Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials
We show how to obtain linear combinations of polynomials in an orthogonal sequence {Pn}n≥0, that characterize quasi-orthogonal polynomials of order k ≤ n-1. The polynomials in the sequence {Qn,k}n≥0 are obtained from Pn, by making use of parameter shifts. We use an algorithmic approach to find thes...
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| citation_txt | Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials / D.D. Tcheutia, A.S. Jooste, W. Koepf // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 27 назв. — англ. |
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| description | We show how to obtain linear combinations of polynomials in an orthogonal sequence {Pn}n≥0, that characterize quasi-orthogonal polynomials of order k ≤ n-1. The polynomials in the sequence {Qn,k}n≥0 are obtained from Pn, by making use of parameter shifts. We use an algorithmic approach to find these linear combinations for each family applicable, and these equations are used to prove the quasi-orthogonality of order k. We also determine the location of the extreme zeros of the quasi-orthogonal polynomials with respect to the endpoints of the interval of orthogonality of the sequence {Pn}n≥0, where possible.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 051, 26 pages
Quasi-Orthogonality of Some Hypergeometric
and q-Hypergeometric Polynomials
Daniel D. TCHEUTIA †, Alta S. JOOSTE ‡ and Wolfram KOEPF †
† Institute of Mathematics, University of Kassel,
Heinrich-Plett Str. 40, 34132 Kassel, Germany
E-mail: duvtcheutia@yahoo.fr, koepf@mathematik.uni-kassel.de
URL: http://www.mathematik.uni-kassel.de/~koepf/
‡ Department of Mathematics and Applied Mathematics, University of Pretoria,
Pretoria 0002, South Africa
E-mail: alta.jooste@up.ac.za
Received January 26, 2018, in final form May 17, 2018; Published online May 23, 2018
https://doi.org/10.3842/SIGMA.2018.051
Abstract. We show how to obtain linear combinations of polynomials in an orthogonal
sequence {Pn}n≥0, such as Qn,k(x) =
k∑
i=0
an,iPn−i(x), an,0an,k 6= 0, that characterize quasi-
orthogonal polynomials of order k ≤ n − 1. The polynomials in the sequence {Qn,k}n≥0
are obtained from Pn, by making use of parameter shifts. We use an algorithmic approach
to find these linear combinations for each family applicable and these equations are used
to prove quasi-orthogonality of order k. We also determine the location of the extreme
zeros of the quasi-orthogonal polynomials with respect to the end points of the interval of
orthogonality of the sequence {Pn}n≥0, where possible.
Key words: classical orthogonal polynomials; quasi-orthogonal polynomials; interlacing of
zeros
2010 Mathematics Subject Classification: 33C05; 33C45; 33F10; 33D15; 12D10
1 Introduction
A sequence of polynomials {Pn}n≥0, where each polynomial Pn has degree n, is orthogonal with
respect to the weight function w(x) > 0 on the finite (or infinite) interval [a, b] if∫ b
a
xmPn(x)w(x)dx = 0, m ∈ {0, 1, . . . , n− 1}.
A consequence of orthogonality is that each polynomial Pn(x) has n real, simple zeros in (a, b).
In order for orthogonality conditions to hold, we often need restrictions on the parameters of the
classical orthogonal polynomials and when the parameters deviate from these restricted values
in an orderly way, the zeros may depart from the interval of orthogonality in a predictable way.
This phenomenon can be explained in terms of the concept of quasi-orthogonality. The sequence
of polynomials {Qn,k}n≥0, where each polynomial Qn,k has degree n, is quasi-orthogonal of order
k ∈ {1, 2, . . . , n− 1} with respect to the weight function w(x) > 0 on [a, b] if∫ b
a
xmQn,k(x)w(x)dx = 0, m ∈ {0, 1, . . . , n− k − 1}. (1.1)
This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applica-
tions (OPSFA14). The full collection is available at https://www.emis.de/journals/SIGMA/OPSFA2017.html
mailto:duvtcheutia@yahoo.fr
mailto:koepf@mathematik.uni-kassel.de
http://www.mathematik.uni-kassel.de/~koepf/
mailto:alta.jooste@up.ac.za
https://doi.org/10.3842/SIGMA.2018.051
https://www.emis.de/journals/SIGMA/OPSFA2017.html
2 D.D. Tcheutia, A.S. Jooste and W. Koepf
It is clear that when k = 0 in (1.1), the sequence {Qn,k}n≥0 is orthogonal with respect to w(x)
on [a, b].
Quasi-orthogonality was first studied by Riesz [25], followed by Fejér [12], Shohat [26], Chi-
hara [4], Dickinson [6], Draux [7], Maroni [22] and Joulak [17]. The quasi-orthogonality of Jacobi,
Gegenbauer and Laguerre sequences is discussed in [1], the quasi-orthogonality of Meixner se-
quences in [16] and of Meixner–Pollaczek, Hahn, dual Hahn and continuous dual Hahn sequences
in [15]. More recently, interlacing of zeros of quasi-orthogonal Meixner, Jacobi, Laguerre and
Gegenbauer polynomials were studied in [8, 9, 10, 11] and in [2] interlacing properties of zeros
of quasi-orthogonal polynomials were used to prove results on Gaussian-type quadrature.
Quasi-orthogonal polynomials are characterized by the following property:
Lemma 1.1 ([1, 4]). Let {Pn}n≥0 be a family of orthogonal polynomials on [a, b] with respect
to the weight function w(x) > 0. A necessary and sufficient condition for a polynomial Qn,k of
degree n to be quasi-orthogonal of order k ≤ n− 1 with respect to w on [a, b], is that
Qn,k(x) =
k∑
i=0
an,iPn−i(x), an,0an,k 6= 0, n > k. (1.2)
We show at the end of this section how to derive equations of type (1.2), where {Pn}n≥0 is
a sequence of orthogonal polynomials in the Askey or q-Askey scheme [18]. The polynomials in
the Askey scheme are defined in terms of the generalized hypergeometric series
pFq
(
a1, . . . , ap
b1, . . . , bq
∣∣∣∣∣x
)
=
∞∑
m=0
(a1)m · · · (ap)m
(b1)m · · · (bq)m
xm
m!
,
and (a)m denotes the Pochhammer symbol (or shifted factorial) defined by
(a)m =
{
1 if m = 0,
a(a+ 1)(a+ 2) · · · (a+m− 1) if m ∈ N.
The polynomials in the q-Askey scheme are defined in terms of the basic hypergeometric se-
ries rφs, given by
rφs
(
a1, . . . , ar
b1, . . . , bs
∣∣∣∣∣ q; z
)
=
∞∑
m=0
(a1, . . . , ar; q)m
(b1, . . . , bs; q)m
(
(−1)mq(
m
2 ))1+s−r zm
(q; q)m
,
where the q-Pochhammer symbol (a1, a2, . . . , ar; q)n is defined by
(a1, . . . , ar; q)m := (a1; q)m · · · (ar; q)m,
with
(ai; q)m =
m−1∏
j=0
(
1− aiqj
)
if m ∈ {1, 2, 3, . . .},
1 if m = 0.
We will discuss the quasi-orthogonality of monic polynomial systems on a q-linear lattice in
Section 2, the quasi-orthogonality of the monic Askey–Wilson and q-Racah polynomials, that
lie on a q-quadratic lattice, in Section 3, the quasi-orthogonality of the monic Wilson, Racah
and continuous Hahn polynomials in Section 4 and in Section 5 we will make some concluding
remarks. Throughout this paper, xn,j , j ∈ {1, 2, . . . , n}, denote the zeros of a polynomial of
degree n in the following order: xn,1 < xn,2 < · · · < xn,n. The following results will be referred
to in this paper and we state them here for the convenience of the reader.
Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials 3
Lemma 1.2 ([1, 26]). If Qn,k is quasi-orthogonal of order k ≥ 1 on [a, b] with respect to w(x) > 0,
then at least (n− k) real, distinct zeros of Qn,k lie in the interval (a, b).
Lemma 1.3 ([1, 17]). Suppose Qn,1(x) = Pn(x)+anPn−1(x), an 6= 0. Let yn,j, j ∈ {1, 2, . . . , n},
be the zeros of Qn,1(x) and let fn(x) = Pn(x)
Pn−1(x) . We have
(i) yn,1 < a if and only if −an < fn(a) < 0;
(ii) b < yn,n if and only if −an > fn(b) > 0;
(iii) Qn,1 has all its zeros in (a, b) if and only if fn(a) < −an < fn(b).
Lemma 1.4 ([1, 17]). Suppose Qn,1(x) = Pn(x)+anPn−1(x), an 6= 0. Let xn,j, j ∈ {1, 2, . . . , n},
denote the zeros of Pn(x) and yn,j, j ∈ {1, 2, . . . , n}, the zeros of Qn,1(x). Then
(i) an < 0 if and only if xn,1 < yn,1 < xn−1,1 < xn,2 < yn,2 < · · · < xn−1,n−1 < xn,n < yn,n;
(ii) an > 0 if and only if yn,1 < xn,1 < xn−1,1 < yn,2 < xn,2 < · · · < xn−1,n−1 < yn,n < xn,n.
In order to find the equations of type (1.2), that are needed to prove quasi-orthogonality, we
consider the structure relation (cf. [13, 20, 23])
Pn(x) = anDPn+1(x) + bnDPn(x) + cnDPn−1(x), (1.3)
where the constants an, bn and cn are explicitly given and D is a derivative or difference operator.
Most of the classical orthogonal polynomials we consider in this paper (see [18, Chapters 9
and 14]) satisfy
DPn(x) = S(n)Pn−1,k(x), k ∈ {−1, 0, 1, 2}, (1.4)
where S(n) does not depend on x and Pn−1,k(x) denotes the polynomial obtained when each of
the parameters on which the polynomial Pn(x) depends, can be shifted by k units in the case
of the classical systems, or, in the case of the q-classical systems, when the parameters can each
be multiplied by qk. If we substitute (1.4) in (1.3), we obtain
Pn(x) = anS(n+ 1)Pn,k(x) + bnS(n)Pn−1,k(x) + cnS(n− 1)Pn−2,k(x)
or, by making a parameter shift,
Pn,−k(x) = a′nS
′(n+ 1)Pn(x) + b′nS
′(n)Pn−1(x) + c′nS
′(n− 1)Pn−2(x),
where a′n, b′n, c′n and S′(n) are the values of the coefficients taking into consideration the pa-
rameter shift, and we have a linear combination of polynomials in an orthogonal sequence as
in (1.2). For the so-called very-classical orthogonal polynomials, the general expression for the
parameters an,i, i ∈ {0, 1, 2}, in
Qn,2(x) = Pn(x) + an,1Pn−1(x) + an,2Pn−2(x)
were given in [21, equation (76)] in terms of the coefficients of the differential equations they
satisfy.
We can also apply the operator D to (1.3) to obtain
DPn(x) = anD
2Pn+1(x) + bnD
2Pn(x) + cnD
2Pn−1(x). (1.5)
Replacing (1.5) in (1.3) and using (1.4) twice, yields
Pn(x) = anan+1S(n+ 2)S(n+ 1)Pn,2k(x) + an(bn + bn+1)S(n+ 1)S(n)Pn−1,2k(x)
4 D.D. Tcheutia, A.S. Jooste and W. Koepf
+
(
ancn+1 + an−1cn + b2n
)
S(n)S(n− 1)Pn−2,2k(x)
+ cn(bn + bn−1)S(n− 1)S(n− 2)Pn−3,2k + cncn−1S(n− 2)S(n− 3)Pn−4,2k(x).
