 0 and a2 < 0 < a1 < b2 give us the known constrains c > 0 and 0 < b < q−1 on the parameters of Mn(x; b, c; q) with orthogonality on {1, q−1, q−2, . . . } in the sense (2.15) where d2n = (q−1 − 1)c2nq−n(2n+1) ( q,−c−1q, bq; q ) n (q,−c; q)∞ (bq; q)∞ . In the literature, this relation can be found as an infinite sum [21, p. 360]. In Fig. 7B, the only possible interval is (b2,∞). An analogous analysis as the one that has been done for the case in Fig. 7A yields ρ → 0 as x → ∞. Moreover, σ1(x, q)ρ(x, q)xk → 0 as x → ∞ for k ∈ N0 by (4.3), then there exists a q-weight function on [b2,∞) supported at the points q−kb2 for k ∈ N0. Thus we have the following result. Theorem 4.10. Let a1 < 0 < a2 ≤ b2 and Λq < 0. Let a = b2 be the zero of σ2(x, q) and b → ∞. Then, there exists a sequence of polynomials (Pn)n for n ∈ N0 orthogonal w.r.t. the weight function (4.9) (see Theorem 4.9) supported on {q−ka}k∈N0 (see (2.15) of Theorem 2.3-5). A typical example of this family is again the q-Meixner polynomials orthogonal on the set of points {1, q−1, q−2, . . . }. They satisfy the q-EHT with the coefficients (4.10) where a1 = bq, a2 = −bc and b2 = 1. This set of q-Meixner polynomials corresponds to the case IIa2 in Chapter 11 of [21, p. 337 and p. 358] and their orthogonality relation is valid in a larger set of parameters. In fact, the conditions a1 < 0 < a2 ≤ b2 and Λq < 0 yield c > 0, b < 0 and 0 < −bc ≤ 1. This was not reported in [21]. To conclude this subsection we summarize the other cases in the following theorem. Theorem 4.11. The other positive definite orthogonal polynomial solutions of the q-EHT (1.1) and (1.3) for the ∅-Jacobi/Laguerre case can be divided into three cases. Case 2(a). a2 ≤ b2 < a1 < 0 and Λq < 0. The polynomials with orthogonality on the union of the sets {a1, a1q, a1q2, . . . } and {q±kα}k∈N0 for arbitrary α > 0 w.r.t. the weight function (see the 1st expression of the ∅-Jacobi/Laguerre case in Table 2) in the sense (2.14) with d2n = (1− q)q−n(2n−1) ( a2b2a −1 1 )2n ( q, a−12 a1, b −1 2 a1; q ) n On the Orthogonality of q-Classical Polynomials of the Hahn Class 21 × (q, a1, qa −1 1 , a−12 b−12 a1, qa2b2a −1 1 ; q)∞ (a−12 a1, b −1 2 a1, a −1 2 , b−12 , qa2, qb2; q)∞ . (4.11) This case coincides with the case VIa2 in Chapter 10 of [21, p. 285 and p. 315]. Case 2(c). a1 < 0, a2, b2 ∈ C and Λq < 0. The polynomials with orthogonality on the union of the sets {a1, a1q, a1q2, . . . } and {q±kα}k∈N0 for arbitrary α > 0 w.r.t. the weight function (see the 1st expression of the ∅-Jacobi/Laguerre case in Table 2) in the sense (2.14) where the norm is given by (4.11). This case coincides with the case VIa1 in Chapter 10 of [21, p. 285 and p. 315]. The orthogonality relation of this OPS has the same form as in the previous Case 2(a) but now the zeros of σ2 are complex numbers. Case 2(a). 