A short survey of duality in special functions
aa r X i v : . [ m a t h . C A ] F e b A short survey of duality in special functions
Tom H. Koornwinder
Abstract
This is a tutorial on duality properties of special functions, mainly of orthogonal poly-nomials in the ( q -)Askey scheme. It is based on the first part of the 2017 R. P. AgarwalMemorial Lecture delivered by the author. Classical orthogonal polynomials p n ( x ) and their generalizations in the Askey and q -Askeyscheme have the property that they are eigenfunctions of some second order operator L witheigenvalues depending on n , which therefore may be called the spectral variable. Moreover, be-ing orthogonal polynomials, the p n ( x ) satisfy a three-term recurrence relation and are therefore,as functions of n , eigenfunctions of a so-called Jacobi operator with eigenvalues x . This dualityphenomenon was also guiding for the author in the companion paper [17], where he derived thedual addition formula for continuous q -ultraspherical polynomials.This paper gives a brief tutorial type survey of duality, mainly for orthogonal polynomials,but also a little bit for transcendental special functions. This paper is based on the first part ofthe R. P. Agarwal Memorial Lecture , which the author delivered on November 2, 2017 duringthe conference ICSFA-2017 held in Bikaner, Rajasthan, India. See [17] for the paper based onthe second part.With pleasure I remember to have met Prof. Agarwal during the workshop on Special Func-tions and Differential Equations held at the Institute of Mathematical Sciences in Chennai,January 1997, where he delivered the opening address [1]. I cannot resist to quote from it thefollowing wise words, close to the end of the article:“
I think that I have taken enough time and I close my discourse- with a word of caution andadvice to the research workers in the area of special functions and also those who use themin physical problems. The corner stones of classical analysis are ‘elegance, simplicity, beautyand perfection.’ Let them not be lost in your work. Any analytical generalization of a specialfunction, only for the sake of a generalization by adding a few terms or parameters here andthere, leads us nowhere. All research work should be meaningful and aim at developing a qualitytechnique or have a bearing in some allied discipline. ”1 ote For definition and notation of ( q -)shifted factorials and ( q -)hypergeometric series see [9, § q = 1 case we will mostly meet terminating hypergeometric series r F s (cid:18) − n, a , . . . , a r b , . . . , b s ; z (cid:19) := n X k =0 ( − n ) k k ! ( a , . . . , a r ) k ( b , . . . , b s ) k z k . (1.1)Here ( b , . . . , b s ) k := ( b ) k . . . ( b s ) k and ( b ) k := b ( b + 1) . . . ( b + k −
1) is the
Pochhammer symbol or shifted factorial . In (1.1) we even allow that b i = − N for some i with N integer ≥ n . Thereis no problem because the sum on the right terminates at k = n ≤ N .In the q -case we will always assume that 0 < q <
1. We will only meet terminating q -hypergeometric series of the form s +1 φ s (cid:18) q − n , a , . . . , a s +1 b , . . . , b s ; q, z (cid:19) := n X k =0 ( q − n ; q ) k ( q ; q ) k ( a , . . . , a s +1 ; q ) k ( b , . . . , b s ; q ) k z k . (1.2)Here ( b , . . . , b s ; q ) k := ( b ; q ) k . . . ( b s ; q ) k and ( b ; q ) k := (1 − b )(1 − qb ) . . . (1 − q k − b ) is the q -Pochhammer symbol or q -shifted factorial . In (1.2) we even allow that b i = q − N for some i with N integer ≥ n .For formulas on orthogonal polynomials in the ( q -)Askey scheme we will often refer to Chap-ters 9 and 14 in [13]. Almost all of these formulas, with different numbering, are available inopen access on http://aw.twi.tudelft.nl/~koekoek/askey/ . With respect to a (positive) measure µ on R with support containing infinitely many points wecan define orthogonal polynomials (OPs) p n ( n = 0 , , , . . . ), unique up to nonzero real constantfactors, as (real-valued) polynomials p n of degree n such that Z R p m ( x ) p n ( x ) dµ ( x ) = 0 ( m, n = 0) . Then the polynomials p n satisfy a three-term recurrence