By making a parameter shift again, we obtain
Pn,−2k(x) = a′na
′
n+1S
′(n+ 2)S′(n+ 1)Pn(x) + a′n(b′n + b′n+1)S′(n+ 1)S′(n)Pn−1(x)
+
(
a′nc
′
n+1 + a′n−1c
′
n + (b′n)2
)
S′(n)S′(n− 1)Pn−2(x)
+ c′n(b′n + b′n−1)S′(n− 1)S′(n− 2)Pn−3 + c′nc
′
n−1S
′(n− 2)S′(n− 3)Pn−4(x).
These induction arguments show that equations of type (1.2) are structurally valid. We also
refer the reader to [13], where we have so-called connection formulae for the classical orthogonal
polynomial of the q-linear lattice from which one can deduce equations of type (1.2).
Using Zeilberger’s algorithm and following the approach in [3, 19], we wrote, using the com-
puter algebra system Maple, a procedure denoted MixRec(F, k, S(n), α, β,m) which finds a re-
currence equation of the form
S(n, α+m,β +m) =
J∑
j=0
σjS(n− j, α, β), J ∈ {1, 2, . . .},
where S(n, α, β) =
∞∑
k=−∞
F , F is a function of k, n, α and β (of hypergeometric type), and m is
an integer, and its q-analogue denoted qMixRec(F, q, k, S(n), α, β,m) which finds a recurrence
equation of the form
S
(
n, αqm, βqm
)
=
J∑
j=0
σjS(n− j, α, β), J ∈ {1, 2, . . .}.
Our algorithms and all the equations used in this paper can be downloaded from http:
//www.mathematik.uni-kassel.de/~koepf/Publikationen. In [27], using the same algorith-
mic approach, the authors provide a procedure to find mixed recurrence equations satisfied by
classical q-orthogonal polynomials with shifted parameters. These equations were used to inves-
tigate interlacing properties of zeros of sequences of q-orthogonal polynomials. Using our MixRec
function for example, we recover (as shown at the end of our Maple file) the equations used in
[1, Section 3] to study the quasi-orthogonality of the Gegenbauer, the Laguerre, the Jacobi
polynomials.
2 Classical orthogonal polynomials on a q-linear lattice
In this section we consider the quasi-orthogonality of sequences of monic orthogonal polynomials
that are defined on a q-linear lattice, as well as the location of the zeros of the quasi-orthogonal
sequences.
2.1 The big q-Jacobi polynomials
Big q-Jacobi polynomials
P̃n(x;α, β, γ; q) =
(αq; q)n(γq; q)n
(αβqqn; q)n
3φ2
(
q−n, αβqn+1, x
αq, γq
∣∣∣∣∣ q; q
)
(2.1)
are orthogonal for 0 < αq < 1, 0 ≤ βq < 1 and γ < 0 with respect to the continuous weight
function w(x) = (α−1x,γ−1x;q)∞
(x,βγ−1x;q)∞
on (γq, αq).
http://www.mathematik.uni-kassel.de/~koepf/Publikationen
http://www.mathematik.uni-kassel.de/~koepf/Publikationen
Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials 5
The first two recurrence equations in the following proposition follow from [27, equations (7a)
and (7b)], with α and β replaced by α
q and β
q , respectively. The big q-Jacobi polynomials are or-
thogonal for γ < 0, and by replacing γ by γ
q , 0 < q < 1, we obtain the polynomial P̃n
(
x;α, β, γq ; q
)
of which all the parameters are still in the regions where orthogonality is guaranteed and we
will therefore not consider a q-shift of γ.
Proposition 2.1.
P̃n
(
x;
α
q
, β, γ; q
)
= P̃n(x;α, β, γ; q)
+
αq (qn − 1) (βqn − 1) (γqn − 1)
(αβq2n − 1) (αβq2n − q)
P̃n−1(x;α, β, γ; q); (2.2a)
P̃n
(
x;α,
β
q
, γ; q
)
= P̃n(x;α, β, γ; q)
− αβqn+1 (qn − 1) (αqn − 1) (γqn − 1)
(αβq2n − 1) (αβq2n − q)
P̃n−1(x;α, β, γ; q); (2.2b)
P̃n
(
x;α,
β
q
,
γ
q
; q
)
= P̃n(x;α, β, γ; q)
− (qn − 1) (αqn − 1) (−αβqn + γ)
(αβq2n − 1) (αβq2n−1 − 1)
P̃n−1(x;α, β, γ; q). (2.2c)
Corollary 2.2.
P̃n
(
x;
α
q
,
β
q
, γ; q
)
= P̃n(x;α, β, γ; q)−
αq
(
αβq2n − βqn+1 − βqn + q
)
(qn − 1) (γqn − 1)
(αβq2n − 1) (αβq2n − q2)
× P̃n−1(x;α, β, γ; q)
− α2β (qn − 1) (βqn − q) (αqn − q) (γqn − 1) (γqn − q) (qn − q) qn+3
(αβq2n − q2)2 (αβq2n − q3) (αβq2n − q)
× P̃n−2(x;α, β, γ; q). (2.3)
Proof. By replacing β with β
q in (2.2a), we obtain an equation involving polynomials
P̃n
(
x; αq ,
β
q , γ; q
)
, P̃n
(
x;α, bq , γ; q
)
and P̃n−1
(
x;α, βq , γ; q
)
. We use (2.2b) to replace the latter
two polynomials and, after simplifying, we obtain (2.3). �
We will start by proving the quasi-orthogonality of the sequence
{
P̃n
(
x; α
qk
, β, γ; q
)}∞
n=0
. In
order to ensure that the parameter α
qk
is not in the region where orthogonality is guaranteed,
we fix α > 1 with 0 < αq < 1, such that α
qk
> 1, k ∈ {1, 2, . . . , n− 1}.
Theorem 2.3. Let k, l,m ∈ N0, α, β, γ ∈ R, 0 < αq < 1, 0 ≤ βq < 1 and γ < 0. The sequence
of big q-Jacobi polynomials
(i)
{
P̃n
(
x; α
qk
, β, γ; q
)}∞
n=0
, α > 1, is quasi-orthogonal of order k ≤ n− 1 with respect to w(x)
on the interval (γq, αq) and the polynomials have at least (n − k) real, distinct zeros in
(γq, αq);
(ii)
{
P̃n
(
x;α, β
qm , γ; q
)}∞
n=0
, β > 1, is quasi-orthogonal of order m ≤ n−1 with respect to w(x)
on (γq, αq) and the polynomials have at least (n−m) real, distinct zeros in (γq, αq);
(iii)
{
P̃n
(
x;α, β
ql
, γ
ql
; q
)}∞
n=0
, β > 1, is quasi-orthogonal of order l ≤ n− 1 with respect to w(x)
on (γq, αq) and the polynomials have at least (n− l) real, distinct zeros in (γq, αq);
6 D.D. Tcheutia, A.S. Jooste and W. Koepf
(iv)
{
P̃n
(
x; α
qk
, β
qm , γ; q
)}∞
n=0
, α, β > 1 is quasi-orthogonal of order k +m ≤ n− 1 with respect
to w(x) on (γq, αq) and the polynomials have at least n − (k + m) real, distinct zeros in
(γq, αq).
Proof. (i) Fix α > 1 such that 0 < αq < 1. From Lemma 1.1 and (2.2a), it follows that
P̃n
(
x; αq , β, γ; q
)
is quasi-orthogonal of order one on (γq, αq) and according to Lemma 1.2,
at least (n − 1) zeros of P̃n
(
x; αq , β, γ; q
)
lie in the interval (γq, αq). By iteration, we can
express P̃n
(
x; α
qk
, β, γ; q
)
as a linear combination of P̃n(x;α, β, γ; q), P̃n−1(x;α, β, γ; q), . . . ,
P̃n−k(x;α, β, γ; q), and from Lemma 1.1 we deduce that P̃n
(
x; α
qk
, β, γ; q
)
is quasi-orthogonal
of order k on (γq, αq). It follows from Lemma 1.2 that at least (n− k) zeros of P̃n
(
x; α
qk
, β, γ; q
)
are in (γq, αq).
(ii)–(iii) Fix β > 1 such that 0 < βq < 1. The proofs follow in exactly the same way as the
proof of (i), by using (2.2b) and (2.2c), together with Lemmas 1.1 and 1.2.
(iv) Fix α > 1, β > 1 such that 0 < αq < 1 and 0 < βq < 1. From (2.3), P̃n
(
x; αq ,
β
q , γ; q
)
can
be written as a linear combination of P̃n(x;α, β, γ; q), P̃n−1(x;α, β, γ; q) and P̃n−2(x;α, β, γ; q),
and it follows from Lemma 1.1 that each polynomial P̃n
(
x; αq ,
β
q , γ; q
)
, n ∈ {1, 2, . . . }, is quasi-
orthogonal of order two on (γq, αq). From Lemma 1.2, we know that at least (n − 2) zeros
of P̃n
(
x; αq ,
β
q , γ; q
)
lie in (γq, αq). By iteration, we can express P̃n
(
x; α
qk
, β
qm , γ; q
)
as a linear
combination of P̃n(x;α, β, γ; q), P̃n−1(x;α, β, γ; q), . . . , P̃n−(k+m)(x;α, β, γ; q), and the results
follow directly from Lemmas 1.1 and 1.2. �
In order to determine the location of the zeros of the order one and order two quasi-orthogonal
systems, we use a q-analogue of the Vandermonde identity [14, equation (1.2.9)], namely [14,
equation (1.5.3)]
2φ1
(
q−n, b
c
∣∣∣∣∣ q; q
)
=
( cb ; q)n
(c; q)n
bn. (2.4)
Theorem 2.4. Let n ∈ N, α, β, γ ∈ R, such that 0 < αq, βq < 1 and γ < 0. Suppose
xn,j, j ∈ {1, 2, . . . , n} denote the zeros of P̃n(x;α, β, γ; q), yn,j, j ∈ {1, 2, . . . , n} the zeros of
P̃n
(
x; αq , β, γ; q
)
, zn,j, j ∈ {1, 2, . . . , n} the zeros of P̃n
(
x;α, βq , γ; q
)
, vn,j, j ∈ {1, 2, . . . , n} the
zeros of P̃n
(
x; αq ,
β
q , γ; q
)
and wn,j, j ∈ {1, 2, . . . , n} the zeros of P̃n
(
x;α, βq ,
γ
q ; q
)
. Then,
(i) when we fix α > 1, such that 0 < αq < 1,
γq < xn,1 < yn,1 < xn−1,1 < xn,2 < yn,2 < xn−1,2 < · · · < xn−1,n−1 < xn,n < yn,n;
(ii) when we fix β > 1, such that 0 < βq < 1,
zn,1 < γq < xn,1 < xn−1,1 < zn,2 < xn,2 < · · · < xn−1,n−1 < zn,n < xn,n < αq;
(iii) when we fix β > 1, such that 0 < βq < 1, we also have
wn,1 < xn,1 < xn−1,1 < wn,2 < xn,2 < · · · < xn−1,n−1 < wn,n < xn,n < αq;
(iv) when we fix α, β > 1, such that 0 < αq, βq < 1, all the zeros of P̃n
(
x; αq ,
β
q , γ; q
)
are real
and distinct and vn,1 < γq.
Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials 7
Proof. (i) From (2.2a), we obtain
an =
αq (qn − 1) (βqn − 1) (γqn − 1)
(αβq2n − q) (αβq2n − 1)
< 0,
and the interlacing result, as well as the location of yn,1, follows from Lemma 1.4(i).
(ii) From (2.2b), we obtain
an = −αβq
n+1 (qn − 1) (αqn − 1) (γqn − 1)
(αβq2n − 1) (αβq2n − q)
,
which is positive when taking into consideration the values of the parameters. The interlacing
result, as well as the location of yn,n, follows from Lemma 1.4(ii).
