Additions to the formula lists in “Hypergeometric orthogonal ...

Additions to the formula lists in “Hypergeometric orthogonal polynomials and their q-analogues” by Koekoek, Lesky and Swarttouw Tom H. Koornwinder June 19, 2015

Abstract This report gives a rather arbitrary choice of formulas for (q-)hypergeometric orthogonal polynomials which the author missed while consulting Chapters 9 and 14 in the book “Hypergeometric orthogonal polynomials and their q-analogues” by Koekoek, Lesky and Swarttouw. The systematics of these chapters will be followed here, in particular for the numbering of subsections and of references.

Introduction This report contains some formulas about (q-)hypergeometric orthogonal polynomials which I missed but wanted to use while consulting Chapters 9 and 14 in the book [KLS]: R. Koekoek, P. A. Lesky and R. F. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, Springer-Verlag, 2010. These chapters form together the (slightly extended) successor of the report R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1998; http://aw.twi.tudelft.nl/~koekoek/askey/. Certainly these chapters give complete lists of formulas of special type, for instance orthogonality relations and three-term recurrence relations. But outside these narrow categories there are many other formulas for (q-)orthogonal polynomials which one wants to have available. Often one can find the desired formula in one of the standard references listed at the end of this report. Sometimes it is only available in a journal or a less common monograph. Just for my own comfort, I have brought together some of these formulas. This will possibly also be helpful for some other users. Usually, any type of formula I give for a special class of polynomials, will suggest a similar formula for many other classes, but I have not aimed at completeness by filling in a formula of such type at all places. The resulting choice of formulas is rather arbitrary, just depending on the formulas which I happened to need or which raised my interest. For each formula I give a suitable reference or I sketch a proof. It is my intention to gradually extend this collection of formulas. 1

Conventions The (x.y) and (x.y.z) type subsection numbers, the (x.y.z) type formula numbers, and the [x] type citation numbers refer to [KLS]. The (x) type formula numbers refer to this manuscript and the [Kx] type citation numbers refer to citations which are not in [KLS]. Some standard references like [DLMF] are given by special acronyms. N is always a positive integer. Always assume n to be a nonnegative integer or, if N is present, to be in {0, 1, . . . , N }. Throughout assume 0 < q < 1. For each family the coefficient of the term of highest degree of the orthogonal polynomial of degree n can be found in [KLS] as the coefficient of pn (x) in the formula after the main formula under the heading “Normalized Recurrence Relation”. If that main formula is numbered as (x.y.z) then I will refer to the second formula as (x.y.zb). In the notation of q-hypergeometric orthogonal polynomials we will follow the convention that the parameter list and q are separated by ‘ | ’ in the case of a q-quadratic lattice (for instance Askey-Wilson) and by ‘;’ in the case of a q-linear lattice (for instance big q-Jacobi). This convention is mostly followed in [KLS], but not everywhere, see for instance little q-Laguerre / Wall.

Acknowledgement Many thanks to Howard Cohl for having called my attention so often to typos and inconsistencies.

2

Contents Introduction Conventions Generalities 9.1 Wilson 9.2 Racah 9.3 Continuous dual Hahn 9.4 Continuous Hahn 9.5 Hahn 9.6 Dual Hahn 9.7 Meixner-Pollaczek 9.8 Jacobi 9.8.1 Gegenbauer / Ultraspherical 9.8.2 Chebyshev 9.9 Pseudo Jacobi (or Routh-Romanovski) 9.10 Meixner 9.11 Krawtchouk 9.12 Laguerre 9.14 Charlier 9.15 Hermite 14.1 Askey-Wilson 14.2 q-Racah 14.3 Continuous dual q-Hahn 14.4 Continuous q-Hahn 14.5 Big q-Jacobi 14.7 Dual q-Hahn 14.8 Al-Salam-Chihara 14.9 q-Meixner-Pollaczek 14.10 Continuous q-Jacobi 14.10.1 Continuous q-ultraspherical / Rogers 14.11 Big q-Laguerre 14.12 Little q-Jacobi 14.14 Quantum q-Krawtchouk 14.16 Affine q-Krawtchouk 14.17 Dual q-Krawtchouk 14.20 Little q-Laguerre / Wall 14.21 q-Laguerre 14.27 Stieltjes-Wigert 14.28 Discrete q-Hermite I 14.29 Discrete q-Hermite II Standard references References from [KLS] Other references 3

Generalities Criteria for uniqueness of orthogonality measure According to Shohat & Tamarkin [K28, p.50] orthonormal polynomials pn have a unique orthogonality measure (up to positive constant factor) if for some z ∈ C we have ∞ X

|pn (z)|2 = ∞.

(1)

n=0

Also (see Shohat & Tamarkin [K28, p.59]), monic orthogonal polynomials pn with three-term recurrence relation xpn (x) = pn+1 (x) + Bn pn (x) + Cn pn−1 (x) (Cn necessarily positive) have a unique orthogonality measure if ∞ X (Cn )−1/2 = ∞. (2) n=1

Furthermore, if orthogonal polynomials have an orthogonality measure with bounded support, then this is unique (see Chihara [146]). Even orthogonality measure If {pn } is a system of orthogonal polynomials with respect to an even orthogonality measure which satisfies the three-term recurrence relation xpn (x) = An pn+1 (x) + Cn pn−1 (x) then

p2n (0) C2n−1 =− . p2n−2 (0) A2n−1

Appell’s bivariate hypergeometric function F4

This is defined by

∞ X (a)m+n (b)m+n m n x y (c)m (c0 )n m! n!

F4 (a, b; c, c0 ; x, y) :=

(3)

1

1

(|x| 2 + |y| 2 < 1),

(4)

m,n=0

see [HTF1, 5.7(9), 5.7(44)] or [DLMF, (16.13.4)]. There is the reduction formula     −x −y a, 1 + a − b a a F4 a, b; b, b; , = (1 − x) (1 − y) 2 F1 ; xy , (1 − x)(1 − y) (1 − x)(1 − y) b see [HTF1, 5.10(7)]. When combined with the quadratic transformation [HTF1, 2.11(34)] (here a − b − 1 should be replaced by a − b + 1), see also [DLMF, (15.8.15)], this yields   −y −x , F4 a, b; b, b; (1 − x)(1 − y) (1 − x)(1 − y)   1 1  (1 − x)(1 − y) a a, 2 (a + 1) 4xy 2 = ; . 2 F1 1 + xy b (1 + xy)2 This can be rewritten as −a

F4 (a, b; b, b; x, y) = (1 − x − y) 1

1 2 F1

1

1 2 a, 2 (a

b

+ 1)

4xy ; (1 − x − y)2

Note that, if x, y ≥ 0 and x 2 + y 2 < 1, then 1 − x − y > 0 and 0 ≤ 4

4xy (1−x−y)2

 .

< 1.

(5)

q-Hypergeometric series of base q −1  r φs

a1 , . . . , ar −1 ;q ,z b1 , . . . bs

By [GR, Exercise 1.4(i)]:

 =

s+1 φs

−1 a−1 qa1 . . . ar z 1 , . . . ar , 0, . . . , 0 ; q, −1 −1 b1 . . . bs b1 , . . . , bs

! (6)

for r ≤ s + 1, a1 , . . . , ar , b1 , . . . , bs 6= 0. In the non-terminating case, for 0 < q < 1, there is convergence if |z| < b1 . . . bs /(qa1 . . . ar ) . A transformation of a terminating 2 φ1 By [GR, Exercise 1.15(i)] we have  −n  −n   q ,b q , c/b, 0 −1 ; q, z = (bz/(cq); q )n 3 φ2 ; q, q . 2 φ1 c c, cq/(bz) Very-well-poised q-hypergeometric series

The notation of [GR, (2.1.11)] will be followed: 

r+1 Wr (a1 ; a4 , a5 , . . . , ar+1 ; q, z)

Theta function

:=

(7)

r+1 φr 

1

1

a1 , qa12 , −qa12 , a4 , . . . , ar+1 1

1

a12 , −a12 , qa1 /a4 , . . . , qa1 /ar+1

 ; q, z  .

(8)

The notation of [GR, (11.2.1)] will be followed:

θ(x; q) := (x, q/x; q)∞ ,

θ(x1 , . . . , xm ; q) := θ(x1 ; q) . . . θ(xm ; q).