relation x p n ( x ) = A n p n +1 ( x ) + B n p n ( x ) + C n p n − ( x ) ( n = 0 , , , . . . ) , (2.1)where the term C n p n − ( x ) is omitted if n = 0, and where A n , B n , C n are real and A n − C n > n = 1 , , . . . ) . (2.2)By Favard’s theorem [8] we can conversely say that if p ( x ) is a nonzero real constant, and the p n ( x ) ( n = 0 , , , . . . ) are generated by (2.1) for certain real A n , B n , C n which satisfy (2.2), thenthe p n are OPs with respect to a certain measure µ on R .With A n , B n , C n as in (2.1) define a Jacobi operator M , acting on infinite sequences { g ( n ) } ∞ n =0 ,by ( M g )( n ) = M n (cid:0) g ( n ) (cid:1) := A n g ( n + 1) + B n g ( n ) + C n g ( n −
1) ( n = 0 , , , . . . ) , C n g ( n −
1) is omitted if n = 0. Then (2.1) can be rewritten as the eigenvalueequation M n (cid:0) p n ( x ) (cid:1) = x p n ( x ) ( n = 0 , , , . . . ) . (2.3)One might say that the study of a system of OPs p n turns down to the spectral theory andharmonic analysis associated with the operator M . From this perspective one can wonder if thepolynomials p n satisfy some dual eigenvalue equation( Lp n )( x ) = λ n p n ( x ) (2.4)for n = 0 , , , . . . , where L is some linear operator acting on the space of polynomials. We willconsider varioua types of operators L together with the corresponding OPs, first in the Askeyscheme and next in the q -Askey scheme. Classical OPs
Bochner’s theorem [5] classifies the second order differentai operators L to-gether with the OPs p n such that (2.4) holds for certain eigenvalues λ n . The resulting clas-sical orthogonal polynomials are essentially the polynomials listed in the table below. Here dµ ( x ) = w ( x ) dx on ( a, b ) and the closure of that interval is the support of µ . Furthermore, w ( x ) occurs in the formula for L to be given after the table.name p n ( x ) w ( x ) w ( x ) w ( x ) ( a, b ) constraints λ n Jacobi P ( α,β ) n ( x ) (1 − x ) α (1 + x ) β − x ( − , α, β > − − n ( n + α + β + 1) Laguerre L ( α ) n ( x ) x α e − x x (0 , ∞ ) α > − − n Hermite H n ( x ) e − x −∞ , ∞ ) − n Then ( Lf )( x ) = w ( x ) − ddx (cid:0) w ( x ) f ′ ( x ) (cid:1) . For these classical OPs the duality goes much further than the two dual eigenvalue equations(2.3) and (2.4). In particular for Jacobi polynomials it is true to a large extent that every formulaor property involving n and x has a dual formula or property where the roles of n and x areinterchanged. We call this the duality principle . If the partner formula or property is not yetknown then it is usually a good open problem to find it (but one should be warned that thereare examples where the duality fails).The Jacobi, Laguerre and Hermite families are connected by limit transitions, as is alreadysuggested by limit transitions for their (rescaled) weight functions: • Jacobi → Laguerre: x α (1 − β − x ) β → x α e − x as β → ∞ ; • Jacobi → Hermite: (cid:0) − α − x (cid:1) α → e − x as α → ∞ ; • Laguerre → Hermite: e α (1 − log α ) (cid:0) (2 α ) x + α (cid:1) α e − (2 α ) x − α → e − x as α → ∞ .Formulas and properties of the three families can be expected to be connected under these limits.Although this is not always the case, this limit principle is again a good source of open problems.3 iscrete analogues of classical OPs Let L be a second order difference operator:( Lf )( x ) := a ( x ) f ( x + 1) + b ( x ) f ( x ) + c ( x ) f ( x − . (2.5)Here as solutions of (2.4) we will also allow OPs { p n } Nn =0 for some finite N ≥
0, which willbe orthogonal with respect to positive weights w k ( k = 0 , , . . . , N ) on a finite set of points x k ( k = 0 , , . . . , N ): N X k =0 p m ( x k ) p n ( x k ) w k = 0 ( m, n = 0 , , . . . , N ; m = n ) . If such a finite system of OPs satisfies (2.4) for n = 0 , , . . . , N with L of the form (2.5) thenthe highest n for which the recurrence relation (2.1) holds is n = N , where the zeros of p N +1 are precisely the points x , x , . . . , x N .The classification of OPs satisfying (2.4) with L of the form (2.5) (first done by O. Lancaster,1941, see [2]) yields the four families of Hahn, Krawtchouk, Meixner