The polynomial P̃n(x;α, β, γ; q) evaluated at x = γq, can be written in terms of a 2φ1-
hypergeometric function. We apply (2.4), and simplify, to obtain
fn(γq) =
P̃n(γq;α, β, γ; q)
P̃n−1(γq;α, β, γ; q)
=
α (αβqn − 1) (βqn − 1) (γqn − 1) qn+1
(αβq2n − q) (αβq2n − 1)
(2.5)
and by taking into account the values of the parameters, this expression is negative. Since
−an − fn(γq) = −α (β − 1) (γqn − 1) qn+1
αβq2n − q
< 0,
the result follows from Lemma 1.3(i).
(iii) From (2.2c), we obtain
an = −(qn − 1) (αqn − 1) (−αβqn + γ) q
(αβq2n − q) (αβq2n − 1)
> 0,
and the interlacing result, as well as the location of wn,n, follows from Lemma 1.4(ii).
(iv) Fix α > 1 and β > 1 such that 0 < αq < 1 and 0 < βq < 1. We use (2.3), with an
the coefficient of P̃n−1(x;α, β, γ; q) and bn the coefficient of P̃n−2(x;α, β, γ; q). By taking into
account the values of the parameters,
bn = −α
2β (qn − 1) (βqn − q) (αqn − q) (γqn − 1) (γqn − q) (qn − q) qn+3
(αβq2n − q2)2 (αβq2n − q3) (αβq2n − q)
< 0,
and it follows from [1, Theorem 4] that vn,j , j ∈ {1, 2, . . . , n}, are real.
In order to determine the location of vn,1 and vn,n, we use [17, Theorem 9]. Since
fn(γq)fn−1(γq) + anfn−1(γq) + bn =
P̃n(γq;α, β, γ; q)
P̃n−1(γq;α, β, γ; q)
+ an
P̃n−1(γq;α, β, γ; q)
P̃n−2(γq;α, β, γ; q)
+ bn
=
α2 (β − 1) (γqn − 1) (βqn − q) (γqn − q) qn+2
(αβq2n − q2) (αβq2n − q3)
< 0,
it follows that vn,1 < γq. Furthermore,
fn(αq)fn−1(αq) + anfn−1(αq) + bn
= −
(αqn − q)
(
α2βq2n − αγqn+1 − αβqn − αqn+1 + γqn+1 + αq
)
(γ − αβ) qn+2
(αβq2n − q3) (αβq2n − q2)
and since the sign of this expression varies as the parameters vary within the regions applicable,
we cannot determine the position of vn,n. �
8 D.D. Tcheutia, A.S. Jooste and W. Koepf
Remark 2.5.
(i) From Theorem 2.3(i) we know that the polynomial P̃n
(
x; αq , β, γ; q
)
, α > 1, is quasi-
orthogonal of order one and an interlacing result is proved in Theorem 2.4(i), but the
location of the extreme zero yn,n, with respect to (γq, αq), is not fixed, since the sign of
−an − fn(αq) = an −
P̃n(αq;α, β, γ; q)
P̃n−1(αq;α, β, γ; q)
= −
(
−α2q2nβ + αβqn + qnαγ + αqn − γqn − α
)
q
αβq2n − q
(2.6)
changes as the parameters vary within the region applicable.
(ii) When β = 0 in the definition of the big q-Jacobi polynomials (2.1), we obtain the big
q-Laguerre polynomials, i.e., P̃n(x;α, 0, γ; q) = P̃n(x;α, γ; q) [18, equation (14.5.13)] and
we can use (2.2a) with β = 0. Let xn,j , j ∈ {1, 2, . . . , n}, be the zeros of P̃n(x;α, γ; q)
and yn,j , j ∈ {1, 2, . . . , n}, the zeros of P̃n
(
x; αq , γ; q
)
. When β = 0 in (2.6), we obtain
−an − fn(αq) = an −
P̃n(αq;α, γ; q)
P̃n−1(αq;α, γ; q)
= γqn(α− 1) + α
(
qn − 1
)
< 0,
taking into consideration that α > 1, 0 < αq < 1 and γ < 0, and
fn(γq) + an =
P̃n(γq;α, γ; q)
P̃n−1(γq;α, γ; q)
+ an = α qn
(
γqn − 1
)
+ an = α
(
γqn − 1
)
< 0.
We thus have fn(γq) < −an < fn(αq) and according to Lemma 1.3(iii), all the zeros of the
order one quasi-orthogonal polynomial P̃n
(
x; αq , γ; q
)
, α > 1, lie in (γq, αq). Furthermore,
since an < 0, it follows from Lemma 1.4(ii) that
γq < xn,1 < yn,1 < xn−1,1 < xn,2 < yn,2 < xn−1,2 < · · · < xn−1,n−1
< xn,n < yn,n < αq.
2.2 The q-Hahn polynomials
The q-Hahn polynomials
Q̃n(x̄;α, β,N |q) =
(αq; q)n(q−N ; q)n
(αβqqn; q)n
3φ2
(
q−n, αβqn+1, x̄
αq, q−N
∣∣∣∣∣ q; q
)
,
with x̄ = q−x and n ∈ {0, 1, . . . , N} are orthogonal on
(
1, q−N
)
with respect to the discrete weight
w(x) = (αq,q−N ;q)x
(q,β−1q−N ;q)x(αβq)x
for 0 < αq < 1 and 0 < βq < 1 or α > q−N and β > q−N . We recall
the definition of quasi-orthogonality with respect to a discrete weight (cf. [5]). A polynomial Qn,k
of exact degree n ≥ k, n ∈ {0, 1, . . . , N}, where N may be infinite, is discrete quasi-orthogonal
of order k with wi the weight at each point xi, i ∈ {0, 1, . . . , N − 1}, if
N−1∑
i=0
(xi)
mQn,k(xi)wi
{
= 0, for m ∈ {0, 1, . . . , n− k − 1},
6= 0, for m = n− k.
We will consider the case 0 < αq, βq < 1. The following equations follow from [27, equa-
tions (8a) and (8b)], with α and β replaced by α
q and β
q , respectively,
Q̃n
(
x̄;
α
q
, β,N |q
)
= Q̃n(x̄;α, β,N |q)
Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials 9
+
α (qn − 1) (βqn − 1)
(
qn − qN+1
)
qN (αβq2n − 1) (αβq2n − q)
Q̃n−1(x̄;α, β,N |q); (2.7a)
Q̃n
(
x̄;α,
β
q
,N |q
)
= Q̃n(x̄;α, β,N |q)
−
αβ
(
qn − qN+1
)
(αqn − 1)(qn − 1)
(αβq2n − 1) (αβq2n − q) qN−n
Q̃n−1(x̄;α, β,N |q). (2.7b)
Corollary 2.6.
Q̃n
(
x̄;
α
q
,
β
q
,N |q
)
= Q̃n(x̄;α, β,N |q)
−
α
(
αβq2n − βqn+1 − βqn + q
)
(qn − 1)
(
qn − qN+1
)
(αβq2n − q2) (αβq2n − 1) qN
Q̃n−1(x̄;α, β,N |q)
−
α2βqn+1 (qn − 1) (βqn − q)
(
qn − qN+1
)
(αqn − q) (qn − q)
(
qn − qN+2
)
(αβq2n − q2)2 (αβq2n − q) (αβq2n − q3) q2N
× Q̃n−2(x̄;α, β,N |q). (2.8)
Theorem 2.7. Let k,m,N ∈ N0, n ∈ {0, 1, 2, . . . , N}, α, β,∈ R. For 0 < αq < 1 and
0 < βq < 1, the sequence of q-Hahn polynomials
(i)
{
Q̃n
(
x̄; α
qk
, β,N |q
)}N
n=0
, with α > 1, is quasi-orthogonal of order k ≤ n− 1 with respect to
the discrete weight w(x) on the interval
(
1, q−N
)
and the polynomials have at least (n−k)
real, distinct zeros in
(
1, q−N
)
;
(ii)
{
Q̃n
(
x̄;α, β
qm , N |q
)}N
n=0
, β > 1, is quasi-orthogonal of order m ≤ n − 1 with respect to
w(x) on
(
1, q−N
)
and the polynomials have at least (n − m) real, distinct zeros in the
interval
(
1, q−N
)
;
(iii)
{
Q̃n
(
x̄; α
qk
, β
qm , N |q
)}N
n=0
, α, β > 1, is quasi-orthogonal of order k+m ≤ n−1 with respect
to w(x) on
(
1, q−N
)
and the polynomials have at least n − (k + m) real, distinct zeros in(
1, q−N
)
.
Proof. (i) Fix α > 1, β ∈ R, such that 0 < αq < 1, 0 < βq < 1. From Lemma 1.1 and (2.7a), it
follows that Q̃n
(
x̄; αq , β,N |q
)
is quasi-orthogonal of order one on
(
1, q−N
)
. From Lemma 1.2 we
know that at least (n− 1) zeros of Q̃n
(
x̄; αq , β,N |q
)
lie in the interval
(
1, q−N
)
. By iteration, we
can express Q̃n
(
x̄; α
qk
, β,N |q
)
as a linear combination of Q̃n(x̄;α, β,N |q), Q̃n−1(x̄;α, β,N |q), . . . ,
Q̃n−k(x̄;α, β,N |q), and the results follow from Lemmas 1.1 and 1.2.
(ii) Fix β > 1, α ∈ R, such that 0 < αq < 1, 0 < βq < 1. The quasi-orthogonality follows in
the same way as in (i), by using (2.7b).
(iii) Fix α > 1 and β > 1 such that 0 < αq < 1 and 0 < βq < 1. From (2.8), we see that
Q̃n
(
x̄; αq ,
β
q , N |q
)
can be written as a linear combination of Q̃n(x̄;α, β,N |q), Q̃n−1(x̄;α, β,N |q)
and Q̃n−2(x̄;α, β,N |q) and it follows from Lemma 1.1 that the sequence Q̃n
(
x̄; αq ,
β
q , γ; q
)
is
quasi-orthogonal of order two on
(
1, q−N
)
. By iteration, we can express Q̃n
(
x̄; α
qk
, β
qm , N |q
)
as
a linear combination of Q̃n(x̄;α, β,N |q), Q̃n−1(x̄;α, β,N |q), . . . , Q̃n−(k+m)(x̄;α, β,N |q), and
the result follows directly from Lemma 1.1. It follows from Lemma 1.2 that at least n− (k+m)
zeros of Q̃n
(
x̄; α
qk
, β
qm , γ; q
)
lie in the interval
(
1, q−N
)
. �
Theorem 2.8. Let N ∈ N0, n ∈ {0, 1, 2, . . . , N}, α, β ∈ R, 0 < αq, βq < 1, and let xn,j,
j ∈ {1, 2, . . . , n}, denote the zeros of Q̃n(x̄;α, β,N |q), yn,j, j ∈ {1, 2, . . . , n}, the zeros of
Q̃n
(
x̄; αq , β,N |q
)
and zn,j, j ∈ {1, 2, . . . , n}, the zeros of Q̃n
(
x̄;α, βq , N |q
)
. Then
10 D.D. Tcheutia, A.S. Jooste and W. Koepf
(i) if α > 1, yn,1 < 1 < xn,1 < xn−1,1 < yn,2 < xn,2 < · · · < xn−1,n−1 < yn,n < xn,n < q−N ;
(ii) if β > 1, 1 < xn,1 < zn,1 < xn−1,1 < xn,2 < zn,2 < · · · < xn−1,n−1 < xn,n < q−N < zn,n.
Proof. (i) From (2.7a) we obtain the value
an =
α (qn − 1) (βqn − 1)
(
qn − qN+1
)
qN (αβq2n − 1) (αβq2n − q)
,
which is positive when we take into consideration the values of the parameters. The interlacing
result, from which we can deduce the location of yn,n, follows from Lemma 1.4(ii).