0 < a2 ≤ b2 < a1 and Λq > 0. The quantum q-Kravchuk polynomials satisfying the q-EHT with σ1(x, q) = −q−2(x− a1), σ2(x, q) = p(x− a2)(x− b2), τ(x, q) = − p 1− q x+ p− q−1 + q−N−1 1− q q and λn(q) = p 1− q [n]q, where a1 = q−N , a2 = p−1q−N−1 and b2 = 1. They are orthogonal w.r.t. the weight function (see the 3th expression of ∅-Jacobi/Laguerre in Table 2) on the set of points {1, q−1, q−2, . . . q−N} with p ≥ q−N−1 in the sense (2.13) where d2n = ( q−1 − 1 ) 1 (p−1q−N ; q)N p−2nq−n(2n+1) ( q, pq, q−N ; q ) n ( q, p−1q−N , qN+1; q ) ∞. In the literature, this relation is usually written as a finite sum [21, p. 362]. This case coincides with the case IIb1 in Chapter 11 of [21, p. 337 and p. 361]. 4.3 q-classical ∅-Jacobi/Hermite polynomials Let the coefficients σ2 and σ1 be quadratic and constant polynomials in x, respectively, such that σ1(0, q)σ2(0, q) 6= 0. If σ1(x, q) = σ1(0, q) 6= 0 then, from (1.4), σ2(x, q) = q [ σ1(x, q) + ( 1− q−1 ) xτ(x, q) ] = (q − 1)τ ′(0, q)x2+ (q − 1)τ(0, q)x+ qσ1(0, q), where τ ′(0, q) 6= 0 by hypothesis. Then the q-Pearson equation (1.5) takes the form f(x, q) := ρ(qx, q) ρ(x, q) = q−1σ2(x, q) σ1(qx, q) = ( 1− q−1 ) τ ′(0, q) σ1(0, q) [x− a2(q)][x− b2(q)] (4.12) provided that the discriminant denoted by ∆q, ∆q := [ (1− q−1) τ(0, q) σ1(0, q) ]2 − 4(1− q−1) τ ′(0, q) σ1(0, q) of f in (4.12) is non-zero. Notice that y-intercept of f is y = 1 since σ2(0, q) = qσ1(0, q). Moreover, x = a2 and x = b2 indicate its zeros which are constant multiples of the roots of σ2. The following straightforward lemma allows us to determine the locations of the zeros of f . Lemma 4.12. Let Λq = τ ′(0,q) σ1(0,q) 6= 0. Then we encounter the following cases for the roots of the equation f(x, q) = 0. Case 1. If Λq > 0, f has two real distinct roots with opposite signs. Case 2. If Λq < 0, there exist three possibilities 22 R. Álvarez-Nodarse, R. Sevinik Adıgüzel and H. Taşeli (a) if ∆q > 0, f has two real roots with same signs, (b) if ∆q = 0, f has a double root, (c) if ∆q < 0, f has a pair of complex conjugate roots. Theorem 4.13. The positive definite orthogonal polynomial solutions of the q-EHT (1.1) and (1.3) for the ∅-Jacobi/Hermite case can be divided into two cases. Case 2(a). 0 < a2 ≤ b2 and Λq < 0. The Al-Salam–Carlitz II polynomials satisfying the q-EHT with σ1(x, q) = aq−1, σ2(x, q) = (x− a2)(x− b2), τ(x, q) = 1 q − 1 x− 1 + a q − 1 and λn(q) = 1 1− q [n]q, where a2 = a, b2 = 1. They are orthogonal w.r.t. the weight function (see expression 2 for the ∅-Jacobi/Hermite case in Table 2) on the set of points {1, q−1, q−2, . . . } with 0 < a ≤ 1 in the sense (2.15) where d2n = ( q−1 − 1 ) anq−n 2 (q; q)n(q; q)∞. In the literature, this relation is usually written as an infinite sum [21, p. 357]. This case coincides with the case Ia1 in Chapter 11 of [21, p. 335 and pp. 355–357]. Case 2(c). a2(q), b2(q) ∈ C and Λq < 0. The discrete q-Hermite II polynomials satisfying the q-EHT with σ1(x, q) = q−1, σ2(x, q) = (x− a2)(x− b2), τ(x, q) = 