(9)

9.1 Wilson Symmetry The Wilson polynomial Wn (y; a, b, c, d) is symmetric in a, b, c, d. This follows from the orthogonality relation (9.1.2) together with the value of its coefficient of y n given in (9.1.5b). Alternatively, combine (9.1.1) with [AAR, Theorem 3.1.1]. As a consequence, it is sufficient to give generating function (9.1.12). Then the generating functions (9.1.13), (9.1.14) will follow by symmetry in the parameters. Hypergeometric representation Wn (x2 ; a, b, c, d) = × 7 F6

In addition to (9.1.1) we have (see [513, (2.2)]):

(a − ix)n (b − ix)n (c − ix)n (d − ix)n (−2ix)n

! 2ix − n, ix − 21 n + 1, a + ix, b + ix, c + ix, d + ix, −n ; 1 . (10) ix − 21 n, 1 − n − a + ix, 1 − n − b + ix, 1 − n − c + ix, 1 − n − d + ix

The symmetry in a, b, c, d is clear from (10). Special value Wn (−a2 ; a, b, c, d) = (a + b)n (a + c)n (a + d)n , and similarly for arguments −b2 , −c2 and −d2 by symmetry of Wn in a, b, c, d. 5

(11)

Uniqueness of orthogonality measure Under the assumptions on a, b, c, d for (9.1.2) or (9.1.3) the orthogonality measure is unique up to constant factor. For the proof assume without loss of generality (by the symmetry in a, b, c, d) that Re a ≥ 0. Write the right-hand side of (9.1.2) or (9.1.3) as hn δm,n . Observe from (9.1.2) and (11) that |Wn (−a2 ; a, b, c, d)|2 = O(n4Re a−1 ) hn

as n → ∞.

Therefore (1) holds, from which the uniqueness of the orthogonality measure follows. By a similar, but necessarily more complicated argument Ismail et al. [281, Section 3] proved the uniqueness of orthogonality measure for associated Wilson polynomials.

9.2 Racah Racah in terms of Wilson In the Remark on p.196 Racah polynomials are expressed in terms of Wilson polynomials. This can be equivalently written as Rn x(x − N + δ); α, β, −N − 1, δ





Wn − (x + 12 (δ − N ))2 ; 21 (δ − N ), α + 1 − 12 (δ − N ), β + 12 (δ + N ) + 1, − 12 (δ + N ) = . (12) (α + 1)n (β + δ + 1)n (−N )n

9.3 Continuous dual Hahn Symmetry The continuous dual Hahn polynomial Sn (y; a, b, c) is symmetric in a, b, c. This follows from the orthogonality relation (9.3.2) together with the value of its coefficient of y n given in (9.3.5b). Alternatively, combine (9.3.1) with [AAR, Corollary 3.3.5]. As a consequence, it is sufficient to give generating function (9.3.12). Then the generating functions (9.3.13), (9.3.14) will follow by symmetry in the parameters. Special value Sn (−a2 ; a, b, c) = (a + b)n (a + c)n ,

(13)

and similarly for arguments −b2 and −c2 by symmetry of Sn in a, b, c. Uniqueness of orthogonality measure Under the assumptions on a, b, c for (9.3.2) or (9.3.3) the orthogonality measure is unique up to constant factor. For the proof assume without loss of generality (by the symmetry in a, b, c, d) that Re a ≥ 0. Write the right-hand side of (9.3.2) or (9.3.3) as hn δm,n . Observe from (9.3.2) and (13) that |Sn (−a2 ; a, b, c)|2 = O(n2Re a−1 ) hn

as n → ∞.

Therefore (1) holds, from which the uniqueness of the orthogonality measure follows. 6

9.4 Continuous Hahn Orthogonality relation and symmetry The orthogonality relation (9.4.2) holds under the more general assumption that Re (a, b, c, d) > 0 and (c, d) = (a, b) or (b, a). Thus, under these assumptions, the continuous Hahn polynomial pn (x; a, b, c, d) is symmetric in a, b and in c, d. This follows from the orthogonality relation (9.4.2) together with the value of its coefficient of xn given in (9.4.4b). As a consequence, it is sufficient to give generating function (9.4.11). Then the generating function (9.4.12) will follow by symmetry in the parameters. Uniqueness of orthogonality measure The coefficient of pn−1 (x) in (9.4.4) behaves as O(n2 ) as n → ∞. Hence (2) holds, by which the orthogonality measure is unique. Special cases

In the following special case there is a reduction to Meixner-Pollaczek: pn (x; a, a + 12 , a, a + 21 ) =

(2a)n (2a + 12 )n (2a) Pn (2x; 21 π). (4a)n

(14)

See [342, (2.6)] (note that in [342, (2.3)] the Meixner-Pollaczek polynonmials are defined different from (9.7.1), without a constant factor in front). For 0 < a < 1 the continuous Hahn polynomials pn (x; a, 1 − a, a, 1 − a) are orthogonal on
−1 (−∞, ∞) with respect to the weight function cosh(2πx) − cos(2πa) (by straightforward 1 computation from (9.4.2)). For a = 4 the two special cases coincide: Meixner-Pollaczek with
−1 weight function cosh(2πx) .

9.5 Hahn Special values Qn (0; α, β, N ) = 1,

Qn (N ; α, β, N ) =

(−1)n (β + 1)n . (α + 1)n

(15)

Use (9.5.1) and compare with (9.8.1) and (34). From (9.5.3) and (3) it follows that Q2n (N ; α, α, 2N ) =

( 12 )n (N + α + 1)n . (−N + 12 )n (α + 1)n

(16)

From (9.5.1) and [DLMF, (15.4.24)] it follows that QN (x; α, β, N ) =

(−N − β)x (α + 1)x 7

(x = 0, 1, . . . , N ).

(17)

Symmetries

By the orthogonality relation (9.5.2): Qn (N − x; α, β, N ) = Qn (x; β, α, N ), Qn (N ; α, β, N )

(18)

It follows from (25) and (20) that QN −n (x; α, β, N ) = Qn (x; −N − β − 1, −N − α − 1, N ) QN (x; α, β, N ) Duality

(x = 0, 1, . . . , N ).

(19)

The Remark on p.208 gives the duality between Hahn and dual Hahn polynomials: Qn (x; α, β, N ) = Rx (n(n + α + β + 1); α, β, N )

(n, x ∈ {0, 1, . . . N }).

(20)

9.6 Dual Hahn Special values By (17) and (20) we have Rn (N (N + γ + δ + 1); γ, δ, N ) =

(−N − δ)n . (γ + 1)n

(21)

It follows from (15) and (20) that RN (x(x + γ + δ + 1); γ, δ, N ) =

Symmetries

(−1)x (δ + 1)x (γ + 1)x

(x = 0, 1, . . . , N ).

(22)

Write the weight in (9.6.2) as 2x + γ + δ + 1 (γ + 1)x wx (α, β, N ) := N ! (x + γ + δ + 1)N +1 (δ + 1)x

  N . x

(23)

Then (δ + 1)N wN −x (γ, δ, N ) = (−γ − N )N wx (−δ − N − 1, −γ − N − 1, N ).

(24)

Hence, by (9.6.2), Rn ((N − x)(N − x + γ + δ + 1); γ, δ, N ) = Rn (x(x−2N −γ −δ −1); −N −δ −1, −N −γ −1, N ). Rn (N (N + γ + δ + 1); γ, δ, N ) (25) Alternatively, (25) follows from (9.6.1) and [DLMF, (16.4.11)]. It follows from (18) and (20) that RN −n (x(x + γ + δ + 1); γ, δ, N ) = Rn (x(x + γ + δ + 1); δ, γ, N ) RN (x(x + γ + δ + 1); γ, δ, N ) 8

(x = 0, 1, . . . , N ).

(26)

Re: (9.6.11).

The generating function (9.6.11) can be written in a more conceptual way as

(1 − t)

x

 2 F1

 N X x − N, x + γ + 1 N! ;t = ωn Rn (λ(x); γ, δ, N ) tn , −δ − N (δ + 1)N

(27)

n=0

where ωn :=

   γ+n δ+N −n , n N −n

(28)

i.e., the denominator on the right-hand side of (9.6.2). By the duality between Hahn polynomials and dual Hahn polynomials (see (20)) the above generating function can be rewritten in terms of Hahn polynomials: (1 − t)

n

 2 F1

n − N, n + α + 1 ;t −β − N

where

 =

N X N! wx Qn (x; α, β, N ) tx , (β + 1)N

(29)

x=0

   α+x β+N −x wx := , x N −x

(30)

i.e., the weight occurring in the orthogonality relation (9.5.2) for Hahn polynomials. Re: (9.6.15).

There should be a closing bracket before the equality sign.

9.7 Meixner-Pollaczek Uniqueness of orthogonality measure The coefficient of pn−1 (x) in (9.7.4) behaves as O(n2 ) as n → ∞. Hence (2) holds, by which the orthogonality measure is unique.