In order to prove that yn,1 does not lie in the interval of orthogonality, i.e., yn,1 < 1, we use
the fact that
3φ2
(
q−n, αβqn+1, 1
αq, q−N
∣∣∣∣∣ q; q
)
= 3φ2
(
q−n, αβqn+1, q−0
αq, q−N
∣∣∣∣∣ q; q
)
= 1
and Lemma 1.3. Consider
fn(1) =
Q̃n(1;α, β,N |q)
Q̃n−1(1;α, β,N |q)
= −
(αqn − 1) (αβqn − 1)
(
qn − qN+1
)
(αβq2n − 1) (αβq2n − q)
,
which is negative for the appropriate parameter values. We thus have
−an − fn(1) =
(α− 1)
(
qn − qN+1
)
(αβq2n − q) qN
< 0,
i.e., −an < fn(1) < 0, and the result follows from Lemma 1.3(i).
(ii) From (2.7b) we obtain
an = −
αβ
(
qn − qN+1
)
(αqn − 1) (qn − 1) qn
(αβq2n − 1) (αβq2n − q) qN
,
which is negative. The interlacing result, from which we can deduce the location of zn,1, follows
from Lemma 1.4(i).
The polynomial Q̃n(x̄;α, β,N |q) evaluated at x̄ = q−N , can be written in terms of a 2φ1-
hypergeometric function. We apply (2.4), and simplify, to obtain
fn
(
q−N
)
=
Q̃n
(
q−N ;α, β,N |q
)
Q̃n−1 (q−N ;α, β,N |q)
= −
α (βqn − 1) (αβqn − 1)
(
−qN+1 + qn
)
qn
(αβq2n − q) (αβq2n − 1) qN
.
When taking into consideration the values of the parameters,
−an − fn
(
q−N
)
= −
α (β − 1)
(
qn − qN+1
)
qn
(αβq2n − q)qN
< 0
and the result follows from Lemma 1.3(ii). �
Remark 2.9. We cannot say anything about the location of the zeros of Q̃n
(
x̄; α
q2
, β,N |q
)
, since
the coefficient of Q̃n−2(x̄;α, β,N |q), in the equation
Q̃n
(
x̄;
α
q2
, β,N |q
)
= Q̃n(x̄;α, β,N |q)
+
α (q + 1) (qn − 1) (βqn − 1)
(
qn − qN+1
)
(αβq2n − q2) (αβq2n − 1) qN
Q̃n−1(x̄;α, β,N |q)
Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials 11
+
(αq)2 (qn − 1) (βqn − q)
(
qn − qN+1
)
(qn − q) (βqn − 1)
(
qn − qN+2
)
(αβq2n − q2)2 (αβq2n − q) (αβq2n − q3) q2N
Q̃n−2(x̄;α, β,N |q),
that can be obtained from (2.7a), is positive (cf. [1, Theorem 4]). The same is true for the
location of the zeros of Q̃n
(
x̄;α, β
q2
, N |q
)
and the equation can be found in the accompanying
Maple file.
Theorem 2.10. Let N ∈ N0, n ∈ {0, 1, 2, . . . , N}, α, β > 1. All the zeros of Q̃n
(
x̄; αq ,
β
q , N |q
)
are real and distinct and if zn,j, j ∈ {1, 2, . . . , n}, are the zeros of Q̃n
(
x̄; αq ,
β
q , N |q
)
, then zn,1 < 1
and q−N < zn,n.
Proof. Fix α > 1 and β > 1 such that 0 < αq < 1 and 0 < βq < 1. We use (2.8) with an
the coefficient of Q̃n−1(x̄;α, β,N |q) and bn the coefficient of Q̃n−2(x̄;α, β,N |q). By taking into
account the values of the parameters, we see that
bn = −
α2β (βqn − q)
(
qn − qN+1
) (
qn − qN+2
)
(αqn − q) (qn − 1) (qn − q) qn+1
(αβq2n − q) (αβq2n − q3) (αβq2n − q2)2 q2N
< 0,
and it follows from [1, Theorem 4] that zn,j , j ∈ {1, 2, . . . , n}, are real.
In order to determine the location of zn,1 and zn,n, we use [17, Theorem 9]. Since
fn(1)fn−1(1) + anfn−1(1) + bn =
Q̃n(1;α, β,N |q)
Q̃n−2(1;α, β,N |q)
+ an
Q̃n−1(1;α, β,N |q)
Q̃n−2(1;α, β,N |q)
+ bn
=
(α− 1) (αqn − q)
(
qn − qN+2
) (
qn − qN+1
)
q
(αβq2n − q2) (αβq2n − q3) q2N
< 0,
it follows that zn,1 < 1. Furthermore,
fn
(
q−N
)
fn−1
(
q−N
)
+ anfn−1
(
q−N
)
+ bn
=
α2 (β − 1) (βqn − q)
(
qn − qN+1
) (
qn − qN+2
)
qn
(αβq2n − q2) (αβq2n − q3) q2N
< 0
and q−N < zn,n. �
Remark 2.11.
(i) When we let β = 0 in the definition of the q-Hahn polynomials, we obtain the affine
q-Krawtchouk polynomials [18, Section 14.16] K̃Aff
n (x̄;α,N ; q), orthogonal on
(
1, q−N
)
if
0 < αq < 1. When we fix α > 1, such that 0 < αq < 1, the quasi-orthogonality of the
polynomials K̃Aff
n
(
x̄; α
qk
, N ; q
)
, k < n, on
(
1, q−N
)
follows directly from (2.8), with β = 0.
If xn,j , j ∈ {1, 2, . . . , n}, denote the zeros of K̃Aff
n (x̄;α,N ; q) and yn,j , j ∈ {1, 2, . . . , n},
the zeros of K̃Aff
n
(
x̄; αq , N ; q
)
, the interlacing result in Theorem 2.8(i) follows.
(ii) Since lim
α→∞
Q̃n(x;α, p,N |q) = K̃qtm
n (x; p,N ; q), [18, Section 14.14], we obtain from (2.7b),
the equation
K̃qtm
n
(
x;
p
q
,N ; q
)
= K̃qtm
n (x; p,N ; q) +
(
qN+1 − qn
)
(qn − 1)
pq2n+N
K̃qtm
n−1(x; p,N ; q).
For q−N < p < q−N+1, the quantum q-Krawtchouk polynomials K̃qtm
n
(
x; p
qk
, N ; q
)
are
quasi-orthogonal of order k < n and the interlacing result in Theorem 2.8(ii) follows, where
xn,j , j ∈ {1, 2, . . . , n}, denote the zeros of K̃qtm
n (x; p,N ; q) and zn,j , j ∈ {1, 2, . . . , n}, the
zeros of K̃qtm
n
(
x; pq , N ; q
)
.
12 D.D. Tcheutia, A.S. Jooste and W. Koepf
2.3 The little q-Jacobi polynomials
The little q-Jacobi polynomials
p̃n(x;α, β|q) = (−1)nq(
n
2)
(αq; q)n
(αβqqn; q)n
2φ1
(
q−n, αβqn+1
αq
∣∣∣∣∣ q; qx
)
are orthogonal with respect to the discrete weight w(x) = (βq;q)x(αq)x
(q;q)x
for 0 < αq < 1, βq < 1 on
(0, 1). Consider the recurrence equations (cf. [27, equations (9a) and (9b)])
p̃n
(
x;
α
q
, β|q
)
= p̃n(x;α, β|q) +
αqn (qn − 1) (βqn − 1)
(αβq2n − 1) (αβq2n − q)
p̃n−1(x;α, β|q); (2.9a)
p̃n
(
x;α,
β
q
|q
)
= p̃n(x;α, β|q)− αβq2n (qn − 1) (αqn − 1)
(αβq2n − 1) (αβq2n − q)
p̃n−1(x;α, β|q). (2.9b)
Corollary 2.12.
p̃n
(
x;
α
q
,
β
q
|q
)
= p̃n(x;α, β|q)
−
αqn
(
αβq2n − qn+1β − βqn + q
)
(qn − 1)
(αβq2n − q2) (αβq2n − 1)
p̃n−1(x;α, β|q)
− α2βq3n+1 (qn − 1) (βqn − q) (αqn − q) (qn − q)
(αβq2n − q2)2 (αβq2n − q) (αβq2n − q3)
p̃n−2(x;α, β|q). (2.10)
Theorem 2.13. Let k,m ∈ N0, α, β ∈ R. For 0 < αq < 1 and 0 < βq < 1, the sequence of
little q-Jacobi polynomials
(i)
{
p̃n
(
x; α
qk
, β|q
)}∞
n=0
, with α > 1, is quasi-orthogonal of order k ≤ n−1 with respect to w(x)
on the interval (0, 1) and the polynomials have at least (n−k) real, distinct zeros in (0, 1);
(ii)
{
p̃n
(
x;α, β
qm |q
)}∞
n=0
, β > 1, is quasi-orthogonal of order m ≤ n − 1 with respect to w(x)
on (0, 1) and the polynomials have at least (n−m) real, distinct zeros in (0, 1);
(iii)
{
p̃n
(
x; α
qk
, β
qm |q
)}∞
n=0
, α, β > 1, is quasi-orthogonal of order k +m ≤ n− 1 with respect to
w(x) on (0, 1) and the polynomials have at least n− (k +m) real, distinct zeros in (0, 1).
Proof. (i) Fix α > 1, β ∈ R, such that 0 < αq < 1, 0 < βq < 1. From Lemma 1.1 and (2.9a), it
follows that p̃n
(
x; αq , β|q
)
is quasi-orthogonal of order one on (0, 1). By iteration, we can express
p̃n
(
x; α
qk
, β|q
)
as a linear combination of p̃n(x;α, β|q), p̃n−1(x;α, β|q), . . . , p̃n−k(x;α, β|q), and
the result follows from Lemma 1.1. The location of the (n−k) real, distinct zeros of p̃n(x; α
qk
, β|q),
k ∈ {1, 2, . . . , n− 1}, follows from Lemma 1.2.
(ii) Fix β > 1, α ∈ R, such that 0 < αq < 1, 0 < βq < 1. The quasi-orthogonality follows in
the same way as in (i), by using (2.9b).
(iii) Fix α > 1 and β > 1 such that 0 < αq < 1 and 0 < βq < 1. From (2.10), we
see that p̃n
(
x; αq ,
β
q |q
)
can be written as a linear combination of p̃n(x;α, β|q), p̃n−1(x;α, β|q)
and p̃n−2(x;α, β|q), and it follows from Lemma 1.1 that the sequence p̃n
(
x; αq ,
β
q |q
)
is quasi-
orthogonal of order two on (0, 1). By iteration, we can express p̃n
(
x; α
qk
, β
qm |q
)
as a linear com-
bination of p̃n(x;α, β|q), p̃n−1(x;α, β|q), . . . , p̃n−(k+m)(x;α, β|q), and the results follow directly
from Lemmas 1.1 and 1.2. �
Theorem 2.14. Let α, β ∈ R, 0 < αq, βq < 1, and suppose xn,j, j ∈ {1, 2, . . . , n}, denote the
zeros of p̃n(x;α, β|q), yn,j, j ∈ {1, 2, . . . , n}, the zeros of p̃n
(
x; αq , β|q
)
and zn,j, j ∈ {1, 2, . . . , n},
the zeros of p̃n
(
x;α, βq |q
)
. Then
Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials 13
(i) if α > 1, yn,1 < 0 < xn,1 < xn−1,1 < yn,2 < · · · < xn−1,n−1 < yn,n < xn,n < 1;
(ii) if β > 1, 0 < xn,1 < zn,1 < xn−1,1 < xn,2 < · · · < xn−1,n−1 < xn,n < 1 < zn,n.
Proof. (i) From (2.9a) we obtain the value
an =
α (qn − 1) (βqn − 1) qn
(αβq2n − 1) (αβq2n − q)
> 0.
The interlacing result, as well as the position of yn,n, follows from Lemma 1.4(ii).
To obtain the position of yn,1 we use Lemma 1.3 and when we consider the given parameter
values,
fn(0) =
p̃n(0;α, β|q)
p̃n−1(0;α, β|q)
= − (αβqn − 1) (αqn − 1) qn
(αβq2n − 1) (αβq2n − q)
< 0.