1 q − 1 x and λn(q) = 1 1− q [n]q, where a2 = −i, b2 = i ∈ C. They are orthogonal w.r.t. the weight function (see expression 1 for the ∅-Jacobi/Hermite in Table 2) on the set of points {±q∓k}k∈N0 in the sense (2.17) where d2n = (1− q)q−n2 (q; q)n (q,−q,−1,−1,−q; q)∞ (i,−i,−iq, iq,−i, i, iq,−iq; q)∞ . This case corresponds to the case Ia1 in Chapter 11 and case Va2 in Chapter 10 of [21, p. 335, pp. 355–356, p. 283 and pp. 314–315]. 4.4 q-classical ∅-Laguerre/Jacobi polynomials Let the coefficients σ2 and σ1 be linear and quadratic polynomials in x, respectively, such that σ1(0, q)σ2(0, q) 6= 0. If σ1 is written in terms of its roots, i.e., σ1(x, q) = 1 2σ ′′ 1(0, q)[x− a1(q)][x− b1(q)], then from (1.4) σ2(x, q) = σ′2(0, q)x+ σ2(0, q) where σ′2(0, q) = −q [ 1 2 σ′′1(0, q)[a1(q) + b1(q)]− (1− q−1)τ(0, q) ] 6= 0 and σ2(0, q) = 1 2 σ′′1(0, q)qa1(q)b1(q) 6= 0 provided that τ ′(0, q) = − 1 2 σ′′1 (0,q) (1−q−1) . Therefore, the q-Pearson equation (1.5) takes the form f(x, q) := ρ(qx, q) ρ(x, q) = − [ a1(q) + b1(q)− (1−q−1)τ(0,q) 1 2 σ′′1 (0,q) ] [x− a2(q)] [qx− a1(q)][qx− b1(q)] , On the Orthogonality of q-Classical Polynomials of the Hahn Class 23 where [ a1(q) + b1(q) − (1−q−1)τ(0,q) 1 2 σ′′1 (0,q) ] a2(q) = a1(q)b1(q). Let us point out that f(x, q) intersects the y-axis at the point y = 1 since σ2(0, q) = qσ1(0, q). On the other hand, we consider the cases depending on the signs of zeros of σ1 and Λq defined by Λq := [ a1(q) + b1(q)− (1− q−1)τ(0, q) 1 2σ ′′ 1(0, q) ] . Case 1. Λq < 0 with a1 < 0 < b1, Case 2. Λq > 0 with 0 < a1 < b1, Case 3. Λq < 0 with 0 < a1 < b1. Theorem 4.14. The positive definite orthogonal polynomial solutions of the q-EHT (1.1) and (1.3) for the ∅-Laguerre/Jacobi case can be divided into two cases. Case 1. a1 < 0 < b1 < a2 and Λq < 0. The big q-Laguerre polynomials satisfying the q-EHT with σ1(x, q) = q−2(x− a1)(x− b1), σ2(x, q) = −abq(x− a2), τ(x, q) = − q−1 q − 1 x+ a+ b− abq q − 1 and λn(q) = q−n q − 1 [n]q, where a1 = bq, b1 = aq and a2 = 1. They are orthogonal w.r.t. the weight function (see the 1st expression of the ∅-Laguerre/Jacobi case in Table 2) on {bq, bq2, bq3, . . . } ⋃ {. . . , aq3, aq2, aq} with b < 0 and 0 < a < q−1 in the sense (2.11) where d2n = (a− b)q(1− q)(−ab)nqn(n+3)/2(q; q)n(aq, bq; q)n (q, a−1bq, ab−1q; q)∞ (aq, bq; q)∞ . This case coincides with the case VIIa1 in Chapter 10 of [21, p. 292 and p. 318]. Case 2. 0 < a1 < a2 < b1 and Λq > 0. The affine q-Kravchuk polynomials satisfying the q-EHT with σ1(x, q) = q−1(x− a1)(x− b1), σ2(x, q) = −pq1−N (x− a2), τ(x, q) = 1 1− q x− pq + q−N − pq1−N 1− q and λn(q) = 1 q − 1 [n]q−1 , where a1 = pq, b1 = q−N and a2 = 1. They are orthogonal w.r.t. the weight function (see the 2nd expression of the ∅-Laguerre/Jacobi case in Table 2) on {1, q−1, q−2, . . . , q−N} with 0 < p < q−1 in the sense (2.13) where d2n = (−1)npn−N ( q−1 − 1 ) q−N(n+1)qn(n+1)/2 ( q, pq, q−N ; q ) n (q, qN+1; q)∞ (pq; q)∞ . In the literature, this relation is usually written as a finite sum [21, p. 364]. This case coincides with the case IIIb3 in Chapter 11 of [21, p. 343 and p. 363]. 