9.8 Jacobi Orthogonality relation Write the right-hand side of (9.8.2) as hn δm,n . Then hn n + α + β + 1 (α + 1)n (β + 1)n = , h0 2n + α + β + 1 (α + β + 2)n n!

h0 =

2α+β+1 Γ(α + 1)Γ(β + 1) , Γ(α + β + 2)

hn

n+α+β+1 (β + 1)n n! = . (α,β) 2n + α + β + 1 (α + 1)n (α + β + 2)n h0 (Pn (1))2

(31)

In (9.8.3) the numerator factor Γ(n + α + β + 1) in the last line should be Γ(β + 1). When thus corrected, (9.8.3) can be rewritten as: Z ∞ (α,β) Pm (x) Pn(α,β) (x) (x − 1)α (x + 1)β dx = hn δm,n , 1

− 1 − β > α > −1, hn n + α + β + 1 (α + 1)n (β + 1)n = , h0 2n + α + β + 1 (α + β + 2)n n!

h0 =

9

m, n < − 21 (α + β + 1),

2α+β+1 Γ(α

+ 1)Γ(−α − β − 1) . Γ(−β)

(32)

Symmetry Pn(α,β) (−x) = (−1)n Pn(β,α) (x).

(33)

Use (9.8.2) and (9.8.5b) or see [DLMF, Table 18.6.1]. Special values Pn(α,β) (1) =

(α + 1)n , n!

Pn(α,β) (−1) =

(α,β)

(−1)n (β + 1)n , n!

Pn

(−1)

(α,β)

Pn

(1)

=

(−1)n (β + 1)n . (34) (α + 1)n

Use (9.8.1) and (33) or see [DLMF, Table 18.6.1]. Generating functions

Formula (9.8.15) was first obtained by Brafman [109].

Bilateral generating functions (see (4)):

For 0 ≤ r < 1 and x, y ∈ [−1, 1] we have in terms of F4

∞ X 1 (α + β + 1)n n! n (α,β) r Pn (x) Pn(α,β) (y) = (α + 1)n (β + 1)n (1 + r)α+β+1 n=0  r(1 − x)(1 − y) r(1 + x)(1 + y)  × F4 21 (α + β + 1), 21 (α + β + 2); α + 1, β + 1; , , (35) (1 + r)2 (1 + r)2 ∞ X 2n + α + β + 1 (α + β + 2)n n! n (α,β) 1−r r Pn (x) Pn(α,β) (y) = n + α + β + 1 (α + 1)n (β + 1)n (1 + r)α+β+2 n=0  r(1 − x)(1 − y) r(1 + x)(1 + y)  × F4 21 (α + β + 2), 21 (α + β + 3); α + 1, β + 1; , . (36) (1 + r)2 (1 + r)2

Formulas (35) and (36) were first given by Bailey [91, (2.1), (2.3)]. See Stanton [485] for a shorter proof. (However, in the second line of [485, (1)] z and Z should be interchanged.) As observed 1 1 d in Bailey [91, p.10], (36) follows from (35) by applying the operator r− 2 (α+β−1) dr ◦ r 2 (α+β+1) to both sides of (35). In view of (31), formula (36) is the Poisson kernel for Jacobi polynomials. The right-hand side of (36) makes clear that this kernel is positive. See also the discussion in Askey [46, following (2.32)]. Quadratic transformations (α+ 21 )

C2n

(α+ 12 )

C2n

(x)

(α,α)

=

(α+ 1 )

(α+ 1 )

C2n+12 (1)

(α,α)

P2n

(1)

C2n+12 (x)

P2n

(x)

(α,− 12 )

=

(1)

=

(α,α)

P2n+1 (1)

(2x2 − 1)

(α,− 12 )

Pn

(α, 21 )

(α,α)

P2n+1 (x)

Pn

=

x Pn

(2x2 − 1)

(α, 21 )

Pn

,

(37)

.

(38)

(1)

(1)

See p.221, Remarks, last two formulas together with (34) and (49). Or see [DLMF, (18.7.13), (18.7.14)]. 10

Differentiation formulas Each differentiation formula is given in two equivalent forms.  d  (1 − x)α Pn(α,β) (x) = −(n + α) (1 − x)α−1 Pn(α−1,β+1) (x),  dx  (39) d (1 − x) − α Pn(α,β) (x) = −(n + α) Pn(α−1,β+1) (x). dx  d  (1 + x)β Pn(α,β) (x) = (n + β) (1 + x)β−1 Pn(α+1,β−1) (x),  dx  d (1 + x) + β Pn(α,β) (x) = (n + β) Pn(α+1,β−1) (x). dx

(40)

Formulas (39) and (40) follow from [DLMF, (15.5.4), (15.5.6)] together with (9.8.1). They also follow from each other by (33). Generalized Gegenbauer polynomials These are defined by (α,β)

(α,β) S2m (x) := const. Pm (2x2 − 1),

(α,β)

(α,β+1) S2m+1 (x) := const. x Pm (2x2 − 1) (λ,µ)

in the notation of [146, p.156] (see also [K5]), while [K9, Section 1.5.2] has Cn (λ− 12 ,µ− 12 )

× Sn

(41)

(x) = const.

(x). For α, β > −1 we have the orthogonality relation Z 1 (α,β) Sm (x) Sn(α,β) (x) |x|2β+1 (1 − x2 )α dx = 0 (m 6= n).

(42)

−1

For β = α − 1 generalized Gegenbauer polynomials are limit cases of continuous q-ultraspherical polynomials, see (159). If we define the Dunkl operator Tµ by (Tµ f )(x) := f 0 (x) + µ

f (x) − f (−x) x

(43)

and if we choose the constants in (41) as (α,β)

S2m (x) =

(α + β + 1)m (α,β) Pm (2x2 − 1), (β + 1)m

(α,β)

S2m+1 (x) =

(α + β + 1)m+1 (α,β+1) x Pm (2x2 − 1) (β + 1)m+1 (44)

then (see [K6, (1.6)]) (α+1,β)

Tβ+ 1 Sn(α,β) = 2(α + β + 1) Sn−1

.

(45)

2

Formula (45) with (44) substituted gives rise to two differentiation formulas involving Jacobi polynomials which are equivalent to (9.8.7) and (40). Composition of (45) with itself gives (α+2,β)

2 (α,β) Tβ+ = 4(α + β + 1)(α + β + 2) Sn−2 1 Sn

,

2

which is equivalent to the composition of (9.8.7) and (40):  2  d 2β + 1 d (α+2,β) + Pn(α,β) (2x2 − 1) = 4(n + α + β + 1)(n + β) Pn−1 (2x2 − 1). dx2 x dx Formula (46) was also given in [322, (2.4)]. 11

(46)

9.8.1 Gegenbauer / Ultraspherical Notation

(λ)

Here the Gegenbauer polynomial is denoted by Cnλ instead of Cn .

Orthogonality relation Write the right-hand side of (9.8.20) as hn δm,n . Then hn λ (2λ)n = , h0 λ + n n!

1

π 2 Γ(λ + 12 ) h0 = , Γ(λ + 1)

Hypergeometric representation

λ n! hn = . λ 2 λ + n (2λ)n h0 (Cn (1))

(47)

Beside (9.8.19) we have also

bn/2c

Cnλ (x)

 1  X (−1)` (λ)n−` − 2 n, − 12 n + 12 1 n−2` n (λ)n (2x) ; 2 . = = (2x) 2 F1 `! (n − 2`)! n! 1−λ−n x

(48)

`=0

See [DLMF, (18.5.10)]. Special value Cnλ (1) =

(2λ)n . n!

(49)

Use (9.8.19) or see [DLMF, Table 18.6.1]. Expression in terms of Jacobi (λ− 1 ,λ− 1 )

Cnλ (x) Pn 2 2 (x) = , (λ− 1 ,λ− 1 ) Cnλ (1) Pn 2 2 (1) Re: (9.8.21) x2 Cnλ (x) =

Cnλ (x) =

(2λ)n (λ− 1 ,λ− 1 ) Pn 2 2 (x). 1 (λ + 2 )n

(50)

By iteration of recurrence relation (9.8.21):

n2 + 2nλ + λ − 1 (n + 1)(n + 2) λ Cn+2 (x) + C λ (x) 4(n + λ)(n + λ + 1) 2(n + λ − 1)(n + λ + 1) n (n + 2λ − 1)(n + 2λ − 2) λ Cn−2 (x). (51) + 4(n + λ)(n + λ − 1)

Bilateral generating functions ∞ X n=0

n! 1 rn Cnλ (x) Cnλ (y) = 2 F1 (2λ)n (1 − 2rxy + r2 )λ

1 1 2 λ, 2 (λ

+ 1) 4r2 (1 − x2 )(1 − y 2 ) ; (1 − 2rxy + r2 )2 λ + 21

!

