Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials

The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronski...

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Datum:	2017
Hauptverfasser:	Odake, S., Sasaki, R.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2017
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1485802019-02-19T01:29:12Z Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials Odake, S. Sasaki, R. The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two different forms of simplified expressions of the multi-indexed Laguerre and Jacobi polynomials are derived based on various identities. The parity transformation property of the multi-indexed Jacobi polynomials is derived based on that of the Jacobi polynomial. 2017 Article Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials / S. Odake, R. Sasaki // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 42C05; 33C45; 34A05 DOI:10.3842/SIGMA.2017.020 http://dspace.nbuv.gov.ua/handle/123456789/148580 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two different forms of simplified expressions of the multi-indexed Laguerre and Jacobi polynomials are derived based on various identities. The parity transformation property of the multi-indexed Jacobi polynomials is derived based on that of the Jacobi polynomial.
format	Article
author	Odake, S. Sasaki, R.
spellingShingle	Odake, S. Sasaki, R. Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Odake, S. Sasaki, R.
author_sort	Odake, S.
title	Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials
title_short	Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials
title_full	Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials
title_fullStr	Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials
title_full_unstemmed	Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials
title_sort	simplified expressions of the multi-indexed laguerre and jacobi polynomials
publisher	Інститут математики НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/148580
citation_txt	Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials / S. Odake, R. Sasaki // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 24 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT odakes simplifiedexpressionsofthemultiindexedlaguerreandjacobipolynomials AT sasakir simplifiedexpressionsofthemultiindexedlaguerreandjacobipolynomials
first_indexed	2025-07-12T19:42:17Z
last_indexed	2025-07-12T19:42:17Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 020, 10 pages Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials Satoru ODAKE and Ryu SASAKI Faculty of Science, Shinshu University, Matsumoto 390-8621, Japan E-mail: odake@azusa.shinshu-u.ac.jp, ryu@yukawa.kyoto-u.ac.jp Received December 30, 2016, in final form March 23, 2017; Published online March 29, 2017 https://doi.org/10.3842/SIGMA.2017.020 Abstract. The multi-indexed Laguerre and Jacobi polynomials form a complete set of or- thogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the ‘holes’ in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux trans- formations. For the ease of applications, two different forms of simplified expressions of the multi-indexed Laguerre and Jacobi polynomials are derived based on various identities. The parity transformation property of the multi-indexed Jacobi polynomials is derived based on that of the Jacobi polynomial. Key words: multi-indexed orthogonal polynomials; Laguerre and Jacobi polynomials; Wron- skian formula; determinant formula 2010 Mathematics Subject Classification: 42C05; 33C45; 34A05 1 Introduction The exceptional and multi-indexed orthogonal polynomials [2, 9, 10, 11, 15, 17, 18, 19, 21, 22] seem to be a focal point of recent research on exactly solvable