and Charlier polynomials,of which Hahn and Krawtchouk are finite systems, and Meixner and Charlier infinite systemswith respect to weights on countably infinite sets. Krawtchouk polynomials [13, (9.11.1)] are given by K n ( x ; p, N ) := F (cid:18) − n, − x − N ; p − (cid:19) ( n = 0 , , , . . . , N ) . (2.6)They satsify the orthogonality relation N X x =0 ( K m K n w )( x ; p, N ) = (1 − p ) N w ( n ; p, N ) δ m,n with weights w ( x ; p, N ) := (cid:18) Nx (cid:19) p x (1 − p ) N − x (0 < p < . By (2.6) they are self-dual : K n ( x ; p, N ) = K x ( n ; p, N ) ( n, x = 0 , , . . . , N ) . The three-term recurrence relation (2.3) immediately implies a dual equation (2.4) for such OPs.The four just mentioned families of discrete OPs are also connected by limit relations. More-over, the classical OPs can be obtained as limit cases of them, but not conversely. For instance,
Hahn polynomials [13, (9.5.1)] are given by Q n ( x ; α, β, N ) := F (cid:18) − n, n + α + β + 1 , − xα + 1 , − N ; 1 (cid:19) ( n = 0 , , . . . , N ) (2.7)and they satisfy the orthogonality relation N X x =0 ( Q m Q n w )( x ; α, β, N ) = 0 ( m, n = 0 , , . . . , N ; m = n ; α, β > − w ( x ; α, β, N ) := ( α + 1) x ( β + 1) N − x x ! ( N − x )! . Then by (2.7) (rescaled) Hahn polynomials tend to (shifted) Jacobi polynomials:lim N →∞ Q n ( N x ; α, β, N ) = F (cid:18) − n, n + α + β + 1 α + 1 ; x (cid:19) = P ( α,β ) n (1 − x ) P ( α,β ) n (1) . (2.8) Continuous versions of Hahn and Meixner polynomials
A variant of the difference operator (2.5) is the operator( Lf )( x ) := A ( x ) f ( x + i ) + B ( x ) f ( x ) + A ( x ) f ( x − i ) ( x ∈ R ) , (2.9)where B ( x ) is real-valued. Then further OPs satisfying (2.4) are the continuous Hahn polyno-mials and the Meixner-Pollaczek polynomials [13, Ch. 9]. Insertion of a quadratic argument
For an operator e L and some polynomial σ of degree 2 we can define an operator L by( Lf ) (cid:0) σ ( x ) (cid:1) := e L x (cid:16) f (cid:0) σ ( x ) (cid:1)(cid:17) , (2.10)Now we look for OPs satisfying (2.4) where e L is of type (2.5) or (2.9). So e L x (cid:16) p n (cid:0) σ ( x ) (cid:1)(cid:17) = λ n p n (cid:0) σ ( x ) (cid:1) . (2.11)The resulting OPs are the Racah polynomials and dual Hahn polynomials for (2.11) with e L oftype (2.5), and Wilson polynomials and continuous dual Hahn polynomials for (2.11) with e L oftype (2.9), see again [13, Ch. 9].The OPs satisfying (2.4) in the cases discussed until now form together the Askey scheme ,see Figure 1. The arrows denote limit transitions.In the Askey scheme we emphasize the self-dual families: Racah, Meixner, Krawtchouk andCharlier for the OPs with discrete orthogonality measure, and Wilson and Meixner-Pollaczekfor the OPs with non-discrete orthogonality measure. We already met perfect self-duality forthe Krawtchouk polynomials, which is also the case for Meixner and Charlier polynomials. Forthe Racah polynomials the dual OPs are still Racah polynomials, but with different values ofthe parameters: R n (cid:0) x ( x + δ − N ); α, β, − N − , δ (cid:1) := F (cid:18) − n, n + α + β + 1 , − x, x + δ − Nα + 1 , β + δ + 1 , − N ; 1 (cid:19) = R x ( n ( n + α + β + 1); − N − , δ, α, β ) ( n, x = 0 , , . . . , N ) . The orthogonality relation for these Racah polynomials involves a weighted sum of terms( R m R n ) (cid:0) x ( x + δ − N ); α, β, − N − , δ (cid:1) over x = 0 , , . . . N .5ilson ✁✁✁✁☛ ❄ Racah ❄ ❆❆❆❆❆❯
Cont.dual Hahn ❄ Cont.Hahn (cid:0)(cid:0)(cid:0)(cid:0)✠ ❄
Hahn ✁✁✁✁✁☛ ❄❅❅❅❅❅❘
Dual Hahn (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✠ ❄
Meixner-Pollaczek ❆❆❆❆❆❯ ✗✖ ✔✕
Jacobi ❄ ❄
Meixner (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✠ ❄
Krawtchouk ✁✁✁✁✁☛ ✗✖ ✔✕
Laguerre ❆❆❆❆❆❯
Charlier (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✠ ✗✖ ✔✕