(r ∈ (−1, 1), x, y ∈ [−1, 1]). (52) For the proof put β := α in (35), then use (5) and (50). The Poisson kernel for Gegenbauer polynomials can be derived in a similar way from (36), or alternatively by applying the operator 12

d r−λ+1 dr ◦ rλ to both sides of (52): ∞ X λ+n n=0

n! 1 − r2 rn Cnλ (x) Cnλ (y) = λ (2λ)n (1 − 2rxy + r2 )λ+1 ! 1 1 2 2 2 2 (λ + 1), 2 (λ + 2) 4r (1 − x )(1 − y ) × 2 F1 ; (1 − 2rxy + r2 )2 λ + 12

(r ∈ (−1, 1), x, y ∈ [−1, 1]). (53)

Formula (53) was obtained by Gasper & Rahman [234, (4.4)] as a limit case of their formula for the Poisson kernel for continuous q-ultraspherical polynomials. Trigonometric expansions By [DLMF, (18.5.11), (15.8.1)]: Cnλ (cos θ)

n X (λ)k (λ)n−k



−n, λ = e =e ; e−2iθ 2 F1 k! (n − k)! n! 1−λ−n k=0   (λ)n − 1 iλπ i(n+λ)θ λ, 1 − λ ie−iθ −λ = λ e 2 e (sin θ) 2 F1 ; 1 − λ − n 2 sin θ 2 n! ∞ (λ)n X (λ)k (1 − λ)k cos((n − k + λ)θ + 12 (k − λ)π) = . n! (1 − λ − n)k k! (2 sin θ)k+λ i(n−2k)θ

inθ (λ)n

 (54) (55) (56)

k=0

In (55) and (56) we require that 16 π < θ < 65 π. Then the convergence is absolute for λ > conditional for 0 < λ ≤ 21 . By [DLMF, (14.13.1), (14.3.21), (15.8.1)]]:

1 2

and

∞ X
(2λ)n (1 − λ)k (n + 1)k 1−2λ (sin θ) sin (2k + n + 1)θ (57) 1 (n + λ + 1)k k! π 2 Γ(λ + 1) (λ + 1)n k=0    2Γ(λ + 12 ) (2λ)n 1 − λ, n + 1 2iθ 1−2λ i(n+1)θ (sin θ) Im e ;e = 1 2 F1 n+λ+1 π 2 Γ(λ + 1) (λ + 1)n    1 2λ Γ(λ + 12 ) (2λ)n λ, 1 − λ eiθ − 2 iλπ i(n+λ)θ −λ = 1 (sin θ) Re e e ; 2 F1 1 + λ + n 2i sin θ π 2 Γ(λ + 1) (λ + 1)n ∞ 22λ Γ(λ + 21 ) (2λ)n X (λ)k (1 − λ)k cos((n + k + λ)θ − 21 (k + λ)π) . (58) = 1 (2 sin θ)k+λ π 2 Γ(λ + 1) (λ + 1)n k=0 (1 + λ + n)k k!

Cnλ (cos θ) =

2Γ(λ + 12 )

We require that 0 < θ < π in (57) and 16 π < θ < 56 π in (58) The convergence is absolute for λ > 12 and conditional for 0 < λ ≤ 21 . For λ ∈ Z>0 the above series terminate after the term with k = λ − 1. Formulas (57) and (58) are also given in [Sz, (4.9.22), (4.9.25)]. Fourier transform Γ(λ + 1) Γ(λ + 21 ) Γ( 12 )

Z

1

−1

1 Cnλ (y) (1 − y 2 )λ− 2 eixy dy = in 2λ Γ(λ + 1) x−λ Jλ+n (x). λ Cn (1)

See [DLMF, (18.17.17) and (18.17.18)]. 13

(59)

Laplace transforms 2 n! Γ(λ)

∞

Z

2

Hn (tx) tn+2λ−1 e−t dt = Cnλ (x).

(60)

0

See Nielsen [K24, p.48, (4) with p.47, (1) and p.28, (10)] (1918) or Feldheim [K10, (28)] (1942). 2 Γ(λ + 12 )

Z 0

1

1 Cnλ (t) 2 2 2 (1 − t2 )λ− 2 t−1 (x/t)n+2λ+1 e−x /t dt = 2−n Hn (x) e−x λ Cn (1)

(λ > − 21 ). (61)

Use Askey & Fitch [K2, (3.29)] for α = ± 12 together with (33), (37), (38), (86) and (87). Addition formula (see [AAR, (9.8.50 )]]) Rn(α,α)

n
X (−1)k (−n)k (n + 2α + 1)k xy + (1 − x ) (1 − y ) t = 22k ((α + 1)k )2 2

1 2

1 2

2

k=0

× (1 −

(α+k,α+k) x2 )k/2 Rn−k (x) (1

(α+k,α+k)

− y 2 )k/2 Rn−k

(α− 21 ,α− 12 )

(y) ωk

(α− 12 ,α− 12 )

Rk

(t), (62)

where R1

α β −1 (1 − x) (1 + x) dx (α,β) (x))2 (1 − x)α (1 + x)β −1 (Rn

Rn(α,β) (x) := Pn(α,β) (x)/Pn(α,β) (1),

ωn(α,β) := R 1

. dx

9.8.2 Chebyshev In addition to the Chebyshev polynomials Tn of the first kind (9.8.35) and Un of the second kind (9.8.36), (− 21 ,− 12 )

Tn (x) :=

Pn

(x)

(− 1 ,− 1 ) Pn 2 2 (1)

= cos(nθ),

( 1 , 21 )

Un (x) := (n + 1)

Pn 2

(x)

(1,1) Pn 2 2 (1)

=

x = cos θ,

sin((n + 1)θ) , sin θ

(63)

x = cos θ,

(64)

we have Chebyshev polynomials Vn of the third kind and Wn of the fourth kind, (− 21 , 12 )

Vn (x) :=

Pn

(x)

(− 1 , 1 ) Pn 2 2 (1)

=

cos((n + 21 )θ) , cos( 12 θ)

( 1 ,− 12 )

x = cos θ,

sin((n + 12 )θ) Wn (x) := (2n + 1) = , ( 1 ,− 1 ) sin( 21 θ) Pn 2 2 (1) Pn 2

(x)

x = cos θ,

(65)

(66)

see [K22, Section 1.2.3]. Then there is the symmetry Vn (−x) = (−1)n Wn (x). 14

(67)

The names of Chebyshev polynomials of the third and fourth kind and the notation Vn (x) are due to Gautschi [K11]. The notation Wn (x) was first used by Mason [K21]. Names and notations for Chebyshev polynomials of the third and fourth kind are interchanged in [AAR, Remark 2.5.3] and [DLMF, Table 18.3.1].

9.9 Pseudo Jacobi (or Routh-Romanovski) In this section in [KLS] the pseudo Jacobi polynomial Pn (x; ν, N ) in (9.9.1) is considered for N ∈ Z≥0 and n = 0, 1, . . . , n. However, we can more generally take − 21 < N ∈ R (so here I overrule my convention formulated in the beginning of this paper), N0 integer such that N − 21 ≤ N0 < N + 21 , and n = 0, 1, . . . , N0 (see [382, §5, case A.4]). The orthogonality relation (9.9.2) is valid for m, n = 0, 1, . . . , N0 . History These polynomials were first obtained by Routh [K27] in 1885, and later, independently, by Romanovski [463] in 1929. Limit relation: See also (142).

Pseudo big q-Jacobi −→ Pseudo Jacobi

References See also [Ism, §20.1], [51], [384], [K17], [K20], [K25].

9.10 Meixner History In 1934 Meixner [406] (see (1.1) and case IV on pp. 10, 11 and 12) gave the orthogonality measure for the polynomials Pn given by the generating function xu(t)

e

f (t) =

∞ X

Pn (x)

n=0

tn , n!

where eu(t) =



1 − βt 1 − αt



k2

1 α−β

,

f (t) =

(1 − βt) β(α−β) k2

(k2 < 0; α > β > 0 or α < β < 0).

(1 − αt) α(α−β)

Then Pn can be expressed as a Meixner polynomial:   x + k2 α−1 Pn (x) = (−k2 (αβ)−1 )n β n Mn − , −k2 (αβ)−1 , βα−1 . α−β In 1938 Gottlieb [K15, §2] introduces polynomials ln “of Laguerre type” which turn out to be special Meixner polynomials: ln (x) = e−nλ Mn (x; 1, e−λ ). Uniqueness of orthogonality measure The coefficient of pn−1 (x) in (9.10.4) behaves as O(n2 ) as n → ∞. Hence (2) holds, by which the orthogonality measure is unique. 15

9.11 Krawtchouk Special values By (9.11.1) and the binomial formula: Kn (N ; p, N ) = (1 − p−1 )n .

Kn (0; p, N ) = 1,

(68)

The self-duality (p.240, Remarks, first formula) Kn (x; p, N ) = Kx (n; p, N )

(n, x ∈ {0, 1, . . . , N })

(69)

(x ∈ {0, 1, . . . , N }).