quantum mechanics. They belong to a new type of orthogonal polynomials which satisfy second-order differential (difference) equations and form complete orthogonal basis in an appropriate Hilbert space. One of their characteristic features is that they do not satisfy three term recurrence relations because of the ‘holes’ in the degrees. This is how they avoid the constraints due to Bochner [3, 23]. They are constructed from the original quantum mechanical systems, the radial oscillator potential and the Pöschl–Teller potential, by multiple application of Darboux transformations [4, 5] in terms of seed solutions called virtual state wavefunctions which are generated by two types of discrete symmetry transformations [17, 18, 19, 21]. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions [17] origi- nating from multiple Darboux transformations [4]. In this note we present two different forms of equivalent determinant expressions without higher derivatives of the Wronskians by using various identities of the Laguerre and Jacobi polynomials [24]. These simplified expressions show explicitly the constituents of the multi-indexed orthogonal polynomials and they are help- ful for deeper understanding. In [6] and [7] Durán employed similar simplified expressions as the starting point of his exposition of the exceptional Laguerre and Jacobi polynomials. See also [8] by Durán and Pérez. In their expressions, the matrix elements of the determinants are polynomials. In the original expressions in [17], the matrix elements of the determinants contain the non-polynomial factors, see (10)–(13). In the simplified expressions presented in this paper in Sections 2.4 and 2.5, they are also all polynomials. This short note is organised as follows. In Section 2.1 the quantum mechanical settings for the original Laguerre and Jacobi polynomials are recapitulated. That is, the Hamiltonians with the radial oscillator potential and the Pöschl–Teller potential are introduced and their eigenval- mailto:odake@azusa.shinshu-u.ac.jp mailto:ryu@yukawa.kyoto-u.ac.jp https://doi.org/10.3842/SIGMA.2017.020 2 S. Odake and R. Sasaki ues and eigenfunctions are presented. Type I and II discrete symmetry transformations for the Hamiltonians of the Laguerre and Jacobi polynomials are explained in Section 2.2. The seed solutions for Darboux transformations, to be called the virtual state wavefunctions of Type I and II, for the Laguerre and Jacobi, are listed explicitly. In Section 2.3 the Wronskian forms of the multi-indexed Laguerre and Jacobi polynomials derived in [17] and [14] are recapitulated as the starting point. Sections 2.4 and 2.5 are the main content of this note. Those who are familiar with the multi-indexed Laguerre and Jacobi polynomials can directly go to this part. The first simplified expressions of the multi-indexed Laguerre and Jacobi polynomials are derived in Sec- tion 2.4 by using various identities of the original Laguerre and Jacobi polynomials. The second simplified expressions, to be derived in Section 2.5, are the consequences of the multi-linearity of determinants and the form of the Schrödinger equation ψ′′(x) = (U(x)− E)ψ(x). Every even order derivative ψ(2m)(x) in the Wronskian can be replaced by (−E)mψ(x). In Section 2.6 the parity transformation formula of the multi-indexed Jacobi polynomials is presented. Section 3 is for a summary and comments. 