We thus have
−an − fn(0) =
qn(α− 1)
αβq2n − q
< 0
and the result follows from Lemma 1.3(i).
(ii) From (2.9b), we obtain the value
an = −αβ (qn − 1) (αqn − 1) q2n
(αβq2n − 1) (αβq2n − q)
< 0.
The interlacing result, as well as the position of zn,1, follows from Lemma 1.4(i).
To obtain the position of zn,n, we use Lemma 1.3, and when we consider the given parameter
values,
fn(1) =
p̃n(1;α, β|q)
p̃n−1(1;α, β|q)
= −α (β − 1) q2n
αβq2n − q
> 0.
We thus have
−an − fn(1) = −α (β − 1) q2n
αβq2n − q
> 0
and it follows from Lemma 1.3(ii) that 1 < zn,n. �
Theorem 2.15. Let α, β > 1. All the zeros of p̃n
(
x; αq ,
β
q |q
)
, denoted by zn,j, j ∈ {1, 2, . . . , n},
are real and distinct and zn,1 < 0 and 1 < zn,n.
Proof. Fix α > 1 and β > 1 such that 0 < αq < 1 and 0 < βq < 1. We use (2.10), with an the
coefficient of p̃n−1(x;α, β|q) and bn the coefficient of p̃n−2(x;α, β|q). By taking into account the
values of the parameters, we see that
bn = −α
2β (βqn − q) (αqn − q) (qn − 1) (qn − q) q3n+1
(αβq2n − q) (αβq2n − q3) (αβq2n − q2)2 < 0,
and it follows from [1, Theorem 4] that zn,j , j ∈ {1, 2, . . . , n}, are real and distinct.
In order to determine the location of zn,1 and zn,n, we use [17, Theorem 9]. Since
fn(0)fn−1(0) + anfn−1(0) + bn =
(α− 1) (αqn − q) q2n+1
(αβq2n − q3) (αβq2n − q2)
< 0,
it follows that zn,1 < 1. Furthermore,
fn(1)fn−1(1) + anfn−1(1) + bn =
α2 (β − 1) (βqn − q) q3n
(αβq2n − q2) (αβq2n − q3)
< 0,
and 1 < zn,n. �
14 D.D. Tcheutia, A.S. Jooste and W. Koepf
Remark 2.16.
(i) From (2.9a) we obtain
p̃n
(
x;
α
q2
, β|q
)
= p̃n(x;α, β|q) +
αqn (q + 1) (qn − 1) (βqn − 1)
(αβq2n − q2) (αβq2n − 1)
× p̃n−1(x;α, β|q) + bnp̃n−2(x;α, β|q),
with
bn =
α2q2n+2 (qn − 1) (βqn − q) (qn − q) (βqn − 1)
(αβq2n − q2)2 (αβq2n − q) (αβq2n − q3)
and
Cn − bn =
q2n+1 (α− q) (qn − q) (βqn − q)α
(αβq2n − q3) (αβq2n − q2)2 ,
where −Cn is the coefficient of p̃n−2(x;α, β|q) in the three-term recurrence equation of the
little q-Jacobi polynomials [18, equation (14.12.4)]. Since Cn < bn, there is an interlacing
between (n − 2) zeros of p̃n
(
x; α
q2
, β|q
)
and the (n − 1) zeros of p̃n−1(x;α, β|q) (cf. [17,
Theorem 15]).
(ii) When β = 0 in the definition of the little q-Jacobi polynomials, we obtain the little q-
Laguerre (or Wall) polynomials p̃n(x;α|q), that are orthogonal on (0, 1) when 0 < αq < 1.
The quasi-orthogonality of
{
p̃n
(
x; α
qk
|q
)}
n≥0
, for k < n, when α > 1, 0 < αq < 1, follows
directly from (2.9a) (with β = 0). The location of the zeros of the order one quasi-
orthogonal polynomial p̃n
(
x; αq |q
)
is given in Theorem 2.14(i), where xn,j , j ∈ {1, 2, . . . , n},
denote the zeros of p̃n(x;α|q) and yn,j , j ∈ {1, 2, . . . , n}, the zeros of p̃n
(
x; αq |q
)
.
2.4 The q-Laguerre polynomials
The q-Laguerre polynomials
L̃(α)
n (x; q) =
(−1)n(qα+1; q)n
qn(n+α) 1φ1
(
q−n
qα+1
∣∣∣∣∣ q;−qn+α+1x
)
are orthogonal for α > −1 on (0,∞) with respect to the weight function w(x) = xα
(−x;q)∞
.
Consider the equation (cf. [24, equation (4.12)] and [27, equation (12a)])
L̃(α−1)
n (x; q) = L̃(α)
n (x; q)− (qn − 1) q
q2n+α
L̃
(α)
n−1(x; q). (2.11)
Theorem 2.17. Let k ∈ N0 and α ∈ R. For −1 < α < 0 and k ∈ {1, 2, . . . , n−1}, the sequence
of q-Laguerre polynomials
{
L̃
(α−k)
n (x; q)
}∞
n=0
is quasi-orthogonal of order k on the interval (0,∞)
with respect to w(x) and the polynomials have at least (n− k) real, distinct zeros in (0,∞).
Proof. Fix −1 < α < 0. From Lemma 1.1 and (2.11) it follows that L̃
(α−1)
n (x; q) is quasi-
orthogonal of order one on (0,∞). By iteration, we can express L̃
(α−k)
n (x; q) as a linear com-
bination of L̃
(α)
n (x; q), L̃
(α)
n−1(x; q), . . . , L̃
(α)
n−k(x; q), and the result follows from Lemma 1.1. The
location of the (n − k) real, distinct zeros of L̃
(α−k)
n (x; q), k ∈ {1, 2, . . . , n − 1}, follows from
Lemma 1.2. �
Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials 15
Theorem 2.18. Let −1 < α < 0 and denote the zeros of L̃
(α)
n (x; q) by xn,j, j ∈ {1, 2, . . . , n},
and the zeros of L̃
(α−1)
n (x; q) by yn,j, j ∈ {1, 2, . . . , n}. Then
yn,1 < 0 < xn,1 < xn−1,1 < yn,2 < xn,2 < · · · < xn−1,n−1 < yn,n < xn,n.
Proof. From (2.11), we obtain the value an = −(qn−1)q
q2n+α
> 0. The interlacing result, as well as
the position of yn,n, follows from Lemma 1.4(ii).
To obtain the position of yn,1, we use Lemma 1.3, and when we consider the given parameter
values,
fn(0) =
(qn+α − 1) q
q2n+α
< 0.
We thus have
−an − fn(0) = −(qα − 1) q
qn+α
< 0
and since −an < fn(0) < 0, the result follows from Lemma 1.3(i). �
2.5 The Al-Salam–Carlitz I polynomials
The Al-Salam–Carlitz I polynomials
Ũ (α)
n (x; q) = (−α)nq(
n
2) 2φ1
(
q−n, x−1
0
∣∣∣∣∣ q; qxα
)
are orthogonal for α < 0 on (α, 1) with respect to the weight function w(x) =
(
qx, qxα ; q
)
∞. The
polynomials Ũ
( α
qk
)
n (x; q), k < n, are orthogonal with respect to
(
qx, q
k+1x
α ; q
)
∞ on the interval(
α
qk
, 1
)
and we will prove that they are quasi-orthogonal with respect to w(x) on (α, 1). We use
the equation
Ũ
(α
q
)
n (x; q) = Ũ (α)
n (x; q) + αq−1
(
qn − 1
)
Ũ
(α)
n−1(x; q). (2.12)
We deduce that
Ũ
( α
q2
)
n (x; q) = Ũ (α)
n (x; q) +
α (qn − 1) (q + 1)
q2
Ũ
(α)
n−1(x; q)
+
α2 (qn − 1) (qn − q)
q4
Ũ
(α)
n−2(x; q). (2.13)
Theorem 2.19. Let k ∈ N0 and α < 0. The sequence of Al-Salam–Carlitz I polynomials{
Ũ
( α
qk
)
n (x; q)
}∞
n=0
is quasi-orthogonal with respect to w(x) on (α, 1) and the polynomials have at
least (n− k) real, distinct zeros in (α, 1).
Proof. From Lemma 1.1 and (2.12) it follows that Ũ
(α
q
)
n (x; q) is quasi-orthogonal of order one on
(α, 1). By iteration, we can express Ũ
( α
qk
)
n (x; q) as a linear combination of Ũ
(α)
n (x; q), Ũ
(α)
n−1(x; q),
. . . , Ũ
(α)
n−k(x; q), and the result follows from Lemma 1.1. The location of the (n−k) real, distinct
zeros of Ũ
( α
qk
)
n (x; q), k ∈ {1, 2, . . . , n− 1}, follows from Lemma 1.2. �
16 D.D. Tcheutia, A.S. Jooste and W. Koepf
Remark 2.20. We can also obtain (2.12) from the generating function [18, equation (14.24.10)]
of the Al-Salam–Carlitz I polynomials
(t, αt; q)∞
(xt; q)∞
=
∞∑
n=0
U
(α)
n (x; q)
(q; q)n
tn, (2.14)
from which it follows that
(t, α
qk
t; q)∞
(xt; q)∞
=
∞∑
n=0
U
( α
qk
)
n (x; q)
(q; q)n
tn, k ∈ {1, 2, . . .}. (2.15)
From the relation
(a; q)∞
(aqn; q)∞
= (a; q)n,
we obtain, when a = α
qk
t and n = k,(
α
qk
t; q
)
∞
=
(
α
qk
t; q
)
k
(αt; q)∞.
By using (2.14), (2.15) becomes
(
α
qk
t; q
)
k
∞∑
n=0
U
(α)
n (x; q)
(q; q)n
tn =
∞∑
n=0
U
( α
qk
t)
n (x; q)
(q; q)n
tn, k ∈ {1, 2, . . .}.
Expanding
(
α
qk
t; q
)
k
and equating powers of t yields U
( α
qk
)
n (x; q) as a linear combination of
U
(α)
n−j(x; q), j ∈ {0, 1, . . . , k}. In particular, for k = 1 and k = 2, we get (2.12) and (2.13),
respectively.
Theorem 2.21. Let α < 0 and denote the zeros of Ũ
(α)
n (x; q) by xn,j, j ∈ {1, 2, . . . , n}, and the
zeros of Ũ
(α
q
)
n (x; q) by yn,j, j ∈ {1, 2, . . . , n}. Then
(i) α
q < yn,1 < xn,1 < xn−1,1 < yn,2 < · · · < xn−1,n−1 < yn,n < xn,n < 1 and, additionally, if
α < qn
qn−1 , then yn,1 < α < xn,1;
(ii) (n− 2) zeros of Ũ
( α
q2
)
n (x; q) interlace with the n zeros of Ũ
(α)
n (x; q) if α < qn+1
qn−1 .
Proof. (i) From (2.12) we obtain the value an = α(qn−1)
q > 0. The interlacing result, as well
as the position of yn,n, follows from Lemma 1.4(ii). The position of yn,1 cannot be determined,
since
fn(α) =
Ũ
(α)
n (α; q)
Ũ
(α)
n−1(α; q)
= −qn−1 < 0
and the sign of
−an − fn(α) = −α(qn − 1)
q
+ qn−1 =
α(1− qn) + qn
q
varies as the parameters vary within the allowed regions. However, if α < qn
qn−1 , then −an <
fn(α) < 0 and from Lemma 1.3(i), it follows that yn,1 < α.
Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials 17
(ii) The coefficient of Ũ
(α)
n−2(x; q) in (2.13) is
bn =
(qn − 1) (qn − q)α2
q4
.