(70)

combined with (68) yields: KN (x; p, N ) = (1 − p−1 )x Symmetry

By the orthogonality relation (9.11.2): Kn (N − x; p, N ) = Kn (x; 1 − p, N ). Kn (N ; p, N )

(71)

By (71) and (69) we have also KN −n (x; p, N ) = Kn (x; 1 − p, N ) KN (x; p, N )

(n, x ∈ {0, 1, . . . , N }),

(72)

and, by (72), (71) and (68),  KN −n (N − x; p, N ) =

p p−1

n+x−N Kn (x; p, N )

(n, x ∈ {0, 1, . . . , N }).

(73)

A particular case of (71) is: Kn (N − x; 12 , N ) = (−1)n Kn (x; 21 , N ).

(74)

K2m+1 (N ; 12 , 2N ) = 0.

(75)

Hence From (9.11.11): ( 12 )m . (−N + 21 )m

(76)

( 21 )m Rm (x2 ; − 21 , − 12 , N ), 1 (−N + 2 )m

(77)

K2m (N ; 12 , 2N ) = Quadratic transformations K2m (x + N ; 12 , 2N ) =

K2m+1 (x + N ; 12 , 2N ) = − K2m (x + N + 1; 12 , 2N + 1) = K2m+1 (x + N + 1;

1 2 , 2N

( 23 )m x Rm (x2 − 1; 12 , 21 , N − 1), N (−N + 21 )m

( 12 )m Rm (x(x + 1); − 12 , 12 , N ), (−N − 12 )m

( 32 )m + 1) = (x + 12 ) Rm (x(x + 1); 12 , − 21 , N ), 1 (−N − 2 )m+1

(78) (79) (80)

where Rm is a dual Hahn polynomial (9.6.1). For the proofs use (9.6.2), (9.11.2), (9.6.4) and (9.11.4). 16

Generating functions N   X N

Km (x; p, N )Kn (x; q, N )z x x x=0       p − z + pz m q − z + qz n (p − z + pz)(q − z + qz) N −m−n = (1 + z) Km n; − ,N . p q z (81)

This follows immediately from Rosengren [K26, (3.5)], which goes back to Meixner [K23].

9.12 Laguerre Notation

(α)

Here the Laguerre polynomial is denoted by Lαn instead of Ln .

Hypergeometric representation   −n (α + 1)n F ; x 1 1 n! α+1   n (−x) −n, −n − α 1 = ;− 2 F0 n! − x n (−x) = Cn (n + α; x), n!

Lαn (x) =

(82) (83) (84)

where Cn in (84) is a Charlier polynomial. Formula (82) is (9.12.1). Then (83) follows by reversal of summation. Finally (84) follows by (83) and (96). It is also the remark on top of p.244 in [KLS], and it is essentially [416, (2.7.10)]. Uniqueness of orthogonality measure The coefficient of pn−1 (x) in (9.12.4) behaves as O(n2 ) as n → ∞. Hence (2) holds, by which the orthogonality measure is unique. Special value Lαn (0) =

(α + 1)n . n!

(85)

Use (9.12.1) or see [DLMF, 18.6.1)]. Quadratic transformations H2n (x) = (−1)n 22n n! L−1/2 (x2 ), n

(86)

H2n+1 (x) = (−1)n 22n+1 n! x Ln1/2 (x2 ).

(87)

See p.244, Remarks, last two formulas. Or see [DLMF, (18.7.19), (18.7.20)]. 17

Fourier transform 1 Γ(α + 1)

Z 0

∞

Lαn (y) −y α ixy yn n e y e dy = i , Lαn (0) (iy + 1)n+α+1

(88)

see [DLMF, (18.17.34)]. Differentiation formulas Each differentiation formula is given in two equivalent forms.   d x + α Lαn (x) = (n + α) Lα−1 n (x). dx

d (xα Lαn (x)) = (n + α) xα−1 Lnα−1 (x), dx



d −x α
e Ln (x) = −e−x Lα+1 n (x), dx

 d − 1 Lαn (x) = −Lα+1 n (x). dx

(89)

(90)

Formulas (89) and (90) follow from [DLMF, (13.3.18), (13.3.20)] together with (9.12.1). Generalized Hermite polynomials by

See [146, p.156], [K9, Section 1.5.1]. These are defined

µ− 1

µ (x) := const. Lm 2 (x2 ), H2m

µ+ 1

µ (x) := const. x Lm 2 (x2 ). H2m+1

(91)

Then for µ > − 12 we have orthogonality relation Z

∞

−∞

2

µ Hm (x) Hnµ (x) |x|2µ e−x dx = 0

(m 6= n).

(92)

Let the Dunkl operator Tµ be defined by (43). If we choose the constants in (91) as µ H2m (x) =

(−1)m (2m)! µ− 21 2 Lm (x ), (µ + 21 )m

µ H2m+1 (x) =

(−1)m (2m + 1)! µ+ 1 x Lm 2 (x2 ) 1 (µ + 2 )m+1

(93)

then (see [K6, (1.6)]) µ Tµ Hnµ = 2n Hn−1 .

(94)

Formula (94) with (93) substituted gives rise to two differentiation formulas involving Laguerre polynomials which are equivalent to (9.12.6) and (89). Composition of (94) with itself gives µ Tµ2 Hnµ = 4n(n − 1) Hn−2 ,

which is equivalent to the composition of (9.12.6) and (89): 

2α + 1 d d2 + 2 dx x dx



Lαn (x2 ) = −4(n + α) Lαn−1 (x2 ).

18

(95)

9.14 Charlier Hypergeometric representation 

 −n, −x 1 ;− Cn (x; a) = 2 F0 − a   −n (−x)n = ;a 1 F1 x−n+1 an n! = Lx−n (a), (−a)n n

(96) (97) (98)

where Lαn (x) is a Laguerre polynomial. Formula (96) is (9.14.1). Then (97) follows by reversal of the summation. Finally (98) follows by (97) and (9.12.1). It is also the Remark on p.249 of [KLS], and it was earlier given in [416, (2.7.10)]. Uniqueness of orthogonality measure The coefficient of pn−1 (x) in (9.14.4) behaves as O(n) as n → ∞. Hence (2) holds, by which the orthogonality measure is unique.

9.15 Hermite Uniqueness of orthogonality measure The coefficient of pn−1 (x) in (9.15.4) behaves as O(n) as n → ∞. Hence (2) holds, by which the orthogonality measure is unique. Fourier transforms 1 √ 2π

Z

∞

1 2

1 2

Hn (y) e− 2 y eixy dy = in Hn (x) e− 2 x ,

(99)

−∞

see [AAR, (6.1.15) and Exercise 6.11]. Z ∞ 1 2 1 2 √ Hn (y) e−y eixy dy = in xn e− 4 x , π −∞

(100)

see [DLMF, (18.17.35)]. in √ 2 π

Z

∞

1 2

2

y n e− 4 y e−ixy dy = Hn (x) e−x ,

(101)

−∞

see [AAR, (6.1.4)].

14.1 Askey-Wilson Symmetry

The Askey-Wilson polynomials pn (x; a, b, c, d | q) are — symmetric in a, b, c, d.

This follows from the orthogonality relation (14.1.2) together with the value of its coefficient of xn given in (14.1.5b). Alternatively, combine (14.1.1) with [GR, (III.15)]. As a consequence, it is sufficient to give generating function (14.1.13). Then the generating functions (14.1.14), (14.1.15) will follow by symmetry in the parameters. 19

Basic hypergeometric representation In addition to (14.1.1) we have (in notation (8)): pn (cos θ; a, b, c, d | q) =

(ae−iθ , be−iθ , ce−iθ , de−iθ ; q)n inθ e (e−2iθ ; q)n
× 8 W7 q −n e2iθ ; aeiθ , beiθ , ceiθ , deiθ , q −n ; q, q 2−n /(abcd) . (102)

This follows from (14.1.1) by combining (III.15) and (III.19) in [GR]. It is also given in [513, (4.2)], but be aware for some slight errors. The symmetry in a, b, c, d is evident from (102). Special value −1 1 2 (a + a ); a, b, c, d | q arguments 21 (b + b−1 ), 12 (c + c−1 )




pn

and similarly for

= a−n (ab, ac, ad; q)n ,

(103)

and 12 (d + d−1 ) by symmetry of pn in a, b, c, d.