2 Multi-indexed Laguerre and Jacobi polynomials The foundation of the theory of multi-indexed orthogonal polynomials is the exactly solvable one-dimensional quantum mechanical system H: Hφn(x) = Enφn(x), n = 0, 1, . . . , H = − d2 dx2 + U(x), (1) and its iso-spectrally deformed system HDφD n(x) = EnφD n(x), n = 0, 1, . . . , HD = − d2 dx2 + UD(x), (2) in terms of multiple application of Darboux transformations [4, 17]. In this note we discuss two systems which have the Laguerre and Jacobi polynomials as the main parts of the eigenfunctions. We follow the notation of [17] with slight modifications for simplification sake. 2.1 Original Laguerre and Jacobi polynomials 2.1.1 Radial oscillator potential The Hamiltonian with the radial oscillator potential U(x) def = x2 + g(g − 1) x2 − 2g − 1, 0 < x <∞, g > 1 2 , (3) has the Laguerre polynomials L (α) n (η) as the main part of the eigenfunctions: φn(x; g) def = φ0(x; g)L (g− 1 2 ) n ( η(x) ) , En def = 4n, η(x) def = x2, φ0(x; g) def = e− 1 2 x2xg, L(α) n (η) = 1 n! n∑ k=0 (−n)k k! (α+ k + 1)n−kη k. Here are some identities of the Laguerre polynomials [24] to be used in Sections 2.4 and 2.5, ∂ηL (α) n (η) = −L(α+1) n−1 (η), (4) L(α) n (η)− L(α−1) n (η) = L (α) n−1(η), (5) ηL (α+1) n−1 (η)− αL(α) n−1(η) = −nL(α−1) n (η). (6) Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials 3 As is clear from (3), the lower boundary, x = 0, is the regular singular point with the character- istic exponents g and 1 − g. The upper boundary point, x = ∞, is an irregular singular point. Here (a)n def = n∏ j=1 (a+ j − 1) is the shifted factorial or the so-called Pochhammer’s symbol. 2.1.2 Pöschl–Teller potential The Hamiltonian with the Pöschl–Teller potential U(x) def = g(g − 1) sin2 x + h(h− 1) cos2 x − (g + h)2, 0 < x < π 2 , g > 1 2 , h > 1 2 , has the Jacobi polynomials P (α,β) n (η) as the main part of the eigenfunctions: φn ( x; (g, h) ) def = φ0 ( x; (g, h) ) P (g− 1 2 ,h− 1 2 ) n ( η(x) ) , η(x) def = cos 2x, φ0 ( x; (g, h) ) def = (sinx)g(cosx)h, En(g, h) def = 4n(n+ g + h), P (α,β) n (η) = (α+ 1)n n! n∑ k=0 1 k! (−n)k(n+ α+ β + 1)k (α+ 1)k ( 1− η 2 )k . Here are some identities of the Jacobi polynomials [24] to be used in in Sections 2.4 and 2.5, ∂ηP (α,β) n (η) = 1 2(n+ α+ β + 1)P (α+1,β+1) n−1 (η), (7) (n+ β)P (α,β−1) n (η)− βP (α−1,β) n (η) = (n+ α+ β)1+η 2 P (α,β+1) n−1 (η), (8) (n+ α)P (α−1,β) n (η)− αP (α,β−1) n (η) = −(n+ α+ β)1−η 2 P (α+1,β) n−1 (η). (9) The two boundary points, x = 0, π2 are the regular singular points with the characteristic exponents g, 1− g and h, 1− h, respectively. 2.2 Discrete symmetry transformations and virtual state wavefunctions The seed solutions (virtual state wavefunctions) for Darboux transformations can be constructed by applying discrete symmetry transformations of the Hamiltonian to the eigenfunctions. They have negative energies and have no node and they are not square integrable, see [17] for more detail. 2.2.1 Laguerre (L) system Type I transformation. It is obvious that the transformation x→ ix is the symmetry of the radial oscillator system. The seed solutions are L1: Hφ̃I v(x) = Ẽ I vφ̃ I v(x), φ̃I v(x) def = φ̃I 0(x)ξI v ( η(x) ) , φ̃I 0(x) def = e 1 2 x2xg, ξI v(η) def = L (g− 1 2 ) v (−η), Ẽ I v def = −4(g + v + 1 2), v ∈ Z≥0. Hereafter we use v for the degree of the seed polynomial ξv in order to distinguish it with the degree n of eigenpolynomial Pn. Type II transformation. The exchange of the characteristic exponents of the regular singular point x = 0, g → 1 − g, is another discrete symmetry transformation. It generates the seed solutions, L2: Hφ̃II v (x) = Ẽ II v φ̃ II v (x), φ̃II v (x) def = φ̃II 0 (x)ξII v ( η(x) ) , φ̃II 0 (x) def = e− 1 2 x2x1−g, ξII v (η) def = L ( 1 2 −g) v (η), Ẽ II v def = −4 ( g − v− 1 2 ) , v = 0, 1, . . . , [ g − 1 2 ]′ , in which [a]′ denotes the greatest integer less than a. 4 S. Odake and R. Sasaki 2.2.2 Jacobi (J) system Type I transformation. The exchange of the characteristic exponents of the regular singular point x = π 2 , h→ 1− h, is a discrete symmetry transformation: J1: Hφ̃I v(x) = Ẽ I vφ̃ I v(x), φ̃I