4.5 q-classical ∅-Hermite/Jacobi polynomials Let the coefficients σ2 and σ1 be constant and quadratic polynomials in x, respectively, such that σ1(0, q)σ2(0, q) 6= 0. If σ1 can be written in terms of its roots, i.e., σ1(x, q) = 1 2σ ′′ 1(0, q)[x− a1(q)][x− b1(q)], then, from (1.4) σ2(x, q) = σ2(0, q) = q 1 2 σ′′1(0, q)a1(q)b1(q) 24 R. Álvarez-Nodarse, R. Sevinik Adıgüzel and H. Taşeli provided that (1 − q−1)τ ′(0, q) = −1 2σ ′′ 1(0, q) and (1 − q−1)τ(0, q) = 1 2σ ′′ 1(0, q)[a1(q) + b1(q)]. Therefore, the q-Pearson equation (1.5) becomes f(x, q) := ρ(qx, q) ρ(x, q) = a1(q)b1(q) [qx− a1(q)][qx− b1(q)] . Notice that the point y = 1 is y-intercept of f . Theorem 4.15. The positive definite orthogonal polynomial solutions of the q-EHT (1.1) and (1.3) for the ∅-Hermite/Jacobi case appear only when a1 < 0 < b1. It corresponds to the Al-Salam–Carlitz I polynomials which satisfy the q-EHT with σ1(x, q) = q−1(x− a1)(x− b1), σ2(x, q) = a, τ(x, q) = 1 1− q x− 1 + a 1− q and λn(q) = q1−n q − 1 [n]q, where a1 = a and b1 = 1. They are orthogonal w.r.t. the weight function (see the ∅-Hermite/Ja- cobi case in Table 2) on the set {a, qa, q2a, . . . } ⋃ {. . . , q2, q, 1} with a < 0 in the sense (2.11) where d2n = (1− a)(−a)nq( n 2 )(1− q)(q; q)n ( q, aq, a−1q; q ) ∞. This case coincides with the case VIIa1 in Chapter 10 of [21, p. 292 and pp. 318–320]. We note that the discrete q-Hermite I polynomials are special case of Al-Salam–Carlitz I polynomials (see [21, p. 320] for further details). 4.6 q-classical 0-Jacobi/Jacobi polynomials Let the coefficients σ2 and σ1 be quadratic polynomials in x such that σ1(0, q) = σ2(0, q) = 0. If σ1 is written as σ1(x, q) = 1 2σ ′′ 1(0, q)x[x − a1(q)] then, from (1.4) σ2(x, q) = 1 2σ ′′ 2(0, q)x2 + σ′2(0, q)x where 1 2 σ′′2(0, q) = q [ 1 2 σ′′1(0, q) + ( 1− q−1 ) τ ′(0, q) ] 6= 0, σ′2(0, q) = q ( 1− q−1 ) τ(0, q)− 1 2 σ′′1(0, q)a1(q). Then it follows from (1.5) that f(x, q) := ρ(qx, q) ρ(x, q) = [ 1 + (1− q−1)τ ′(0, q) 1 2σ ′′ 1(0, q) ] [x− a2(q)] q[qx− a1(q)] , x 6= 0 (4.13) provided that [ 1 + (1−q−1)τ ′(0,q) 1 2 σ′′1 (0,q) ] a2(q) = [ a1(q)− (1−q−1)τ(0,q) 1 2 σ′′1 (0,q) ] . Let us point out that Λq defined in (4.2) is also horizontal asymptote of f(x, q) in (4.13). Moreover, f intersects the y-axis at the point y := y0 = q−1 [ 1− (1− q−1) a1(q) τ(0, q) 1 2σ ′′ 1(0, q) ] . In the zero cases notice that one of the boundary of (a, b) interval could be zero. Therefore for such a case we need to know the behavior of ρ at the origin. Lemma 4.16. If 0 < y0 < 1, then ρ(z, q)→ 0 as z → 0. Otherwise it diverges to ∓∞. On the Orthogonality of q-Classical Polynomials of the Hahn Class 25 Proof. From (2.10), we write ρ(qkx, q) = ρ(x, q) k−1∏ i=0 q−1σ2(qix,q) σ1(qi+1x,q) from which ρ(qkx, q) = q−k [ 1− (1− q−1) a1(q) τ(0, q) 1 2σ ′′ 1(0, q) ]k (x/a2(q); q)k (qx/a1(q); q)k ρ(x, q) is obtained. Taking k →∞ the result follows. � Here we introduce the following two cases which include all possibilities. Case 1. Λq > 0 with (a) 0 < y0 < 1 or (b) y0 > 1 or (c) y0 < 0. Case 2. Λq < 0 with (a) 0 < y0 < 1 or (b) y0 > 1 or (c) y0 < 0. Theorem 4.17. The positive definite orthogonal polynomial solutions of the q-EHT (1.1) and (1.3) for the 0-Jacobi/Jacobi case can be divided into three cases. Case 1(a). 0 < a1 < a2, 0 < q2Λq < 1 and 0 < qy0 < 1. The little q-Jacobi polynomials satisfying the q-EHT with σ1(x, q) = q−2x(x− a1), σ2(x, q) = abqx(x− a2), τ(x, q) = 1− abq2 (1− q)q x+ aq − 1 (1− q)q and λn(q) = −q−n[n]q 1− abqn+1 1− q , (4.14) where a1 = 1 and a2 = b−1q−1. They are orthogonal w.r.t. the weight function (see equation 3 in Table 1) on {. . . , q2, q, 1} with 0 < a < q−1 and 0 < b < q−1 the sense (2.12) where d2n = anqn 2 (1− q) (q, abq; q)n (abq, abq2; q)2n (aq, bq; q)n (q, abq2; q)∞ (aq, bq; q)∞ . In the literature, this relation can be found as an infinite sum [21, p. 312]. This case corresponds to the case IVa3 in Chapter 10 of [21, p. 312]. Case 2(a). a2 < 0 < a1, q2Λq < 0 and 0 < qy0 < 1. The little q-Jacobi polynomials with orthogonality on the set of points {. . . , q2, q, 1} w.r.t. the weight function (see equation 3 in Table 1) with 0 < a < q−1 and b < 0. They satisfy the q-EHT with the coefficients in (4.14) where a1 = 1 and a2 = b−1q−1. This case coincides with the case IVa4 in Chapter 10 of [21, p. 278 and p. 312]. This extends the orthogonality relation of the little q-Jacobi polynomials for 0 < a < q−1 and 0 < b < q−1 to a larger set of the parameters 0 < a < q−1 and b < 0. Notice that combining this with the previous Case 1(a) one can obtain the orthogonality relation of the little q-Jacobi polynomials for 0 < aq < 1, bq < 1. Case 2(c). 0 < a2 < a1, q2Λq < 0 and qy0 < 0. The q-Kravchuk polynomials satisfying the q-EHT with σ1(x, q) = q−2x(x− a1), σ2(x, q) = −px(x− a2), τ(x, q) = 1 + pq (1− q)q x− p+ q−N−1 1− q and λn(q) = −q−n[n]q 1 + pqn 1− q , where a1 = q−N and a2 = 1 They are orthogonal w.r.t. the weight function (see equation 4 in Table 1) on {1, q−1, q−2, . . . , q−N} with p > 0 in the sense (2.13) where d2n = ( q−1 − 1 ) p−Nq−( N+1 2 ) ( −q−Np )n qn 2 1 + p 1 + pq2n (−pq; q)N ( q, qN+1; q ) ∞ (q,−pqN+1; q)n (−p, q−N ; q)n . In the literature, this relation is usually written as a finite sum [22, p. 98]. This case is not mentioned in [21] for 0 < q < 1. However the q-Kravchuk polynomials with this set of parameters are described in [22, p. 98]. 