Hermite self-dual familydual familiesdiscrete OPs ✗✖ ✔✕ classical OPsFigure 1: The Askey schemeFor Wilson polynomials we have also self-duality with a change of parameters but the self-duality is not perfect, i.e., not related to the orthogonality relation:const . W n ( x ; a, b, c, d ) := F (cid:18) − n, n + a + b + c + d − , a + ix, a − ixa + b, a + c, a + d ; 1 (cid:19) = const . W − ix − a (cid:16)(cid:0) i ( n + a ′ ) (cid:1) ; a ′ , b ′ , c ′ , d ′ (cid:17) , (2.12)where a ′ = ( a + b + c + d − a ′ + b ′ = a + b , a ′ + c ′ = a + c , a ′ + d ′ = a + d . The duality(2.12) holds for − ix − a = 0 , , , . . . , while the orthogonality relation for the Wilson polynomialsinvolves a weighted integral of ( W m W n )( x ; a, b, c, d ) over x ∈ [0 , ∞ ).As indicated in Figure 1, the dual Hahn polynomials R n (cid:0) x ( x + α + β + 1); α, β, N (cid:1) := F (cid:18) − n, − x, x + α + β + 1 α + 1 , − N ; 1 (cid:19) ( n = 0 , , . . . , N )are dual to the Hahn polynomials (2.7): Q n ( x ; α, β, N ) = R x (cid:0) n ( n + α + β + 1); α, β, N (cid:1) ( n, x = 0 , , . . . , N ) . q -Askey scheme The families of OPs in the q -Askey scheme [13, Ch. 14] result from the classification [11], [10],[12], [24] of OPs satisfying (2.4), where L is defined in terms of the operator e L and the function σ by (2.10), where e L is of type (2.5) or (2.9), and where σ ( x ) = q x or equal to a quadraticpolynomial in q x . This choice of σ ( x ) is the new feature deviating from what we discussed aboutthe Askey scheme. And here q enters, with 0 < q < q -Askey schemeis considerably larger than the Askey scheme, but many features of the Askey scheme returnhere, in particular it has arrows denoting limit relations. Moreover, the q -Askey scheme is quiteparallel to the Askey scheme in the sense that OPs in th q -Askey scheme, after suitable rescaling,tend to OPs in the Askey scheme as q ↑
1. Parallel to Wilson and Racah polynomials at the topof the Askey scheme there are Askey–Wilson polynomials [3] and q -Racah polynomials at thetop of the q -Askey scheme. These are again self-dual families, with the self-duality for q -Racahbeing perfect.The guiding principles discussed before about formulas or properties related by duality orlimit transitions now extend to the q -Askey scheme: both within the q -Askey scheme and inrelation to the Askey scheme by letting q ↑
1. For instance, one can hope to find as many dualpairs of significant formulas and properties of Askey–Wilson polynomials as possible which havemutually dual limit cases for Jacobi polynomials. In fact, we realize this in [17] with the additionand dual addition formula by taking limits from the continuous q -ultraspherical polynomials (aself-dual one-parameter subclass of the four-parameter class of Askey–Wilson polynomials) tothe ultraspherical polynomials (a one-parameter subclass of the two-parameter class of Jacobipolynomials).One remarkable aspect of duality in the two schemes concerns the discrete OPs living there.Leonard (1982) classified all systems of OPs p n ( x ) with respect to weights on a countable set { x ( m ) } for which there is a system of OPs q m ( y ) on a countable set { y ( n ) } such that p n (cid:0) x ( m ) (cid:1) = q m (cid:0) y ( n ) (cid:1) . See http://homepage.tudelft.nl/11r49/pictures/large/q-AskeyScheme.jpg q -Askey scheme which are orthogonal with respect toweights on a countable set together with their limit cases for q ↑ q ↓ − − < q < q -Askey scheme). The q ↓ − For Bessel functions J α see [23, Ch. 10] and references given there. It is convenient to use adifferent standardization and notation: J α ( x ) := Γ( α + 1) (2 /x ) α J α ( x ) . Then (see [23, (10.16.9)]) J α ( x ) = ∞ X k =0 ( − x ) k ( α + 1) k k ! = F (cid:18) − α + 1 ; − x (cid:19) ( α > − . J α is an even entire analytic function. Some special cases are J − / ( x ) = cos x, J / ( x ) = sin xx . (2.13)The Hankel transform pair [23, § f in a suitable function class, is given by b f ( λ ) = Z ∞ f ( x ) J α ( λx ) x α +1 dx,f ( x ) = 12 α +1 Γ( α + 1) Z ∞ b f ( λ ) J α ( λx ) λ α +1 dλ. In view of (2.13) the Hankel transform contains the Fourier-cosine and Fourier-sine transformas special cases for α = ± .The functions x