Trivial symmetry pn (−x; a, b, c, d | q) = (−1)n pn (x; −a, −b, −c, −d | q).

(104)

Both (103) and (104) are obtained from (14.1.1). Re: (14.1.5)

Let pn (x) :=

pn (x; a, b, c, d | q) = xn + e kn xn−1 + · · · . 2n (abcdq n−1 ; q)n

(105)

Then

(1 − q n )(a + b + c + d − (abc + abd + acd + bcd)q n−1 ) e kn = − . (106) 2(1 − q)(1 − abcdq 2n−2 )
This follows because k˜n − k˜n+1 equals the coefficient 21 a + a−1 − (An + Cn ) of pn (x) in (14.1.5).

Generating functions

Rahman [449, (4.1), (4.9)] gives:

∞ X (abcdq −1 ; q)n an n t pn (cos θ; a, b, c, d | q) (ab, ac, ad, q; q)n

n=0

1

1

1

1

(abcdtq −1 ; q)∞ (abcdq −1 ) 2 , −(abcdq −1 ) 2 , (abcd) 2 , −(abcd) 2 , aeiθ , ae−iθ = ; q, q 6 φ5 (t; q)∞ ab, ac, ad, abcdtq −1 , qt−1 +

!

(abcdq −1 , abt, act, adt, aeiθ , ae−iθ ; q)∞ (ab, ac, ad, t−1 , ateiθ , ate−iθ ; q)∞ 1

× 6 φ5

1

1

1

t(abcdq −1 ) 2 , −t(abcdq −1 ) 2 , t(abcd) 2 , −t(abcd) 2 , ateiθ , ate−iθ ; q, q abt, act, adt, abcdt2 q −1 , qt

! (|t| < 1). (107)

In the limit (108) the first term on the right-hand side of (107) tends to the left-hand side of (9.1.15), while the second term tends formally to 0. The special case ad = bc of (107) was earlier given in [236, (4.1), (4.6)]. 20

Limit relations Askey-Wilson −→ Wilson Instead of (14.1.21) we can keep a polynomial of degree n while the limit is approached: pn (1 − 21 x(1 − q)2 ; q a , q b , q c , q d | q) = Wn (x; a, b, c, d). q→1 (1 − q)3n lim

(108)

For the proof first derive the corresponding limit for the monic polynomials by comparing (14.1.5) with (9.4.4). Askey-Wilson −→ Continuous Hahn Instead of (14.4.15) we can keep a polynomial of degree n while the limit is approached: pn cos φ − x(1 − q) sin φ; q a eiφ , q b eiφ , q a e−iφ , q b e−iφ | q lim q↑1 (1 − q)2n




= (−2 sin φ)n n! pn (x; a, b, a, b)

(0 < φ < π). (109)

Here the right-hand side has a continuous Hahn polynomial (9.4.1). For the proof first derive the corresponding limit for the monic polynomials by comparing (14.1.5) with (9.1.5). In fact, define the monic polynomial
pn cos φ − x(1 − q) sin φ; q a eiφ , q b eiφ , q a e−iφ , q b e−iφ | q . pen (x) := (−2(1 − q) sin φ)n (abcdq n−1 ; q)n Then it follows from (14.1.5) that x pen (x) = pen+1 (x) +

en + C en en−1 C en (1 − q a )eiφ + (1 − q −a )e−iφ + A A pen−1 (x), pen (x) + 2(1 − q) sin φ (1 − q)2 sin2 φ

en and C en are as given after (14.1.3) with a, b, c, d replaced by q a eiφ , q b eiφ , q a e−iφ , q b e−iφ . where A Then the recurrence equation for pen (x) tends for q ↑ 1 to the recurrence equation (9.4.4) with c = a, d = b. Askey-Wilson −→ Meixner-Pollaczek Instead of (14.9.15) we can keep a polynomial of degree n while the limit is approached:
pn cos φ − x(1 − q) sin φ; q λ eiφ , 0, q λ e−iφ , 0 | q lim = n! Pn(λ) (x; π − φ) (0 < φ < π). (110) q↑1 (1 − q)n Here the right-hand side has a Meixner-Pollaczek polynomial (9.7.1). For the proof first derive the corresponding limit for the monic polynomials by comparing (14.1.5) with (9.7.4). In fact, define the monic polynomial
pn cos φ − x(1 − q) sin φ; q λ eiφ , 0, q λ e−iφ , 0 | q . pen (x) := (−2(1 − q) sin φ)n 21

Then it follows from (14.1.5) that x pen (x) = pen+1 (x) +

en en + C en en−1 C (1 − q λ )eiφ + (1 − q −λ )e−iφ + A A pen (x) + pen−1 (x), 2(1 − q) sin φ (1 − q)2 sin2 φ

en and C en are as given after (14.1.3) with a, b, c, d replaced by q λ eiφ , 0, q λ e−iφ , 0. Then where A the recurrence equation for pen (x) tends for q ↑ 1 to the recurrence equation (9.7.4). References See also Koornwinder [K18].

14.2 q-Racah Symmetry Rn (x; α, β, q −N −1 , δ | q) =

(βq, αδ −1 q; q)n n δ Rn (δ −1 x; β, α, q −N −1 , δ −1 | q). (αq, βδq; q)n

(111)

This follows from (14.2.1) combined with [GR, (III.15)]. In particular, Rn (x; α, β, q −N −1 , −1 | q) =

(βq, −αq; q)n (−1)n Rn (−x; β, α, q −N −1 , −1 | q), (αq, −βq; q)n

(112)

and Rn (x; α, α, q −N −1 , −1 | q) = (−1)n Rn (−x; α, α, q −N −1 , −1 | q),

(113)

Trivial symmetry Clearly from (14.2.1): Rn (x; α, β, γ, δ | q) = Rn (x; βδ, αδ −1 , γ, δ | q) = Rn (x; γ, αβγ −1 , α, γδα−1 | q).

(114)

For α = q −N −1 this shows that the three cases αq = q −N or βδq = q −N or γq = q −N of (14.2.1) are not essentially different. Duality

It follows from (14.2.1) that

Rn (q −y + γδq y+1 ; q −N −1 , β, γ, δ | q) = Ry (q −n + βq n−N ; γ, δ, q −N −1 , β | q)

(n, y = 0, 1, . . . , N ). (115)

14.3 Continuous dual q-Hahn The continuous dual q-Hahn polynomials are the special case d = 0 of the Askey-Wilson polynomials: pn (x; a, b, c | q) := pn (x; a, b, c, 0 | q). Hence all formulas in §14.3 are specializations for d = 0 of formulas in §14.1. 22

14.4 Continuous q-Hahn The continuous q-Hahn polynomials are the special case of Askey-Wilson polynomials with parameters aeiφ , beiφ , ae−iφ , be−iφ : pn (x; a, b, φ | q) := pn (x; aeiφ , beiφ , ae−iφ , be−iφ | q). In [72, (4.29)] and [GR, (7.5.43)] (who write pn (x; a, b | q), x = cos(θ + φ)) and in [KLS, §14.4] (who writes pn (x; a, b, c, d; q), x = cos(θ + φ)) the parameter dependence on φ is incorrectly omitted. Since all formulas in §14.4 are specializations of formulas in §14.1, there is no real need to give these specializations explicitly. In particular, the limit (14.4.15) is in fact a limit from Askey-Wilson to continuous q-Hahn. See also (109).