v(x) def = φ̃I 0(x)ξI v ( η(x) ) , φ̃I 0(x) def = (sinx)g(cosx)1−h, ξI v(η) def = P (g− 1 2 , 1 2 −h) v (η), Ẽ I v def = −4 ( g + v + 1 2 )( h− v− 1 2 ) , v = 0, 1, . . . , [ h− 1 2 ]′ . Type II transformation. Likewise the exchange of the characteristic exponents of the regular singular point x = 0, g → 1− g, is a discrete symmetry transformation: J2: Hφ̃II v (x) = Ẽ II v φ̃ II v (x), φ̃II v (x) def = φ̃II 0 (x)ξII v ( η(x) ) , φ̃II 0 (x) def = (sinx)1−g(cosx)h, ξII v (η) def = P ( 1 2 −g,h− 1 2 ) v (η), Ẽ II v def = −4 ( g − v− 1 2 )( h+ v + 1 2 ) , v = 0, 1, . . . , [ g − 1 2 ]′ . 2.3 Wronskian forms of the multi-indexed Laguerre and Jacobi polynomials We deform the original Hamiltonian system (1) by applying multiple Darboux transforma- tions [4, 17] in terms of MI Type I seed solutions and MII Type II seed solutions specified by the degree set D def = {d1, . . . , dM} (ordered set), M def = MI +MII, MI def = #{dj \| dj ∈ D, Type I}, MII def = #{dj \| dj ∈ D, Type II}. The multi-indexed orthogonal polynomials {PD,n(η)} are the main parts of the eigenfunctions {φD n(x)} of the deformed Hamiltonian system HD (2): UD(x) def = U(x)− 2∂2 x log ∣∣W[φ̃d1 , . . . , φ̃dM ](x) ∣∣, φD n(x) def = W [ φ̃d1 , . . . , φ̃dM , φn ] (x) W [ φ̃d1 , . . . , φ̃dM ] (x) def = cMF ψD(x)PD,n ( η(x) ) , n = 0, 1, . . . , ψD(x) def = φ̂0(x) ΞD ( η(x) ) , in which φ̂0(x) and cF are defined by φ̂0(x) def = { φ0(x; g +MI −MII), L, φ0 ( x; (g +MI −MII, h−MI +MII) ) , J, cF def = { 2, L, −4, J. The superscript I and II of the seed solutions are suppressed for simplicity of notation. The Wronskian of n-functions {f1, . . . , fn} is defined by formula W[f1, . . . , fn](x) def = det ( dj−1fk(x) dxj−1 ) 1≤j,k≤n . There are two different but equivalent Wronskian definitions of the denominator polyno- mial ΞD(η) and the multi-indexed orthogonal polynomial PD,n(η). The first is based on the Wronskians of the ‘polynomials’ [17]: ΞD(η) def = W[µd1 , . . . , µdM ](η)× { η(MI+g− 1 2 )MIIe−MIη, L,(1−η 2 )(MI+g− 1 2 )MII (1+η 2 )(MII+h− 1 2 )MI , J, (10) Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials 5 PD,n(η) def = W[µd1 , . . . , µdM , Pn](η)× { η(MI+g+ 1 2 )MIIe−MIη, L,(1−η 2 )(MI+g+ 1 2 )MII (1+η 2 )(MII+h+ 1 2 )MI , J, (11) µv(η) def =  eηL (g− 1 2 ) v (−η), L, v Type I, η 1 2 −gL ( 1 2 −g) v (η), L, v Type II,(1+η 2 ) 1 2 −h P (g− 1 2 , 1 2 −h) v (η), J, v Type I,(1−η 2 ) 1 2 −g P ( 1 2 −g,h− 1 2 ) v (η), J, v Type II, (12) Pn(η) def = L (g− 1 2 ) n (η), L, P (g− 1 2 ,h− 1 2 ) n (η), J. (13) The second is based on the Wronskians of the virtual state wavefunctions φ̃v(x) and the eigen- function φn(x) [14]: ΞD(η) = c − 1 2 M(M−1) F W [ φ̃d1 , . . . , φ̃dM ] (x) × { η−M ′(M ′+g− 1 2 )e−M ′η, L,(1−η 2 )−M ′(M ′+g− 1 2 )(1+η 2 )−M ′(M ′−h+ 1 2 ) , J, (14) PD,n(η) = c − 1 2 M(M+1) F W [ φ̃d1 , . . . , φ̃dM , φn ] (x) × { η−(M ′+ 1 2 )(M ′+g)e−(M ′− 1 2 )η, L,(1−η 2 )−(M ′+ 1 2 )(M ′+g)(1+η 2 )−(M ′− 1 2 )(M ′−h) , J, (15) in which M ′ def = 1 2(MI −MII) and η = η(x). 2.4 Simplified forms of the multi-indexed Laguerre and Jacobi polynomials, A In this and the subsequent subsections we will derive simplified expressions of the multi-indexed Laguerre and Jacobi polynomials PD,n(η) and the corresponding denominator polynomials ΞD(η) starting from the Wronskian expressions (10)–(15) in the previous subsection. In this subsection we simplify the Wronskians of the ‘polynomials’ (10)–(13). Let us first transform the higher derivatives of the ‘seed polynomials’ µv(η) in (12). For the L system, we obtain L1: ∂η ( eηL (g− 1 2 ) v (−η) ) = eηL (g+1− 1 2 ) v (−η), L2: ∂η ( η 1 2 −gL ( 1 2 −g) v (η) ) = ( v− g + 1 2 ) η 1 2 −(g+1)L ( 1 2 −(g+1)) v (η), by using (4)–(5) and (4)–(6), respectively. By repeating these we arrive at for