26 R. Álvarez-Nodarse, R. Sevinik Adıgüzel and H. Taşeli 4.7 q-classical 0-Jacobi/Laguerre polynomials Let σ2 and σ1 be quadratic and linear polynomials in x, respectively, such that σ2(0, q) = σ1(0, q) = 0. If σ1(x, q) = σ′1(0, q)x, then from (1.4), σ2(x, q) = 1 2σ ′′ 2(0, q)x2 + σ′2(0, q)x where 1 2 σ′′2(0, q) = q ( 1− q−1 ) τ ′(0, q) 6= 0 and σ′2(0, q) = q [ σ′1(0, q) + ( 1− q−1 ) τ(0, q) ] 6= 0 provided that (1− q−1)τ(0, q) 6= −σ′1(0, q). For this case the q-Pearson equation reads f(x, q) := ρ(qx, q) ρ(x, q) = q−1 ( 1− q−1 ) τ ′(0, q) σ′1(0, q) [x− a2(q)], (4.15) where −(1− q−1) τ ′(0,q) σ′1(0,q) a2(q) = 1 + (1−q−1)τ(0,q) σ′1(0,q) . Let us point out that f intersects the y-axis at the point y := y0 = q−1 [ 1 + (1− q−1)τ(0, q) σ′1(0, q) ] . Notice that for the zero cases one of the boundary of (a, b) interval could be zero. This requires to find the behavior of ρ at the origin. Lemma 4.18. If 0 < y0 < 1, then ρ(z, q)→ 0 as z → 0. Otherwise it diverges to ∓∞. Proof. From (4.15) it follows that ρ(qkx, q) = q−k [ 1 + (1− q−1)τ(0, q) σ′1(0, q) ]k (x/a2(q); q)kρ(x, q) from where the result follows. � Again we identify the cases depending on σ2, Λq := τ ′(0,q) σ′1(0,q) and y0. Case 1. Λq > 0, a2 > 0 and y0 > 1, Case 2. Λq < 0, a2 < 0 and 0 < y0 < 1, Case 3. Λq < 0, a2 > 0 and y0 < 0. Theorem 4.19. The positive definite orthogonal polynomial solutions of the q-EHT (1.1) and (1.3) for the 0-Jacobi/Laguerre case can be divided into two cases. Case 2. a2 < 0, Λq < 0 and 0 < qy0 < 1. The q-Laguerre polynomials satisfying the q-EHT with σ1(x, q) = q−2x, σ2(x, q) = qαx(x− a2), τ(x, q) = − qα 1− q x+ q−1 − qα 1− q and λn(q) = [n]q qα 1− q , where a2 = −1. They are orthogonal w.r.t. the weight function (see the 1st expression of the 0-Jacobi/Laguerre case in Table 2) on {q±k}k∈N0 with α > −1 in the sense (2.16) where d2n = q−n(1− q)(qα+1; q)n (q; q)n (q,−qα+1,−q−α; q)∞ (qα+1,−q,−q; q)∞ . This case coincides with the case IIIa2 in Chapter 10 of [21, p. 272 and pp. 309–311]. Case 3. a2 > 0, Λq < 0 and qy0 < 0. The q-Charlier polynomials satisfying the q-EHT with σ1(x, q) = aq−2x, σ2(x, q) = x(x− a2), On the Orthogonality of q-Classical Polynomials of the Hahn Class 27 τ(x, q) = − 1 1− q x+ a+ q (1− q)q and λn(q) = [n]q 1 1− q , where a2 = 1. They are orthogonal w.r.t. the weight function (see the 2nd expression of the 0-Jacobi/Laguerre case in Table 2) on {1, q−1, q−2, . . . } with a > 0 in the sense (2.15) where d2n = a2nq−n(2n+1) ( −a−1q, q; q ) n (−a, q; q)∞. In the literature, this relation is usually written as an infinite sum [21, p. 360]. This case coincides with the case IIa2 in Chapter 11 of [21, p. 337 and pp. 358–360]. 4.8 q-classical 0-Bessel/Jacobi polynomials Let σ2 and σ1 be quadratic polynomials in x, respectively, such that σ′2(0, q) = 0 and σ2(0, q) = σ1(0, q) = 0. If σ1(x, q) = 1 2σ ′′ 1(0, q)x[x−a1(q)], τ ′(0,q) 1 2 σ′′1 (0,q) 6= − 1 (1−q−1) and τ(0,q) 1 2 σ′′1 (0,q) = a1(q) (1−q−1) , then from (1.4) we have σ2(x, q) = 1 2σ ′′ 2(0, q)x2 = q [ 1 2σ ′′ 1(0, q) + (1− q−1)τ ′(0, q) ] x2. As a result, the q-Pearson equation (1.5) becomes f(x, q) := ρ(qx, q) ρ(x, q) = [ 1 + (1−q−1)τ ′(0,q) 1 2 σ′′1 (0,q) ] x q[qx− a1(q)] . Let us point out that f(x, q) passes through the origin and the line y = Λq := q−2 [ 1 + (1− q−1)τ ′(0, q) 1 2σ ′′ 1(0, q) ] 6= 0 is its horizontal asymptote. Theorem 4.20. The positive definite orthogonal polynomial solutions of the q-EHT (1.1) and (1.3) for the 0-Bessel/Jacobi case appear only when a1 > 0 and q2Λq < 0. This corre- sponds to the alternative q-Charlier (q-Bessel) polynomials which satisfy the q-EHT with σ1(x, q) = −q−2x(x− a1), σ2(x, q) = ax2, τ(x, q) = − 1 + aq (1− q)q x+ 1 (1− q)q and λn(q) = q−n[n]q 1 + aqn 1− q , where a1 = 1. They are orthogonal w.r.t. the weight function (see the 0-Bessel/Jacobi case in Table 2) on {. . . , q2, q, 1} with a > 0 in the sense (2.12) where d2n = anqn(3n−1)/2(−aq, q; q)∞ (q,−a; q)n (−a,−aq; q)2n . In the literature, this relation can be found as an infinite sum [21, p. 314]. This case coincides with the case IVa5 in Chapter 10 of [21, p. 278 and p. 313]. 4.9 q-classical 0-Bessel/Laguerre polynomials Let σ2 and σ1 be quadratic and linear polynomials in x, respectively, such that σ′2(0, q) = 0 and σ2(0, q) = σ1(0, q) = 0. If σ1(x, q) = σ′1(0, q)x, then, from (1.4) σ2(x, q) = 1 2σ ′′ 2(0, q)x2 = q(1 − q−1)τ ′(0, q)x2 provided that (1 − q−1)τ(0, q) = −σ′1(0, q). So the q-Pearson equation is now f(x, q) := ρ(qx, q) ρ(x, q) = q−1 ( 1− q−1 ) τ ′(0, q) σ′1(0, q) x. Clearly, f passes through the origin. According to the sign of Λq := τ ′(0,q) σ′1(0,q) we have only one possible case. 28 R. Álvarez-Nodarse, R. Sevinik Adıgüzel and H. Taşeli Theorem 4.21. The positive definite orthogonal polynomial solutions of the q-EHT (1.1) and (1.3) for the 0-Bessel/Laguerre case appear only when a2 = 0, Λq < 0 and qy0 = 0. This corresponds to the Stieltjes–Wigert polynomials which satisfy the q-EHT with σ1(x, q) = q−2x, σ2(x, q) = x2, τ(x, q) = − 1 1− q x+ 1 (1− q)q and λn(q) = [n]q 1 1− q . They are orthogonal w.r.t. the weight function (see the 0-Bessel/Laguerre case in Table 2) on {q∓k}k∈N0 in the sense (2.16) where d2n = q−n(1− q)(−tq,−1/t, q; q)∞ (q2; q)n . This case coincides with the case IIIa2 in Chapter 10 of [21, p. 272 and p. 309]. 4.10 q-classical 0-Laguerre/Jacobi polynomials Let σ2 and σ1 be linear and quadratic polynomials in x, respectively, such that σ2(0, q) = σ1(0, q) = 0. If σ1(x, q) = 1 2σ ′′ 1(0, q)x[x− a1(q)] and τ ′(0,q) 1 2 σ′′1 (0,q) = − 1 (1−q−1) , then from (1.4) we get σ2(x, q) = σ′2(0, q)x = q [ (1− q−1)τ(0, q)− 1 2σ ′′ 1(0, q)a1(q) ] x. Therefore, the q-Pearson equation has the form f(x, q) := ρ(qx, q) ρ(x, q) = (1− q−1) τ(0,q) 1 2 σ′′1 (0,q) − a1(q) q[qx− a1(q)] . Notice that y = 0 is the horizontal asymptote of f(x, q), and its y-intercept is y := y0 = q−1 [ 1− (1− q−1) a1(q) τ(0, q) 1 2σ ′′ 1(0, q) ] . Theorem 4.22. The positive definite orthogonal polynomial solutions of the q-EHT (1.1) and (1.3) for the 0-Laguerre/Jacobi case appear only