14.5 Big q-Jacobi Different notation See p.442, Remarks: −1

Pn (x; a, b, c, d; q) := Pn (qac

−1

x; a, b, −ac

 d; q) = 3 φ2

 q −n , q n+1 ab, qac−1 x ; q, q . qa, −qac−1 d

(116)

Furthermore, Pn (x; a, b, c, d; q) = Pn (λx; a, b, λc, λd; q),

(117)

Pn (x; a, b, c; q) = Pn (−q −1 c−1 x; a, b, −ac−1 , 1; q)

(118)

Orthogonality relation (equivalent to (14.5.2), see also [K19, (2.42), (2.41), (2.36), (2.35)]). Let c, d > 0 and either a ∈ (−c/(qd), 1/q), b ∈ (−d/(cq), 1/q) or a/c = −b/d ∈ / R. Then Z c (qx/c, −qx/d; q)∞ Pm (x; a, b, c, d; q)Pn (x; a, b, c, d; q) dq x = hn δm,n , (119) (qax/c, −qbx/d; q)∞ −d where 1 hn = q 2 n(n−1) h0



q 2 a2 d c

and h0 = (1 − q)c

n

1 − qab (q, qb, −qbc/d; q)n 2n+1 1−q ab (qa, qab, −qad/c; q)n

(q, −d/c, −qc/d, q 2 ab; q)∞ . (qa, qb, −qbc/d, −qad/c; q)∞

(120)

(121)

Other hypergeometric representation and asymptotics  −n −n −1 −1  (−qbd−1 x; q)n q , q b , cx Pn (x; a, b, c, d; q) = φ ; q, q (122) 3 2 (−q −n a−1 cd−1 ; q)n qa, −q −n b−1 dx−1  −n −n −1  (qb, cx−1 ; q)n q , q a , −qbd−1 x −1 n n+1 −1 = (qac x) ; q, −q ac d (123) 3 φ2 (qa, −qac−1 d; q)n qb, q 1−n c−1 x n 1 (qb, q; q)n X (cx−1 ; q)n−k (−qbd−1 x; q)k = (qac−1 x)n (−1)k q 2 k(k−1) (−dx−1 )k . −1 (−qac d; q)n (q, qa; q)n−k (qb, q; q)k k=0

(124) 23

Formula (122) follows from (116) by [GR, (III.11)] and next (123) follows by series inversion [GR, Exercise 1.4(ii)]. Formulas (122) and (124) are also given in [Ism, (18.4.28), (18.4.29)]. It follows from (123) or (124) that (see [298, (1.17)] or [Ism, (18.4.31)]) lim (qac−1 x)−n Pn (x; a, b, c, d; q) =

n→∞

(cx−1 , −dx−1 ; q)∞ , (−qac−1 d, qa; q)∞

(125)

uniformly for x in compact subsets of C\{0}. (Exclusion of the spectral points x = cq m , dq m (m = 0, 1, 2, . . .), as was done in [298] and [Ism], is not necessary. However, while (125) yields 0 at these points, a more refined asymptotics at these points is given in [298] and [Ism].) For the proof of (125) use that −1

lim (qac

n→∞

x)

−n

  (qb, cx−1 ; q)n −qbd−1 x −1 Pn (x; a, b, c, d; q) = ; q, −dx , 1 φ1 (qa, −qac−1 d; q)n qb

(126)

which can be evaluated by [GR, (II.5)]. Formula (126) follows formally from (123), and it follows rigorously, by dominated convergence, from (124). Symmetry

(see [K19, §2.5]). Pn (−x; a, b, c, d; q) = Pn (x; b, a, d, c; q). Pn (−d/(qb); a, b, c, d; q)

(127)

Special values Pn (c/(qa); a, b, c, d; q) = 1,   ad n (qb, −qbc/d; q)n Pn (−d/(qb); a, b, c, d; q) = − , bc (qa, −qad/c; q)n  n 1 (−qbc/d; q)n ad n(n+1) Pn (c; a, b, c, d; q) = q 2 , c (−qad/c; q)n 1 (qb; q)n Pn (−d; a, b, c, d; q) = q 2 n(n+1) (−a)n . (qa; q)n

(128) (129) (130) (131)

Quadratic transformations (see [K19, (2.48), (2.49)] and (161)). These express big q-Jacobi polynomials Pm (x; a, a, 1, 1; q) in terms of little q-Jacobi polynomials (see §14.12). pn (x2 ; q −1 , a2 ; q 2 ) , pn ((qa)−2 ; q −1 , a2 ; q 2 ) qax pn (x2 ; q, a2 ; q 2 ) P2n+1 (x; a, a, 1, 1; q) = . pn ((qa)−2 ; q, a2 ; q 2 ) P2n (x; a, a, 1, 1; q) =

24

(132) (133)

Hence, by (14.12.1), [GR, Exercise 1.4(ii)] and (161),  −n −n+1  q ,q (qa2 ; q 2 )n n 2 −2 Pn (x; a, a, 1, 1; q) = (qax) 2 φ1 −2n+1 −2 ; q , (ax) q a (qa2 ; q)n

(134)

1

[ 2 n] X (q; q)n (qa2 ; q 2 )n−k n k k(k−1) = (qa) (−1) q xn−2k . (qa2 ; q)n (q 2 ; q 2 )k (q; q)n−2k

(135)

k=0

1

1

q-Chebyshev polynomials In (116), with c = d = 1, the cases a = b = q − 2 and a = b = q 2 can be considered as q-analogues of the Chebyshev polynomials of the first and second kind, respectively (§9.8.2) because of the limit (14.5.17). The quadratic relations (132), (133) can also be specialized to these cases. The definition of the q-Chebyshev polynomials may vary by normalization and by dilation of argument. They were considered in [K4]. By [24, p.279] and (132), (133), the Al-Salam-Ismail polynomials Un (x; a, b) (q-dependence suppressed) in the case a = q can be expressed as q-Chebyshev polynomials of the second kind: 1

Un (x, q, b) = (q −3 b) 2 n

1 1 1 1 − q n+1 Pn (b− 2 x; q 2 , q 2 , 1, 1; q). 1−q

Similarly, by [K7, (5.4), (5.1), (5.3)] and (132), (133), Cigler’s q-Chebyshev polynomials Tn (x, s, q) and Un (x, s, q) can be expressed in terms of the q-Chebyshev cases of (116): 1

1

1

1

Tn (x, s, q) = (−s) 2 n Pn ((−qs)− 2 x; q − 2 , q − 2 , 1, 1; q), 1

Un (x, s, q) = (−q −2 s) 2 n

1 1 1 1 − q n+1 Pn ((−qs)− 2 x; q 2 , q 2 , 1, 1; q). 1−q

Limit to Discrete q-Hermite I lim a−n Pn (x; a, a, 1, 1; q) = q n hn (x; q).

(136)

a→0

Here hn (x; q) is given by (14.28.1). For the proof of (136) use (122). ∞ Pseudo big q-Jacobi polynomials Let a, b, c, d ∈ C, z+ > 0, z− < 0 such that (ax,bx;q) (cx,dx;q)∞ > 0 for x ∈ z− q Z ∪ z+ q Z . Then (ab)/(qcd) > 0. Assume that (ab)/(qcd) < 1. Let N be the largest nonnegative integer such that q 2N > (ab)/(qcd). Then

Z Pm (cx; c/b, d/a, c/a; q) Pn (cx; c/b, d/a, c/a; q) z−

q Z ∪z

+

qZ

(ax, bx; q)∞ dq x = hn δm,n (cx, dx; q)∞ (m, n = 0, 1, . . . , N ), (137)

where hn = (−1)n h0



c2 ab

n

1

q 2 n(n−1) q 2n

(q, qd/a, qd/b; q)n 1 − qcd/(ab) (qcd/(ab), qc/a, qc/b; q)n 1 − q 2n+1 cd/(ab) 25

(138)

and Z

(ax, bx; q)∞ (q, a/c, a/d, b/c, b/d; q)∞ θ(z− /z+ , cdz− z+ ; q) dq x = (1 − q)z+ . (cx, dx; q) (ab/(qcd); q)∞ θ(cz− , dz− , cz+ , dz+ ; q) ∞ z− + (139) See Groenevelt & Koelink [K16, Prop. 2.2]. Formula (139) was first given by Slater [K29, (5)] as an evaluation of a sum of two 2 ψ2 series. The same formula is given in Slater [471, (7.2.6)] and in [GR, Exercise 5.10], but in both cases with the same slight error, see [K16, 2nd paragraph after Lemma 2.1] for correction. The theta function is given by (9). Note that h0 =

q Z ∪z

qZ

Pn (cx; c/b, d/a, c/a; q) = Pn (−q −1 ax; c/b, d/a, −a/b, 1; q).

(140)

In [K14] the weights of the pseudo big q-Jacobi polynomials occur in certain measures on the space of N -point configurations on the so-called extended Gelfand-Tsetlin graph. Limit relations Pseudo big q-Jacobi −→ Discrete Hermite II 1 hn (x; q). lim in q 2 n(n−1) Pn (q −1 a−1 ix; a, a, 1, 1; q) = e

a→∞

(141)

For the proof use (135) and (196). Note that Pn (q −1 a−1 ix; a, a, 1, 1; q) is obtained from the right-hand side of (141) by replacing a, b, c, d by −ia−1 , ia−1 , i, −i. Pseudo big q-Jacobi −→ Pseudo Jacobi 1

lim Pn (iq 2 (−N −1+iν) x; −q −N −1 , −q −N −1 , q −N +iν−1 ; q) = q↑1

Pn (x; ν, N ) . Pn (−i; ν, N )

(142)

Here the big q-Jacobi polynomial on the left-hand side equals Pn (cx; c/b, d/a, c/a; q) with 1 1 1 1 a = iq 2 (N +1−iν) , b = −iq 2 (N +1+iν) , c = iq 2 (−N −1+iν) , d = −iq 2 (−N −1−iν) .