j ∈ Z≥1, ∂j−1 η ( eηL (g− 1 2 ) v (−η) ) = eηL (g+j− 3 2 ) v (−η), ∂j−1 η ( η 1 2 −gL ( 1 2 −g) v (η) ) = (−1)j−1 ( g − 1 2 − v ) j−1 η 3 2 −g−jL ( 3 2 −g−j) v (η) = η 1 2 −g−K(−1)j−1 ( g − 1 2 − v ) j−1 ηK+1−jL ( 3 2 −g−j) v (η), in which K is a positive integer. For the J system, we obtain J1: ∂η ((1+η 2 ) 1 2 −h P (g− 1 2 , 1 2 −h) v (η) ) = 1 2 ( v− h+ 1 2 )(1+η 2 ) 1 2 −(h+1) P (g+1− 1 2 , 1 2 −(h+1)) v (η), J2: ∂η ((1−η 2 ) 1 2 −g P ( 1 2 −g,h− 1 2 ) v (η) ) = −1 2 ( v− g + 1 2 )(1−η 2 ) 1 2 −(g+1) P ( 1 2 −(g+1),h+1− 1 2 ) v (η), 6 S. Odake and R. Sasaki by using (7), (8) and (7), (9), respectively. Repeated applications of these formulas lead to ∂j−1 η ((1+η 2 ) 1 2 −h P (g− 1 2 , 1 2 −h) v (η) ) = (−1)j−1 2j−1 ( h− 1 2 − v ) j−1 (1+η 2 ) 3 2 −h−j P (g+j− 3 2 , 3 2 −h−j) v (η) = (1+η 2 ) 1 2 −h−K (−1)j−1 2j−1 ( h− 1 2 − v ) j−1 (1+η 2 )K+1−j P (g+j− 3 2 , 3 2 −h−j) v (η), ∂j−1 η ((1−η 2 ) 1 2 −g P ( 1 2 −g,h− 1 2 ) v (η) ) = 1 2j−1 ( g − 1 2 − v ) j−1 (1−η 2 ) 3 2 −g−j P ( 3 2 −g−j,h+j− 3 2 ) v (η) = (1−η 2 ) 1 2 −g−K 1 2j−1 ( g − 1 2 − v ) j−1 (1−η 2 )K+1−j P ( 3 2 −g−j,h+j− 3 2 ) v (η). The higher derivatives of the eigenpolynomials Pn(η) in (13) are replaced simply through (4) and (7), respectively, ∂j−1 η L (g− 1 2 ) n (η) = (−1)j−1L (g+j− 3 2 ) n+1−j (η), ∂j−1 η P (g− 1 2 ,h− 1 2 ) n (η) = 1 2j−1 (n+ g + h)j−1P (g+j− 3 2 ,h+j− 3 2 ) n+1−j (η), in which we adopt the convention L (α) n (η) = P (α,β) n (η) = 0, n ∈ Z<0. Let us define M -dimensional column vectors ~X (M) v = ( X (M) v,j )M j=1 and ~Z (M) n = ( Z (M) n,j )M j=1 by X (M) v,j (η) def =  L (g+j− 3 2 ) v (−η), L, v Type I, (−1)j−1 ( g − 1 2 − v ) j−1 ηM−jL ( 3 2 −g−j) v (η), L, v Type II, (−1)j−1 2j−1 ( h− 1 2 − v ) j−1 (1+η 2 )M−j P (g+j− 3 2 , 3 2 −h−j) v (η), J, v Type I, 1 2j−1 ( g − 1 2 − v ) j−1 (1−η 2 )M−j P ( 3 2 −g−j,h+j− 3 2 ) v (η), J, v Type II, Z (M) n,j (η) def = (−1)j−1L (g+j− 3 2 ) n+1−j (η), L, 1 2j−1 (n+ g + h)j−1P (g+j− 3 2 ,h+j− 3 2 ) n+1−j (η), J. The Wronskians in (10)–(11) are replaced by ordinary determinants consisting of these column vectors: W[µd1 , . . . , µdM ](η) = ∣∣ ~X(M) d1 (η) · · · ~X(M) dM (η) ∣∣ × {( eη )MI ( η 3 2 −g−M)MII , L,((1+η 2 ) 3 2 −h−M)MI ((1−η 2 ) 3 2 −g−M)MII , J, W[µd1 , . . . , µdM , Pn](η) = ∣∣ ~X(M+1) d1 (η) · · · ~X(M+1) dM (η)~Z(M+1) n (η) ∣∣ × {( eη )MI ( η 1 2 −g−M)MII , L,((1+η 2 ) 1 2 −h−M)MI ((1−η 2 ) 1 2 −g−M)MII , J. We arrive at the main result, the simple expressions of ΞD(η) and PD,n(η) ΞD(η) = A ∣∣ ~X(M) d1 (η) · · · ~X(M) dM (η) ∣∣, (16) PD,n(η) = A ∣∣ ~X(M+1) d1 (η) · · · ~X(M+1) dM (η)~Z(M+1) n (η) ∣∣, (17) A = { η−MII(MII−1), L,(1+η 2 )−MI(MI−1)(1−η 2 )−MII(MII−1) , J. It should be stressed that the components of the matrices in (16) and (17) are all polyno- mials in η. This is a good contrast with the starting Wronskians W[µd1 , . . . , µdM ](η) and W[µd1 , . . . , µdM , Pn](η) in (10), (11), in which µdj ’s have non-polynomial factors (12). Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials 7 2.5 Simplified forms of the multi-indexed Laguerre and Jacobi polynomials, B Here we will simplify the Wronskians of the virtual state wavefunctions and the eigenpolyno- mials (14), (15). By replacing the even order derivatives of the virtual state wavefunctions and eigenfunctions in the Wronskians (14), (15), by the rule ψ(2m)(x)→ (−E)mψ(x), we obtain W [ φ̃d1 , . . . , φ̃dM , φn ] (x) = det(aj,k)1≤j,k≤M+1, (18) a2l−1,k = (−Ẽdk)l−1φ̃dk(x), 1 ≤ k ≤M, 1 ≤ l ≤ [ M+2 2 ] , a2l−1,M+1 = (−En)l−1φn(x), 1 ≤ l ≤ [ M+2 2 ] , a2l,k = (−Ẽdk)l−1φ̃′dk(x), 1 ≤ k ≤M, 1 ≤ l ≤ [ M+1 2 ] , a2l,M+1 = (−En)l−1φ′n(x), 1 ≤ l ≤ [ M+1 2 ] , in which [a] denotes the greatest integer not exceeding a. The first derivatives in the 2l-th row can be simplified by adding −φ′0(x) φ0(x) × (2l − 1)-th row, φ′n(x)→ ( d dx − φ′0(x) φ0(x) ) φn(x) = cF η′(x) φ0(x)ζn ( η(x) ) = φ0(x)ζn ( η(x) ) A, φ̃′v(x)→ ( d dx − φ′0(x) φ0(x) ) φ̃v(x) = cF η′(x) φ̃0(x)ζ̃v ( η(x) ) = φ̃0(x)ζ̃v ( η(x) ) A, A = cF η′(x) = { x−1, L, (sinx cosx)−1, J, (19) in which ζn(η) and ζ̃v(η) are polynomials in η defined by ζn(η) def = −2ηL (g+ 1 2 ) n−1 (η), L, −1 2(n+ g + h) ( 1− η2 ) P (g+ 1 2 ,h+ 1 2 ) n−1 (η), J, ζ̃v(η) def =  2ηL (g+ 1 2 ) v (−η), L, I, −2 ( g − 1 2 − v ) L (−g− 1 2 ) v (η), L, II,( h− 1 2 − v ) (1− η)P (g+ 1 2 ,−h− 1 2 ) v (η), J, I, − ( g − 1 2 − v ) (1 + η)P (−g− 1 2 ,h+ 1 2 ) v (η), J, II. Use is made of (4)–(6) for L and (7)–(9) for J. By extracting the functions φ0(x), φ̃0(x) from each column of the matrix aj,k (18) and the factor A of (19) from the even rows, we arrive at another set of simplified determinant expressions for the multi-indexed polynomials: PD,n(η) = c − 1 2 M(M+1) F det(aj,k)1≤j,k≤M+1A, (20) a2l−1,k = (−Ẽdk)l−1ξdk(η), 1 ≤ k ≤M, 1 ≤ l ≤ [ M+2 2 ] , a2l−1,M+1 = (−En)l−1Pn(η), 1 ≤ l ≤ [ M+2 2 ] , a2l,k = (−Ẽdk)l−1ζ̃dk(η), 1 ≤ k ≤M, 1 ≤ l ≤ [ M+1 2 ] , a2l,M+1 = (−En)l−1ζn(η), 1 ≤ l ≤ [ M+1 2 ] , A = { η−([M ′]+1)([M ′]+M−2[M 2 ]), L,(1−η 2 )−([M ′]+1)([M ′]+M−2[M 2 ])(1+η 2 )−([−M ′]+1)([−M ′]+M−2[M 2 ]) , J. 8 S. Odake and R. Sasaki In a similar way, we obtain ΞD(η) = c − 1 2 M(M−1) F det(aj,k)1≤j,k≤MA, (21) a2l−1,k = (−Ẽdk)l−1ξdk(η), 1 ≤ l ≤ [ M+1 2 ] , a2l,k = (−Ẽdk)l−1ζ̃dk(η), 1 ≤ l ≤ [ M 2 ] , A = { η−[M ′]([M ′]+M−2[M 2 ]), L,(1−η 2 1+η 2 )−[M ′]([M ′]+M−2[M 2 ]) , J. Again all the components of the matrices aj,k in (20) and (21) are polynomials in η. 2.6 Parity transformation of the multi-indexed Jacobi polynomials The Jacobi polynomial has the parity transformation property [24] P (α,β) n (−x) = (−1)nP (β,α) n (x). (22) We will show that this property is inherited by the multi-indexed Jacobi polynomials. It is based on the property of the Wronskian W[f1, . . . , fn](−η) = (−1) 1 2 n(n−1)W[g1, . . . , gn](η), gk(η) def = fk(−η). In this subsection, we indicate the types of the virtual states explicitly by (v, t), in which t stands for Type I or II. Based on (22), we obtain µ(v,t) ( −η; (g, h) ) = (−1)vµ(v,̄t) ( η; (h, g) ) , t̄ def = { II, t = I, I, t = II, Pn ( −η; (g, h) ) = (−1)nPn ( η; (h, g) ) . For the multi-index set D = {(d1, t1), . . . , (dM , tM )} of the virtual state wavefunctions, let us define the ‘mirror reflected’ multi-index set D′ def = {(d1, t̄1), . . . , (dM , t̄M )}. Corresponding to MI def = #{dj \| (dj , I) ∈ D}, MII def = #{dj \| (dj , II) ∈ D}, we have M ′I def = #{dj \| (dj , I) ∈ D′} = MII, M ′II def = #{dj \| (dj , II) ∈ D′} = MI. By parity transformation η → −η, the multi-indexed Jacobi polynomial PD,n ( η; (g, h) ) is mapped to PD′,n ( η; (h, g) ) with the ‘mirror reflected’ multi-index set D′: PD,n ( −η; (g, h) ) = (1−η 2 )(MII+h+ 1 2 )MI (1+η 2 )(MI+g+ 1 2 )MIIW [ µ(d1,t1), . . . , µ(dM ,tM ), Pn ]( −η; (g, h) ) = (1−η 2 )(MII+h+ 1 2 )MI (1+η 2 )(MI+g+ 1 2 )MII(−1) 1 2 M(M+1) ×W [ (−1)d1µ(d1 ,̄t1), . . . , (−1)dMµ(dM ,̄tM ), (−1)nPn ]( η; (h, g) ) = (1−η 2 )(MII+h+ 1 2 )MI (1+η 2 )(MI+g+ 1 2 )MII(−1) 1 2 M(M+1)(−1)d1+···+dM+n ×W [ µ(d1 ,̄t1), . . . , µ(dM ,̄tM ), Pn ]( η; (h, g) ) = (−1) n+ 1 2 M(M+1)+ M∑ k=1 dk PD′,n ( η; (h, g) ) . Similarly, the denominator polynomial ΞD ( η; (g, h) ) is mapped to ΞD′ ( η; (h, g) ) with a sign factor: ΞD ( −η; (g, h) ) = (−1) 1 2 M(M−1)+ M∑ k=1 dk ΞD′ ( η; (h, g) ) . Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials 9 For the special case of ‘mirror symmetric’ multi-index set D′ = D (as a set), i.e., {dj \| (dj , I) ∈ D} = {dj \| (dj , II) ∈ D} (as a set), we have PD′,n(η) = ±PD,n(η). In fact, this formula turns out to be PD′,n(η) = (−1)(M 2 )2PD,n(η). For this special case the parity transformation gives PD,n ( −η; (g, h) ) = (−1)nPD,n ( η; (h, g) ) , ΞD ( −η; (g, h) ) = ΞD ( η; (h, g) ) . 