7→ J α ( λx ) satisfy the eigenvalue equation [23, (10.13.5)] (cid:18) ∂ ∂x + 2 α + 1 x ∂∂x (cid:19) J α ( λx ) = − λ J α ( λx ) . (2.14)Obviously, then also (cid:18) ∂ ∂λ + 2 α + 1 λ ∂∂λ (cid:19) J α ( λx ) = − x J α ( λx ) . (2.15)The differential operator in (2.15) involves the spectral variable λ of (2.14), while the eigenvaluein (2.15) involves the x -variable in the differential operator in (2.14).The Bessel functions and the Hankel transform are closely related to the Jacobi polynomials(2.8) and their orthogonality relation. Indeed, we have the limit formulaslim n →∞ P ( α,β ) n (cid:0) cos( n − x ) (cid:1) P ( α,β ) n (1) = J α ( x ) , lim ν →∞ νλ =1 , ,... P ( α,β ) n (cid:0) cos( ν − x ) (cid:1) P ( α,β ) n (1) = J α ( λx ) . q -)Askey scheme, see for instance [16], [14].In 1986 Duistermaat & Gr¨unbaum [6] posed the question if the pair of eigenvalue equations(2.14), (2.15) could be generalized to a pair L x (cid:0) φ λ ( x ) (cid:1) = − λ φ λ ( x ) ,M λ (cid:0) φ λ ( x ) (cid:1) = τ ( x ) φ λ ( x ) (2.16)for suitable differential operators L x in x and M λ in λ and suitable functions φ λ ( x ) solving thetwo equations. Here the functions φ λ ( x ) occur as eigenfunctions in two ways: for the operator L x with eigenvalue depending on λ and for the operator M λ with eigenvalue depending on x .Since the occurring eigenvalues of an operator form its spectrum, a phenomenon as in (2.16) iscalled bispectrality . For the case of a second order differential operator L x written in potentialform L x = d /dx − V ( x ) they classified all possibilities for (2.16). Beside the mentioned Besselcases and a case with Airy functions (closely related to Bessel functions) they obtained twoother families where M λ is a higher than second order differential operator. These could beobtained by successive Darboux transformations applied to L x in potential form. A Darbouxtransformation produces a new potential from a given potential V ( x ) by a formula which involvesan eigenfunction of L x with eigenvalue 0. Their two new families get a start by the applicationof a Darboux transformation to the Bessel differential equation (2.14), rewritten in potentialform φ ′′ λ ( x ) − ( α − ) x − φ λ ( x ) = − λ φ λ ( x ) , φ λ ( x ) = ( λx ) α + J α ( λx ) . Here α should be in Z + for a start of the first new family or in Z for a start of the secondnew family. For other values of α one would not obtain a dual eigenvalue equation with M λ afinite order differential operator.Just as higher order differential operators M λ occur in (2.16), there has been a lot of workon studying OPs satisfying (2.4) with L a higher order differential operator. See a classificationin [20], [19]. All occurring OPs, the so-called Jacobi type and
Laguerre type polynomials , have aJacobi or Laguerre orthogonality measure with integer values of the parameters, supplementedby mass points at one or both endpoints of the orthogonality interval. Some of the Bessel typefunctions in the second new class in [6] were obtained in [7] as limit cases of Laguerre typepolynomials.
The self-duality property of the family of Askey-Wikson polynomials is reflected in Zhedanov’s
Askey–Wilson algebra [25]. A larger algebraic structure is the double affine Hecke algebra (DAHA), introduced by Cherednik and extended by Sahi. The related special functions areso-called non-symmetric special functions. They are functions in several variables and associ-ated with root systems. Again there is a duality, both in the DAHA and for the related specialfunctions. For the (one-variable) case of the non-symmetric Askey–Wilson polynomials this istreated in [22]. In [18] limit cases in the q -Askey scheme are also considered.9inally we should mention the manuscript [15]. Here the author extended the duality [17,(4.2)] for continuous q -ultraspherical polynomials to Macdonald polynomials and thus obtainedthe so-called Pieri formula [21, § VI.6] for these polynomials.
Acknowledgement
I thank Prof. M. A. Pathan and Prof. S. A. Ali for the invitation todeliver the 2017 R. P. Agarwal Memorial Lecture and for their cordiality during my trip to Indiaon this occasion.
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T. H. Koornwinder, Korteweg-de Vries Institute, University of Amsterdam,P.O. Box 94248, 1090 GE Amsterdam, The Netherlands;email: [email protected]@xs4all.nl