when a1 > 0 and 0 < qy0 < 1. This corresponds to the little q-Laguerre (Wall) polynomials which satisfy the q-EHT with σ1(x, q) = q−2x(a1 − x), σ2(x, q) = ax, τ(x, q) = − 1 (1− q)q x+ 1− aq (1− q)q and λn(q) = q−n 1− q [n]q, where a1 = 1. They are orthogonal w.r.t. the weight function (see the 0-Laguerre/Jacobi case in Table 2) on the set of points {. . . , q2, q, 1} with 0 < a < q−1 in the sense (2.12) where d2n = anqn 2 (q; q)∞ (aq; q)∞ (q, aq; q)n. In the literature, this relation can be found as an infinite sum [21, p. 312]. This case coincides with the case IVa4 in Chapter 10 of [21, p. 278 and p. 312]. On the Orthogonality of q-Classical Polynomials of the Hahn Class 29 5 Concluding remarks The q-polynomials of the Hahn class have been revisited by use of a direct and very simple geometrical approach based on the qualitative analysis of solutions of the q-Pearson (1.5) and the extended q-Pearson (4.3) equations. By this way, it is shown that it is possible to introduce in a unified manner all orthogonal polynomial solutions of the q-EHT, which are orthogonal w.r.t. a measure supported on some set of points in certain intervals. In this review article we are able to extend the well known orthogonality relations for the big q-Jacobi polynomials (see Theorems 4.3 and 4.4), q-Hahn polynomials (see Theorems 4.5 and 4.6), and for the q-Meixner polynomials (see Theorem 4.10) to a larger set of their parameters. 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Classical and quan- tum groups and special functions, Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Aca- demic Publishers Group, Dordrecht, 1992. http://dx.doi.org/10.1016/0377-0427(93)E0241-D http://dx.doi.org/10.1017/CBO9780511526251 http://dx.doi.org/10.1023/A:1012922627100 http://dx.doi.org/10.1007/978-1-4613-0071-7 http://dx.doi.org/10.1007/978-3-642-05014-5 http://aw.twi.tudelft.nl/~koekoek/askey/ http://dx.doi.org/10.1080/10652469408819050 http://dx.doi.org/10.1016/S0377-0427(00)00584-7 http://dx.doi.org/10.1016/S0377-0427(00)00584-7 http://dx.doi.org/10.1080/10652469308819023 http://dx.doi.org/10.1070/RM1989v044n02ABEH002045 http://dx.doi.org/10.1070/RM1989v044n02ABEH002045 1 Introduction 2 The orthogonality and preliminary results 3 The q-weight function 4 The orthogonality of q-polynomials 4.1 q-classical -Jacobi/Jacobi polynomials 4.2 q-classical -Jacobi/Laguerre polynomials 4.3 q-classical -Jacobi/Hermite polynomials 4.4 q-classical -Laguerre/Jacobi polynomials 4.5 q-classical -Hermite/Jacobi polynomials 4.6 q-classical 0-Jacobi/Jacobi polynomials 4.7 q-classical 0-Jacobi/Laguerre polynomials 4.8 q-classical 0-Bessel/Jacobi polynomials 4.9 q-classical 0-Bessel/Laguerre polynomials 4.10 q-classical 0-Laguerre/Jacobi polynomials 5 Concluding remarks References

On the Orthogonality of q-Classical Polynomials of the Hahn Class

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