From the three-term recurrence equation of the Al-Salam–Carlitz I polynomials [18, equa-
tion (14.24.4)]
Ũ (α)
n (x; q) =
(
x− qn (α+ 1)
q
)
Ũ
(α)
n−1(x; q)− αqn (qn − q)
q3
Ũ
(α)
n−2(x; q),
we obtain Cn = αqn(qn−q)
q3
and since
bn − Cn =
α (qn − q)
(
αqn − qn+1 − α
)
q4
> 0
when α < qn+1
qn−1 , the interlacing result follows from [17, Theorem 15(ii)]. �
3 Classical orthogonal polynomials on a q-quadratic lattice
In this section we consider the quasi-orthogonality of the monic Askey–Wilson and q-Racah
polynomials, that are defined on a q-quadratic lattice.
3.1 The Askey–Wilson polynomials
The Askey–Wilson polynomials
p̃n(x; a, b, c, d|q) =
(ab, ac, ad; q)n
(2a)n(abcdqn−1; q)n
4φ3
(
q−n, abcdqn−1, aeiθ, ae−iθ
ab, ac, ad
∣∣∣∣∣ q; q
)
, x = cos θ,
with a, b, c, d either real, or they occur in complex conjugate pairs, and max(|a|, |b|, |c|, |d|) < 1,
are orthogonal on (−1, 1) with respect to
w(x; a, b, c, d|q) =
1√
1− x2
∣∣∣∣ (e2iθ; q)∞
(aeiθ, beiθ, ceiθ, deiθ; q)∞
∣∣∣∣2 . (3.1)
The weight function is independent of the order of the parameters a, b, c and d and by shifting b
to b/q, c to c/q or d to d/q, we obtain the same interlacing results as by shifting a to a/q. We
will now fix a > 0 such that q < |a| < 1 and for these values of a, the polynomial p̃n(x; a, b, c, d|q)
is orthogonal on (−1, 1) with respect to w(x; a, b, c, d|q). In what follows, we assume that |a| =
max(|a|, |b|, |c|, |d|) < 1. Should this not be the case, the order in which the parameters occur,
can be changed.
We will thus only consider the equations in which a is shifted to a
qk
> 1 (or a
qk
< −1 should
a < 0), and we will prove that the polynomials p̃n
(
x; a
qk
, b, c, d|q
)
, k ∈ {1, 2, . . . , n − 1}, are
quasi-orthogonal of order k on (−1, 1). We use the equation [27, equation (17a)]
p̃n
(
x;
a
q
, b, c, d|q
)
= p̃n(x; a, b, c, d|q)
− aq (qn − 1) (cdqn − q) (bdqn − q) (bcqn − q)
2 (abcdq2n − q3) (abcdq2n − q2)
p̃n−1(x; a, b, c, d|q). (3.2)
18 D.D. Tcheutia, A.S. Jooste and W. Koepf
Theorem 3.1. Let a, b, c, d be real, or they occur in complex conjugate pairs if complex,
and max(|a|, |b|, |c|, |d|) < 1, and let w(x; a, b, c, d|q) be as defined in (3.1). For a such that
q < |a| < 1, the sequence of Askey–Wilson polynomials
{
p̃n
(
x; a
qk
, b, c, d|q
)}∞
n=0
is quasi-ortho-
gonal of order k < n with respect to the weight w(x; a, b, c, d|q) on the interval (−1, 1) and the
polynomials have at least (n− k) real, distinct zeros in (−1, 1).
Proof. Suppose q < |a| < 1. From Lemma 1.1 and (3.2), it follows that p̃n
(
x; aq , b, c, d|q
)
is quasi-orthogonal of order one on (−1, 1). By iteration, we can express p̃n
(
x; a
qk
, b, c, d|q
)
as
a linear combination of p̃n(x; a, b, c, d|q), p̃n−1(x; a, b, c, d|q), . . . , p̃n−k(x; a, b, c, d|q) and the result
follows from Lemma 1.1. The location of the (n − k) real, distinct zeros of p̃n
(
x; a
qk
, b, c, d|q
)
,
k ∈ {1, 2, . . . , n− 1}, follows from Lemma 1.2. �
Theorem 3.2. Let a, b, c, d be real, or they occur in complex conjugate pairs if complex.
Suppose |a| = max(|a|, |b|, |c|, |d|) < 1, q < |a| < 1 and let xn,i, i ∈ {1, 2, . . . , n}, denote the
zeros of p̃n(x; a, b, c, d|q) and yn,i, i ∈ {1, 2, . . . , n}, the zeros of p̃n
(
x; aq , b, c, d|q
)
. Then
(i) if a > 0, −1 < xn,1 < yn,1 < xn−1,1 < xn,2 < yn,2 < · · · < xn−1,n−1 < xn,n < yn,n;
(ii) if a < 0, yn,1 < xn,1 < xn−1,1 < yn,2 < xn,2 < · · · < xn−1,n−1 < yn,n < xn,n < 1.
Proof. Suppose |a| = max(|a|, |b|, |c|, |d|) < 1. The coefficient of p̃n−1(x; a, b, c, d|q) in (3.2) is
an = −aq (qn − 1) (cdqn − q) (bdqn − q) (bcqn − q)
2 (abcdq2n − q3) (abcdq2n − q2)
.
(i) Consider the case a > 0 and fix a such that q < a < 1. Then an < 0 for the given parameter
values and the interlacing result, as well as the position of yn,1, follows from Lemma 1.4(i).
(ii) Now we consider the case a < 0 and fix a such that −1 < a < −q. Then an > 0 and the
interlacing result, as well as the position of yn,n, follows from Lemma 1.4(ii). �
3.2 The q-Racah polynomials
The q-Racah polynomials
R̃n(µ(x);α, β, γ, δ|q) =
(αq, βδq, γq; q)n
(αβqn+1; q)n
4φ3
(
q−n, αβqn+1, q−x, γδqx+1
αq, βδq, γq
∣∣∣∣∣ q; q
)
,
with µ(x) = q−x + γδqx+1, are orthogonal for n ∈ {0, 1, . . . , N}, with respect to the discrete
weight function
w(x) =
(αq, βδq, γq, γδq; q)x(1− γδq2x+1)(
q, γδqα , γqβ , δq; q
)
x
(αβq)x(1− γδq)
for αq = q−N or βδq = q−N or γq = q−N , where N is a nonnegative integer. Shifting the
parameter γ or δ will change µ(x) and we will only consider shifts of α and β. From [27,
equations (18a) and (18b)] we obtain
R̃n
(
µ(x);
α
q
, β, γ, δ|q
)
= R̃n(µ(x);α, β, γ, δ|q)
− αq(1− qn)(1− βqn)(1− γqn)(1− βδqn)
(1− αβq2n)(q − αβq2n)
R̃n−1(µ(x);α, β, γ, δ|q); (3.3a)
R̃n
(
µ(x);α,
β
q
, γ, δ|q
)
= R̃n(µ(x);α, β, γ, δ|q)
+
βq(1− qn)(1− αqn)(1− γqn)(αqn − δ)
(1− αβq2n)(q − αβq2n)
R̃n−1(µ(x);α, β, γ, δ|q). (3.3b)
Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials 19
Theorem 3.3. Let k ∈ {1, 2, . . . , n− 1}. The sequence of q-Racah polynomials
(i)
{
R̃n
(
µ(x); α
qk
, β, γ, δ|q
)}N
n=0
, with α = q−N−1, is quasi-orthogonal of order k with respect
to the weight w(x) on (µ(0), µ(N)) and the polynomials have at least (n− k) real, distinct
zeros on (µ(0), µ(N));
(ii)
{
R̃n
(
µ(x);α, βq , γ, δ|q
)}N
n=0
, with β = q−N−1
δ , is quasi-orthogonal of order k with respect to
the weight w(x) on (µ(0), µ(N)) and the polynomials have at least (n − k) real, distinct
zeros on (µ(0), µ(N)).
Proof. (i) Let α = q−N−1. From Lemma 1.1 and (3.3a), it follows that R̃n
(
µ(x); αq , β, γ, δ|q
)
is
quasi-orthogonal of order one on (µ(0), µ(N)). By iteration, we can express R̃n
(
µ(x); α
qk
, β, γ, δ|q
)
as a linear combination of R̃n(µ(x);α, β, γ, δ|q), R̃n−1(µ(x);α, β, γ, δ|q), . . . , R̃n−k(µ(x);α, β,
γ, δ|q) and the result follows from Lemma 1.1. Furthermore, from Lemma 1.2 we know that at
least (n− k) real, distinct zeros of R̃n(µ(x); α
qk
, β, γ, δ), k ∈ {1, 2, . . . , n− 1}, lie in (µ(0), µ(N)).
(ii) Let β = q−N−1
δ . The result follow in the same way from (3.3b) and Lemmas 1.1
and 1.2. �
For values of n larger than N
2 + 1, we obtain the following interlacing results.
Theorem 3.4. Consider n ≤ N and let xn,i, i ∈ {1, 2, . . . , n}, denote the zeros of R̃n(µ(x);α, β,
γ, δ|q), yn,i, i ∈ {1, 2, . . . , n}, the zeros of R̃n
(
µ(x); αq , β, γ, δ|q
)
and zn,i, i ∈ {1, 2, . . . , n}, the
zeros of R̃n
(
µ(x);α, βq , γ, δ|q
)
. Then, for n > N
2 + 1,
(i) if α = q−N−1 and βq < 1, γq < 1, βδq < 1,
µ(0) < xn,1 < yn,1 < xn−1,1 < xn,2 < yn,2 < · · · < xn−1,n−1 < xn,n < yn,n;
(ii) if β = q−N−1
δ and αq < 1, γq < 1, α
δ q < 1, we have
µ(0) < xn,1 < zn,1 < xn−1,1 < xn,2 < zn,2 < · · · < xn−1,n−1 < xn,n < zn,n.
Proof. Under the above hypotheses, the coefficients of R̃n−1(µ(x);α, β, γ, δ|q) in (3.3a) and
(3.3b) are negative and the interlacing results follow from Lemma 1.4(i). �
4 Quasi-orthogonality of polynomials
on a linear and quadratic lattice
In this section we consider the quasi-orthogonality of the monic Wilson and Racah polynomials,
that are defined on a quadratic lattice. The quasi-orthogonality of the dual Hahn and continuous
dual Hahn polynomials, that also fall in this category, was discussed in [15]. We also prove the
quasi-orthogonality of the monic continuous Hahn polynomials that are defined on a linear
lattice.
4.1 The Wilson polynomials
The Wilson polynomials
W̃n
(
x2; a, b, c, d
)
=
(−1)n(a+ b, a+ c, a+ d)n
(n+ a+ b+ c+ d− 1)n
20 D.D. Tcheutia, A.S. Jooste and W. Koepf
× 4F3
(
−n, n+ a+ b+ c+ d− 1, a+ ix, a− ix
a+ b, a+ c, a+ d
∣∣∣∣∣ 1
)
,
are orthogonal on (0,∞) with respect to
w(x; a, b, c, d) =
∣∣∣∣Γ(a+ ix)Γ(b+ ix)Γ(c+ ix)Γ(d+ ix)
Γ(2ix)
∣∣∣∣2 , (4.1)
for Re(a, b, c, d) > 0 and non-real parameters occur in conjugate pairs. Furthermore, as in the
case of the Askey–Wilson polynomials, the weight function is clearly independent of the order
in which the parameter a, b, c and d occur. We note that the polynomial Wn
(
x2; a, b, c, d
)
has n
zeros in (0,∞), namely (xn,1)2, (xn,2)2, . . . , (xn,n)2. Let W̃n
(
x2
)
= W̃n
(
x2; a, b, c, d
)
.
Proposition 4.1.