3 Summary and comments The multi-indexed Laguerre and Jacobi polynomials are defined by the Wronskian expressions originating from multiple Darboux transformations. Two simplified determinant expressions of them, (16), (17) and (20), (21), which do not contain derivatives, are derived based on the pro- perties of the Wronskian and identities of the Laguerre and Jacobi polynomials. For (20), (21), the Schrödinger equation is used. For (16), (17), various identities of the Laguerre and Jacobi polynomials are used, which are essentially forward shift relations. Although the calculation in Section 2.4 is performed for polynomials, it can be done for wavefunctions just like [20], in which simplified determinant expressions are presented for the multi-indexed polynomials obtained by multiple Darboux transformations with pseudo virtual states wavefunctions as seed solutions. The parity transformation property of the multi-indexed Jacobi polynomials is also derived. Multi-indexed orthogonal polynomials have been constructed for the classical orthogonal polynomials in the Askey scheme [1, 12], i.e., the Wilson, Askey–Wilson, Meixner, little q-Jacobi and (q-)Racah polynomials, etc. [18, 19, 21]. These polynomials belong to ‘discrete’ quantum mechanics [16], in which the Schrödinger equations are second-order difference equations. The Casoratian expressions of these multi-indexed polynomials can also be simplified by using various identities as demonstrated here. These simplifications will be published elsewhere [13]. Acknowledgements S.O. is supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), No. 25400395. References [1] Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999. [2] Bagchi B., Quesne C., Roychoudhury R., Isospectrality of conventional and new extended potentials, second- order supersymmetry and role of PT symmetry, Pramana J. Phys. 73 (2009), 337–347, arXiv:0812.1488. [3] Bochner S., Über Sturm–Liouvillesche Polynomsysteme, Math. Z. 29 (1929), 730–736. [4] Crum M.M., Associated Sturm–Liouville systems, Quart. J. Math. Oxford Ser. (2) 6 (1955), 121–127, physics/9908019. [5] Darboux G., Sur une proposition relative aux équations linéaires, Comptes Rendus Acad. Sci. 94 (1882), 1456–1459. [6] Durán A.J., Exceptional Meixner and Laguerre orthogonal polynomials, J. Approx. Theory 184 (2014), 176–208, arXiv:1310.4658. [7] Durán A.J., Exceptional Hahn and Jacobi orthogonal polynomials, J. Approx. Theory 214 (2017), 9–48, arXiv:1510.02579. [8] Durán A.J., Pérez M., Admissibility condition for exceptional Laguerre polynomials, J. Math. Anal. Appl. 424 (2015), 1042–1053, arXiv:1409.4901. https://doi.org/10.1017/CBO9781107325937 https://doi.org/10.1007/s12043-009-0126-4 http://arxiv.org/abs/0812.1488 https://doi.org/10.1007/BF01180560 https://doi.org/10.1093/qmath/6.1.121 http://arxiv.org/abs/physics/9908019 https://doi.org/10.1016/j.jat.2014.05.009 http://arxiv.org/abs/1310.4658 https://doi.org/10.1016/j.jat.2016.11.003 http://arxiv.org/abs/1510.02579 https://doi.org/10.1016/j.jmaa.2014.11.035 http://arxiv.org/abs/1409.4901 10 S. Odake and R. Sasaki [9] Gómez-Ullate D., Kamran N., Milson R., An extended class of orthogonal polynomials defined by a Sturm– Liouville problem, J. Math. Anal. Appl. 359 (2009), 352–367, arXiv:0807.3939. [10] Gómez-Ullate D., Kamran N., Milson R., An extension of Bochner’s problem: exceptional invariant sub- spaces, J. Approx. Theory 162 (2010), 987–1006, arXiv:0805.3376. [11] Gómez-Ullate D., Kamran N., Milson R., Two-step Darboux transformations and exceptional Laguerre polynomials, J. Math. Anal. Appl. 387 (2012), 