W̃n
(
x2; a− 1, b, c, d
)
= W̃n
(
x2
)
+
n(c+ d+ n− 1)(b+ d+ n− 1)(b+ c+ n− 1)
(2n+ a+ b+ c+ d− 2)(2n+ a+ b+ c+ d− 3)
W̃n−1
(
x2
)
; (4.2a)
W̃n
(
x2; a, b− 1, c, d
)
= W̃n
(
x2
)
+
n(c+ d+ n− 1)(a+ d+ n− 1)(a+ c+ n− 1)
(2n+ a+ b+ c+ d− 2)(2n+ a+ b+ c+ d− 3)
W̃n−1
(
x2
)
; (4.2b)
W̃n
(
x2; a, b, c− 1, d
)
= W̃n
(
x2
)
+
n(a+ d+ n− 1)(b+ d+ n− 1)(a+ b+ n− 1)
(2n+ a+ b+ c+ d− 2)(2n+ a+ b+ c+ d− 3)
W̃n−1
(
x2
)
; (4.2c)
W̃n
(
x2; a, b, c, d− 1
)
= W̃n
(
x2
)
+
n(a+ c+ n− 1)(b+ c+ n− 1)(a+ b+ n− 1)
(2n+ a+ b+ c+ d− 2)(2n+ a+ b+ c+ d− 3)
W̃n−1
(
x2
)
. (4.2d)
Theorem 4.2. Let a, b, c and d be such that Re(a, b, c, d) > 0. Consider k1, k2, k3, k4 ∈ {0, 1, . . . ,
n−1}, such that k1+k2+k3+k4 ≤ n−1. The sequence
{
W̃n
(
x2; a−k1, b−k2, c−k3, d−k4
)}∞
n=0
,
with 0 < Re(a) < 1 (if k1 6= 0), 0 < Re(b) < 1 (if k2 6= 0), 0 < Re(c) < 1 (if k3 6= 0) and
0 < Re(d) < 1 (if k4 6= 0), is quasi-orthogonal of order k1 + k2 + k3 + k4 ≤ n− 1 with respect to
the weight w(x) on (0,∞) and the polynomials have at least n− (k1 + k2 + k3 + k4) real, distinct
zeros in (0,∞).
Proof. Fix a such that 0 < Re(a) < 1. From Lemma 1.1 and (4.2a), it follows that W̃n(x2; a−1,
b, c, d) is quasi-orthogonal of order one on (0,∞). By iteration, we can express W̃n
(
x2; a − k,
b, c, d
)
as a linear combination of W̃n
(
x2; a, b, c, d
)
, W̃n−1
(
x2; a, b, c, d
)
, . . . , W̃n−k
(
x2; a, b, c, d
)
and from Lemma 1.1 it follows that W̃n
(
x2; a− k, b, c, d
)
, 0 < Re(a) < 1, is quasi-orthogonal of
order k ≤ n − 1 on (0,∞). Furthermore, from Lemma 1.2 we know that at least (n − k) real,
distinct zeros of W̃n
(
x2; a− k, b, c, d
)
, k ∈ {1, 2, . . . , n− 1}, lie in (0,∞), i.e., at least (n− k) of
the zeros (xn,1)2, (xn,2)2, . . . , (xn,n)2, lie in (0,∞).
When we fix the parameter b, (or c, d) such that 0 < Re(b) < 1 (or 0 < Re(c) < 1,
0 < Re(d) < 1), we can prove in the same way, using (4.2b) (or (4.2c), (4.2d)), that the
polynomial W̃n
(
x2; a, b− k, c, d
)
(alternatively W̃n
(
x2; a, b, c− k, d
)
, or W̃n
(
x2; a, b, c, d− k
)
) is
quasi-orthogonal of order k on (0,∞). Using an iteration process, we can write W̃n
(
x2; a− k1,
b− k2, c− k3, d− k4
)
with 0 < Re(a) < 1 (if k1 6= 0), 0 < Re(b) < 1 (if k2 6= 0), 0 < Re(c) < 1
(if k3 6= 0) and 0 < Re(d) < 1 (if k4 6= 0), in the form of (1.2) and the results follow from
Lemmas 1.1 and 1.2. �
Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials 21
Theorem 4.3. Consider a, b, c, d, such that Re(b, c, d) > 0, 0 < Re(a) < 1 and non-real param-
eters occur in conjugate pairs. Let x2
n,i, i ∈ {1, 2, . . . , n}, denote the zeros of W̃n
(
x2; a, b, c, d
)
and y2
n,i, i ∈ {1, 2, . . . , n}, the zeros of W̃n
(
x2; a− 1, b, c, d
)
. Then
y2
n,1 < x2
n,1 < x2
n−1,1 < y2
n,2 < x2
n,2 < · · · < x2
n−1,n−1 < y2
n,n < x2
n,n.
Proof. From (4.2a), we obtain
an =
n(c+ d+ n− 1)(b+ d+ n− 1)(b+ c+ n− 1)
(2n+ a+ b+ c+ d− 2)(2n+ a+ b+ c+ d− 3)
,
which is positive and the interlacing result, as well as the position of y2
n,n, follows from Lem-
ma 1.4(ii). �
4.2 The Racah polynomials
The Racah polynomials
R̃n(λ(x);α, β, γ, δ) =
(α+ 1, β + δ + 1, γ + 1)n
(n+ α+ β + 1)n
× 4F3
(
−n, n+ α+ β + 1,−x, x+ γ + δ + 1
α+ 1, β + δ + 1, γ + 1
∣∣∣∣∣ 1
)
,
n ∈ {0, 1, 2, . . . , N}, with λ(x) = x(x+ γ + δ + 1), are orthogonal on (0, N) with respect to the
weight function
w(x) =
(α+ 1)x(β + δ + 1)x(γ + 1)x(γ + δ + 1)x((γ + δ + 3)/2)x
(−α+ γ + δ + 1)x(−b+ γ + 1)x(δ + 1)x((γ + δ + 1)/2)x
if α+ 1 = −N or β + δ + 1 = −N or γ + 1 = −N with N a nonnegative integer. Since shifting
γ or δ will change λ(x), we will only consider shifts in α and β.
Proposition 4.4.
R̃n(λ(x);α− 1, β, γ, δ) = R̃n(λ(x);α, β, γ, δ)
− (β + n) (β + δ + n) (γ + n)n
(2n+ α+ β) (2n+ α+ β − 1)
R̃n−1(λ(x);α, β, γ, δ); (4.3a)
R̃n(λ(x);α, β − 1, γ, δ) = R̃n(λ(x);α, β, γ, δ)
− (α+ n) (α− δ + n) (γ + n)n
(2n+ α+ β) (2n+ α+ β − 1)
R̃n−1(λ(x);α, β, γ, δ). (4.3b)
Theorem 4.5. Let k ∈ {1, 2, . . . , n− 1}. The sequence of Racah polynomials
(i)
{
R̃n(λ(x);α−k, β, γ, δ)
}N
n=0
, with α = −N −1, is quasi-orthogonal of order k with respect
to the weight w(x) on (0, λ(N)) and the polynomials have at least (n − k) real, distinct
zeros in (0, λ(N));
(ii)
{
R̃n(λ(x);α, β − k, γ, δ)
}N
n=0
, with β = −N − δ − 1, is quasi-orthogonal of order k with
respect to the weight w(x) on (0, λ(N)) and the polynomials have at least (n − k) real,
distinct zeros in (0, λ(N)).
22 D.D. Tcheutia, A.S. Jooste and W. Koepf
Proof. (i) Let α = −N−1. From Lemma 1.1 and (4.3a), it follows that R̃n(λ(x);α−1, β, γ, δ) is
quasi-orthogonal of order one on (0, λ(N)). By iteration, we can express R̃n(λ(x);α− k, β, γ, δ)
as a linear combination of R̃n(λ(x);α, β, γ, δ), R̃n−1(λ(x);α, β, γ, δ), . . . , R̃n−k(λ(x);α, β, γ, δ)
and the result follows from Lemma 1.1. Furthermore, from Lemma 1.2 we know that at least
(n− k) real, distinct zeros of R̃n(λ(x);α− k, β, γ, δ), k ∈ {1, 2, . . . , n− 1}, lie in (0, λ(N)).
(ii) Let β = −N − δ − 1. The result follow in the same way from (4.3b) and Lemmas 1.1
and 1.2. �
As in the case of the q-Racah polynomials, we obtain different interlacing results for values
of n larger than N
2 + 1, that we show in the next theorem, than for n < N
2 + 1.
Theorem 4.6. Consider n ≤ N and let xn,i, i ∈ {1, 2, . . . , n} denote the zeros of R̃n(λ(x);α, β,
γ, δ), yn,i, i ∈ {1, 2, . . . , n}, the zeros of R̃n(λ(x);α − 1, β, γ, δ) and zn,i, i ∈ {1, 2, . . . , n}, the
zeros of R̃n(λ(x);α, β − 1, γ, δ). Then, for n > N
2 + 1,
(i) if α = −N − 1 and β > 0, δ > 0, γ > 0, we have
0 < xn,1 < yn,1 < xn−1,1 < xn,2 < yn,2 < · · · < xn−1,n−1 < xn,n < yn,n;
(ii) if β = −N − δ − 1 and α > 0, γ > 0, α− δ > 0, we have
0 < xn,1 < zn,1 < xn−1,1 < xn,2 < zn,2 < · · · < xn−1,n−1 < xn,n < zn,n.
Proof. Under the above hypotheses, the coefficients of R̃n(λ(x);α, β, γ, δ) in (4.3a) and (4.3b)
are negative and the interlacing results follow from Lemma 1.4(i). �
4.3 The continuous Hahn polynomials
The continuous Hahn polynomials
P̃n(x; a, b, c, d) =
in(a+ c, a+ d)n
(n+ a+ b+ c+ d− 1)n
3F2
(
−n, n+ a+ b+ c+ d− 1, a+ ix
a+ c, a+ d
∣∣∣∣∣ 1
)
are orthogonal on R with respect to
w(x) = Γ(a+ ix)Γ(b+ ix)Γ(c− ix)Γ(d− ix)
for Re(a, b, c, d) > 0, c = ā and d = b̄.
Proposition 4.7.
P̃n(x; a− 1, b, c, d) = P̃n(x; a, b, c, d)
+
i (b+ c+ n− 1) (b+ d+ n− 1)n
(2n+ a+ b+ c+ d− 3) (2n+ a+ b+ c+ d− 2)
P̃n−1(x; a, b, c, d); (4.4a)
P̃n(x; a, b− 1, c, d) = P̃n(x; a, b, c, d)
+
i (a− 1 + d+ n) (a− 1 + c+ n)n
(2n+ a+ b+ c+ d− 3) (2n+ a+ b+ c+ d− 2)
P̃n−1(x; a, b, c, d); (4.4b)
P̃n(x; a, b, c− 1, d) = P̃n(x; a, b, c, d)
− i (b+ d+ n− 1) (a− 1 + d+ n)n
(2n+ a− 3 + b+ c+ d) (2n+ a− 2 + b+ c+ d)
P̃n−1(x; a, b, c, d); (4.4c)
P̃n(x; a, b, c, d− 1) = P̃n(x; a, b, c, d)
− i (b+ c+ n− 1) (a− 1 + c+ n)n
(2n+ a− 3 + b+ c+ d) (2n+ a− 2 + b+ c+ d)
P̃n−1(x; a, b, c, d). (4.4d)
Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials 23
Corollary 4.8.