410–418, arXiv:1103.5724. [12] Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. [13] Odake S., New determinant expressions of the multi-indexed orthogonal polynomials in discrete quantum mechanics, arXiv:1702.03078. [14] Odake S., Recurrence relations of the multi-indexed orthogonal polynomials. III, J. Math. Phys. 57 (2016), 023514, 24 pages, arXiv:1509.08213. [15] Odake S., Sasaki R., Infinitely many shape invariant potentials and new orthogonal polynomials, Phys. Lett. B 679 (2009), 414–417, arXiv:0906.0142. [16] Odake S., Sasaki R., Discrete quantum mechanics, J. Phys. A: Math. Theor. 44 (2011), 353001, 47 pages, arXiv:1104.0473. [17] Odake S., Sasaki R., Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials, Phys. Lett. B 702 (2011), 164–170, arXiv:1105.0508. [18] Odake S., Sasaki R., Multi-indexed (q-)Racah polynomials, J. Phys. A: Math. Theor. 45 (2012), 385201, 21 pages, arXiv:1203.5868. [19] Odake S., Sasaki R., Multi-indexed Wilson and Askey–Wilson polynomials, J. Phys. A: Math. Theor. 46 (2013), 045204, 22 pages, arXiv:1207.5584. [20] Odake S., Sasaki R., Krein–Adler transformations for shape-invariant potentials and pseudo virtual states, J. Phys. A: Math. Theor. 46 (2013), 245201, 24 pages, arXiv:1212.6595. [21] Odake S., Sasaki R., Multi-indexed Meixner and little q-Jacobi (Laguerre) polynomials, J. Phys. A: Math. Theor. 50 (2017), 165204, 23 pages, arXiv:1610.09854. [22] Quesne C., Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry, J. Phys. A: Math. Theor. 41 (2008), 392001, 6 pages, arXiv:0807.4087. [23] Routh E.J., On some properties of certain solutions of a differential equation of the second order, Proc. London Math. Soc. S1-16 (1884), 245–261. [24] Szegő G., Orthogonal polynomials, American Mathematical Society, Colloquium Publications, Vol. 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975. https://doi.org/10.1016/j.jmaa.2009.05.052 http://arxiv.org/abs/0807.3939 https://doi.org/10.1016/j.jat.2009.11.002 http://arxiv.org/abs/0805.3376 https://doi.org/10.1016/j.jmaa.2011.09.014 http://arxiv.org/abs/1103.5724 https://doi.org/10.1007/978-3-642-05014-5 http://arxiv.org/abs/1702.03078 https://doi.org/10.1063/1.4941087 http://arxiv.org/abs/1509.08213 https://doi.org/10.1016/j.physletb.2009.08.004 https://doi.org/10.1016/j.physletb.2009.08.004 http://arxiv.org/abs/0906.0142 https://doi.org/10.1088/1751-8113/44/35/353001 http://arxiv.org/abs/1104.0473 https://doi.org/10.1016/j.physletb.2011.06.075 http://arxiv.org/abs/1105.0508 https://doi.org/10.1088/1751-8113/45/38/385201 http://arxiv.org/abs/1203.5868 https://doi.org/10.1088/1751-8113/46/4/045204 http://arxiv.org/abs/1207.5584 https://doi.org/10.1088/1751-8113/46/24/245201 http://arxiv.org/abs/1212.6595 https://doi.org/10.1088/1751-8121/aa6496 https://doi.org/10.1088/1751-8121/aa6496 http://arxiv.org/abs/1610.09854 https://doi.org/10.1088/1751-8113/41/39/392001 https://doi.org/10.1088/1751-8113/41/39/392001 http://arxiv.org/abs/0807.4087 https://doi.org/10.1112/plms/s1-16.1.245 https://doi.org/10.1112/plms/s1-16.1.245 1 Introduction 2 Multi-indexed Laguerre and Jacobi polynomials 2.1 Original Laguerre and Jacobi polynomials 2.1.1 Radial oscillator potential 2.1.2 Pöschl–Teller potential 2.2 Discrete symmetry transformations and virtual state wavefunctions 2.2.1 Laguerre (L) system 2.2.2 Jacobi (J) system 2.3 Wronskian forms of the multi-indexed Laguerre and Jacobi polynomials 2.4 Simplified forms of the multi-indexed Laguerre and Jacobi polynomials, A 2.5 Simplified forms of the multi-indexed Laguerre and Jacobi polynomials, B 2.6 Parity transformation of the multi-indexed Jacobi polynomials 3 Summary and comments References

Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials

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