P̃n(x; a− 1, b, c− 1, d)
= P̃n(x)− i (a+ d− b− c) (b+ d+ n− 1)n
(2n+ a− 4 + b+ c+ d) (2n+ a− 2 + b+ c+ d)
P̃n−1(x) (4.5a)
+
(b+ d+ n− 2) (a− 2 + d+ n) (n− 1) (b+ c− 2 + n) (b+ d+ n− 1)n
(2n− 5 + a+ b+ c+ d) (2n+ a− 4 + b+ c+ d)2 (2n+ a− 3 + b+ c+ d)
P̃n−2(x);
P̃n(x; a, b− 1, c, d− 1)
= P̃n(x) +
i (a+ d− b− c) (a− 1 + c+ n)n
(2n− 4 + a+ b+ c+ d) (2n+ a− 2 + b+ c+ d)
P̃n−1(x) (4.5b)
+
n (a+ d+ n− 2) (a− 1 + c+ n) (b+ c+ n− 2) (a+ c+ n− 2) (n− 1)
(2n− 5 + a+ b+ c+ d) (2n− 4 + a+ b+ c+ d)2 (2n+ a− 3 + b+ c+ d)
P̃n−2(x).
Theorem 4.9. Consider a, b, c, d such that Re(a, b, c, d) > 0, c = ā and d = b̄. Consider
k1, k2, k3, k4 ∈ {0, 1, . . . , n− 1}, such that k1 + k2 + k3 + k4 ≤ n− 1. The sequence of continuous
Hahn polynomials {P̃n(x; a − k1, b − k2, c − k3, d − k4)}∞n=0, with 0 < Re(a) = Re(c) < 1 (if
k1 6= 0), 0 < Re(b) = Re(d) < 1 (if k2 6= 0), 0 < Re(a) = Re(c) < 1 (if k3 6= 0) and
0 < Re(b) = Re(d) < 1 (if k4 6= 0), is quasi-orthogonal of order k1 + k2 + k3 + k4 ≤ n− 1 with
respect to the weight w(x) on R and the polynomials have at least n− (k1 + k2 + k3 + k4) real,
distinct zeros.
Proof. Fix a and c such that 0 < Re(a) = Re(c) < 1. From Lemma 1.1 and (4.4a), it follows
that P̃n(x; a − 1, b, c, d) is quasi-orthogonal of order one on R. By iteration, we can express
P̃n(x; a − k, b, c, d) as a linear combination of P̃n(x; a, b, c, d), P̃n−1(x; a, b, c, d), . . . , P̃n−k(x; a, b,
c, d) and it follows from Lemma 1.1 that P̃n(x; a−1, b, c, d) is quasi-orthogonal of order one on R.
By using an iteration process, we can write P̃n(x; a− k, b, c, d) as a linear combination of ortho-
gonal continuous Hahn polynomials and it is quasi-orthogonal of order k ≤ n− 1. Furthermore,
from Lemma 1.2 we know that at least (n− k) zeros of P̃n(x; a− k, b, c, d), k ∈ {1, 2, . . . , n− 1},
are real and distinct. In the same way, using (4.4c), we can prove that P̃n(x; a, b, c − k, d),
0 < Re(a) = Re(c) < 1 is quasi-orthogonal of k ≤ n − 1 on R. By fixing b and d such
that 0 < Re(b) = Re(d) < 1, we can prove the quasi-orthogonality of P̃n(x; a, b − k, c, d) and
P̃n(x; a, b, c, d− k), using (4.4b), (4.4d) and Lemma 1.1.
Using an iteration process, we can write P̃n(x; a− k1, b− k2, c− k3, d− k4), with 0 < Re(a) =
Re(c) < 1 (if k1 6= 0), 0 < Re(b) = Re(d) < 1 (if k2 6= 0), 0 < Re(a) = Re(c) < 1 (if k3 6= 0)
and 0 < Re(b) = Re(d) < 1 (if k4 6= 0), as a linear combination of orthogonal continuos Hahn
polynomials and the results follow from Lemmas 1.1 and 1.2. �
Theorem 4.10. Consider a, b, c, d such that Re(a, b, c, d) > 0, c = ā and d = b̄.
(i) Let 0 < Re(a) = Re(c) < 1. Then (n− 2) zeros of P̃n(x; a− 1, b, c− 1, d) interlace with the
zeros of P̃n−1(x; a, b, c, d);
(ii) Let 0 < Re(b) = Re(d) < 1. Then (n− 2) zeros of P̃n(x; a, b− 1, c, d− 1) interlace with the
zeros of P̃n−1(x; a, b, c, d).
Proof. In this proof −Cn refers to the coefficient of P̃n−2(x; a, b, c, d) in the three-term recur-
rence equation of the continuous Hahn polynomials (cf. [18, equation (9.4.3)]), involving the
polynomials P̃n(x; a, b, c, d), P̃n−1(x; a, b, c, d) and P̃n−2(x; a, b, c, d).
(i) Let 0 < Re(a),Re(c) < 1. We consider the coefficient bn of P̃n−2(x; a, b, c, d) in (4.5a).
Then
Cn − bn =
(a+ c− 2) (n− 2 + b+ d) (a+ d+ n− 2) (n− 1) (b+ c+ n− 2)
(2n− 5 + a+ b+ c+ d) (2n− 4 + a+ b+ c+ d)2 ,
24 D.D. Tcheutia, A.S. Jooste and W. Koepf
and when we take into consideration the specific restrictions on the parameters, we observe that
Cn < bn and the result follows from [17, Theorem 15(ii)].
(ii) Let 0 < Re(b),Re(d) < 1 and let bn be the coefficient of P̃n−2(x; a, b, c, d) in (4.5b). Then
Cn − bn =
(b+ d− 2) (n− 1) (b+ c+ n− 2) (a+ d+ n− 2) (a+ c+ n− 2)
(2n− 5 + a+ b+ c+ d) (2n− 4 + a+ b+ c+ d)2 < 0,
when we take into consideration the specific restrictions on the parameters and the result follows
from [17, Theorem 15(ii)]. �
5 Concluding remarks
The q-Meixner polynomials, defined by
M̃n(x̄;β, γ; q) = (−1)nq−n
2
γn(βq; q)n 2φ1
(
q−n, x̄
βq
∣∣∣∣∣ q;−qn+1
γ
)
,
with x̄ = q−x, are orthogonal with respect to the discrete weight (βq;q)xγxq
(x2)
(q,−βγq;q)x , when 0 ≤ βq < 1,
γ > 0, x̄ ∈ (1,∞), and satisfy
M̃n
(
x̄;
β
q
, γ; q
)
=
(qnx+ βγ)
(βγ + x)qn
M̃n(x̄;β, γ; q)− βγ (qn + γ) (qn − 1) q
(βγ + x)q3n
M̃n(x̄;β, γ; q).
The polynomial M̃n
(
x̄; β
qk
, γ; q
)
, k < n, is not quasi-orthogonal with respect to (βq;q)xγxq
(x2)
(q,−βγq;q)x ,
on (1,∞), since it cannot be written as a linear combination of the polynomials M̃n(x̄;β, γ; q),
M̃n−1(x̄;β, γ; q), . . . , M̃n−k(x̄;β, γ; q). Since γ > 0, we also have γ
q > 0 or γq > 0 and the
sequences M̃n
(
x̄;β, γq ; q
)
or M̃n(x̄;β, γq; q), are orthogonal on (1,∞) for 0 ≤ βq < 1. We
therefore do not consider q-shifts of γ.
The same is true for the Al-Salam–Carlitz II polynomials [18, Section 14.25], that satisfy
Ṽ
(α
q
)
n (x; q) = −(qnx− α)
(α− x)qn
Ṽ (α)
n (x; q)− αq (qn − 1)
(α− x)q2n
Ṽ
(α)
n−1(x; q)
and the Bessel polynomials [18, Section 9.13], that satisfy the equation
ỹn(x;α+ 1) =
α2x+ 4αnx+ 4n2x+ αx+ 2nx− 2n
x(α+ 1 + 2n)(α+ 2n)
ỹn(x;α)
+
4n(α+ n)
x(α+ 1 + 2)(α+ 2n− 1)(α+ 2n)2
ỹn−1(x;α).
The q-Krawtchouk polynomials
K̃n(x̄; p,N ; q) =
(q−N ; q)n
(−pqn; q)n
3φ2
(
q−n, x̄,−pqn
q−N , 0
∣∣∣∣∣ q; q
)
,
with x̄ = q−x and n ∈ {0, 1, . . . , N}, are orthogonal for p > 0 with respect to the discrete weight
w(x) = (q−N ;q)x(−p)x
(q;q)x
on
(
1, q−N
)
. The polynomials K̃n
(
x̄; p
qk
, N ; q
)
are orthogonal for p > 0
with respect to
(q−N ;q)x(−p
qk
)x
(q;q)x
on
(
1, q−N
)
. By iterating the equation
Kn
(
x̄;
p
q
,N ; q
)
= Kn(x̄; p,N ; q)−
p (qn − 1)
(
qn − qN+1
)
qn+1
(q2np+ q) (q2np+ q2) qN
Kn−1(x̄; p,N ; q),
Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials 25
we can write K̃n(x̄; p
qk
, N ; q) as a linear combination of the polynomials K̃n−j(x̄; p,N ; q), j ∈
{0, 1, . . . , k}, and the polynomials K̃n
(
x̄; p
qk
, N ; q
)
are also quasi-orthogonal for p > 0 on
(
1, q−N
)
with respect to w(x).
The same is true for the q-Charlier [18, Section 14.23] and alternative q-Charlier (or q-Bessel)
polynomials [18, Section 14.22].
Acknowledgments
The authors thank the referees for the valuable comments and suggestions which considerably
improved the manuscript. This work has been supported by the Institute of Mathematics of the
University of Kassel (Germany) for D.D. Tcheutia.
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1 Introduction
2 Classical orthogonal polynomials on a q-linear lattice
2.1 The big q-Jacobi polynomials
2.2 The q-Hahn polynomials
2.3 The little q-Jacobi polynomials
2.4 The q-Laguerre polynomials
2.5 The Al-Salam–Carlitz I polynomials
3 Classical orthogonal polynomials on a q-quadratic lattice
3.1 The Askey–Wilson polynomials
3.2 The q-Racah polynomials
4 Quasi-orthogonality of polynomials on a linear and quadratic lattice
4.1 The Wilson polynomials
4.2 The Racah polynomials
4.3 The continuous Hahn polynomials
5 Concluding remarks
References
|
| id | nasplib_isofts_kiev_ua-123456789-209521 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-03T04:30:58Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Tcheutia, D.D. Jooste, A.S. Koepf, W. 2025-11-24T10:31:20Z 2018 Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials / D.D. Tcheutia, A.S. Jooste, W. Koepf // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C05; 33C45; 33F10; 33D15; 12D10 arXiv: 1805.08954 https://nasplib.isofts.kiev.ua/handle/123456789/209521 https://doi.org/10.3842/SIGMA.2018.051 We show how to obtain linear combinations of polynomials in an orthogonal sequence {Pn}n≥0, that characterize quasi-orthogonal polynomials of order k ≤ n-1. The polynomials in the sequence {Qn,k}n≥0 are obtained from Pn, by making use of parameter shifts. We use an algorithmic approach to find these linear combinations for each family applicable, and these equations are used to prove the quasi-orthogonality of order k. We also determine the location of the extreme zeros of the quasi-orthogonal polynomials with respect to the endpoints of the interval of orthogonality of the sequence {Pn}n≥0, where possible. The authors thank the referees for the valuable comments and suggestions, which considerably improved the manuscript. This work has been supported by the Institute of Mathematics of the University of Kassel (Germany) for D.D. Tcheutia. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials Article published earlier |
| spellingShingle | Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials Tcheutia, D.D. Jooste, A.S. Koepf, W. |
| title | Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials |
| title_full | Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials |
| title_fullStr | Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials |
| title_full_unstemmed | Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials |
| title_short | Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials |
| title_sort | quasi-orthogonality of some hypergeometric and q-hypergeometric polynomials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209521 |
| work_keys_str_mv | AT tcheutiadd quasiorthogonalityofsomehypergeometricandqhypergeometricpolynomials AT joosteas quasiorthogonalityofsomehypergeometricandqhypergeometricpolynomials AT koepfw quasiorthogonalityofsomehypergeometricandqhypergeometricpolynomials |