14.7 Dual q-Hahn Orthogonality relation More generally we have (14.7.2) with positive weights in any of the following cases: (i) 0 < γq < 1, 0 < δq < 1; (ii) 0 < γq < 1, δ < 0; (iii) γ < 0, δ > q −N ; (iv) γ > q −N , δ > q −N ; (v) 0 < qγ < 1, δ = 0. This also follows by inspection of the positivity of the coefficient of pn−1 (x) in (14.7.4). Case (v) yields Affine q-Krawtchouk in view of (14.7.13). Symmetry
n (δ −1 q −N ; q)n γδq N +1 Rn (γ −1 δ −1 q −1−N x; δ −1 q −N −1 , γ −1 q −N −1 , N | q). (γq; q)n (143) This follows from (14.7.1) combined with [GR, (III.11)]. Rn (x; γ, δ, N | q) =

26

14.8 Al-Salam-Chihara Symmetry The Al-Salam-Chihara polynomials Qn (x; a, b | q) are symmetric in a, b. This follows from the orthogonality relation (14.8.2) together with the value of its coefficient of xn given in (14.8.5b). q −1 -Al-Salam-Chihara For x ∈ Z≥0 :

Re: (14.8.1)


1 Qn ( 21 (aq −x + a−1 q x );a, b | q −1 ) = (−1)n bn q − 2 n(n−1) (ab)−1 ; q n  −n −x −2 x  q ,q ,a q n −1 × 3 φ1 ; q, q ab (ab)−1   −1 q −x , a−2 q x −1 x − 21 x(x+1) (qba ; q)x n+1 = (−ab ) q ; q, q 2 φ1 (a−1 b−1 ; q)x qba−1 1 (qba−1 ; q)x px (q n ; ba−1 , (qab)−1 ; q). = (−ab−1 )x q − 2 x(x+1) −1 −1 (a b ; q)x

(144) (145) (146)

Formula (144) follows from the first identity in (14.8.1). Next (145) follows from [GR, (III.8)]. Finally (146) gives the little q-Jacobi polynomials (14.12.1). See also [79, §3]. Orthogonality ∞ X (1 − q 2x a−2 )(a−2 , (ab)−1 ; q)x x=0

(1 −

a−2 )(q, bqa−1 ; q) =

x

2

(ba−1 )x q x (Qm Qn )( 21 (aq −x + a−1 q x ); a, b | q −1 )

(qa−2 ; q)∞ 2 (q, (ab)−1 ; q)n (ab)n q −n δm,n (ba−1 q; q)∞

(ab > 1, qb < a). (147)

This follows from (146) together with (14.12.2) and the completeness of the orthogonal system of the little q-Jacobi polynomials, See also [79, §3]. An alternative proof is given in [64]. There combine (3.82) with (3.81), (3.67), (3.40). Normalized recurrence relation xpn (x) = pn+1 (x) + 21 (a + b)q −n pn (x) + 41 (q −n − 1)(abq −n+1 − 1)pn−1 (x),

(148)

where Qn (x; a, b | q −1 ) = 2n pn (x).

14.9 q-Meixner-Pollaczek The q-Meixner-Pollaczek polynomials are the special case of Askey-Wilson polynomials with parameters aeiφ , 0, ae−iφ , 0: Pn (x; a, φ | q) :=

1 pn (x; aeiφ , 0, ae−iφ , 0 | q) (q; q)n 27

(x = cos(θ + φ)).

In [KLS, §14.9] the parameter dependence on φ is incorrectly omitted. Since all formulas in §14.9 are specializations of formulas in §14.1, there is no real need to give these specializations explicitly. See also (110). There is an error in [KLS, (14.9.6), (14.9.8)]. Read x = cos(θ + φ) instead of x = cos θ.

14.10 Continuous q-Jacobi Symmetry 1

Pn(α,β) (−x | q) = (−1)n q 2 (α−β)n Pn(β,α) (x | q).

(149)

This follows from (104) and (14.1.19).

14.10.1 Continuous q-ultraspherical / Rogers Re: (14.10.17) 1

1

1

1

q − 2 n , βq 2 n , β 2 eiθ , β 2 e−iθ 1 1 (β 2 ; q)n − 1 n Cn (cos θ; β | q) = β 2 4 φ3 ;q2,q2 1 1 1 1 (q; q)n −β, β 2 q 4 , −β 2 q 4

! ,

(150)

see [GR, (7.4.13), (7.4.14)]. Special value (see [63, (3.23)]) Cn

1 1 2 2 (β


(β 2 ; q)n − 1 n 1 + β − 2 ); β | q = β 2 . (q; q)n

(151)

(another q-difference equation). Let Cn [eiθ ; β | q] := Cn (cos θ; β | q).

Re: (14.10.21)

1 1 1 1 1 − βz 2 1 − βz −2 2 z; β | q] + C [q Cn [q − 2 z; β | q] = (q − 2 n + q 2 n β) Cn [z; β | q], n 2 −2 1−z 1−z

(152)

see [351, (6.10)]. Re: (14.10.23)

This can also be written as 1

1

1

Cn [q 2 z; β | q] − Cn [q − 2 z; β | q] = q − 2 n (β − 1)(z − z −1 )Cn−1 [z; qβ | q].

(153)

Two other shift relations follow from the previous two equations: 1

1

1

1

(β + 1)Cn [q 2 z; β | q] = (q − 2 n + q 2 n β)Cn [z; β | q] + q − 2 n (β − 1)(z − βz −1 )Cn−1 [z; qβ | q], (154) (β + 1)Cn [q

− 21

z; β | q] = (q

− 12 n

+q

1 n 2

β)Cn [z; β | q] + q

28

− 21 n

(β − 1)(z −1 − βz)Cn−1 [z; qβ | q]. (155)

Trigonometric representation (see p.473, Remarks, first formula) Cn (cos θ; β | q) =

n X (β; q)k (β; q)n−k

(q; q)k (q; q)n−k

k=0

Limit for q ↓ −1

ei(n−2k)θ .

(156)

(see [63, pp. 74–75]). By (156) and (54) we obtain 1

(λ+1)

2 lim C2m (x; −q λ | − q) = Cm

q↑1

1

1

(λ+1)

2 (2x2 − 1) + Cm−1

(λ+1)

2 lim C2m+1 (x; −q λ | − q) = 2x Cm

q↑1

(2x2 − 1),

(2x2 − 1).

By (50) and [HTF2, 10.6(36)] this can be rewritten as (λ)m ( 21 λ, 12 λ−1) (2x2 − 1), Pm q↑1 ( 12 λ)m (λ + 1)m ( 1 λ, 1 λ) lim C2m+1 (x; −q λ | − q) = 2 1 x Pm2 2 (2x2 − 1). q↑1 ( 2 λ + 1)m lim C2m (x; −q λ | − q) =

(157) (158)

By (41) the limits (157), (158) imply that ( 1 λ, 21 λ−1)

lim Cn (x; −q λ | − q) = const. Sn2 q↑1

(x),

(159)

where the right-hand side gives a one-parameter subclass of the generalized Gegenbauer polynomial. Note that in [K13, Section 7.1] the generalized Gegenbauer polynomials are also observed as fitting in the q = −1 Askey scheme, but the limit (159) is not observed there.

14.11 Big q-Laguerre Symmetry

The big q-Laguerre polynomials Pn (x; a, b; q) are symmetric in a, b.

This follows from (14.11.1). As a consequence, it is sufficient to give generating function (14.11.11). Then the generating function (14.1.12) will follow by symmetry in the parameters.

14.12 Little q-Jacobi Notation

Here the little q-Jacobi polynomial is denoted by pn (x; a, b; q) instead of pn (x; a, b | q).

Special values (see [K19, §2.4]). pn (0; a, b; q) = 1,

(160) 1

(qb; q)n , (qa; q)n (qb; q)n . (qa; q)n

pn (q −1 b−1 ; a, b; q) = (−qb)−n q − 2 n(n−1) 1

pn (1; a, b; q) = (−a)n q 2 n(n+1)

29

(161) (162)

14.14 Quantum q-Krawtchouk q-Hypergeometric representation

For n = 0, 1, . . . , N (see (14.14.1) and use (7)):  −n  q ,y qtm n+1 Kn (y; p, N ; q) = 2 φ1 ; q, pq q −N  −n −N  q , q /y, 0 N +1 = (pyq ; q)n 3 φ2 −N −N −n ; q, q . q ,q /(py)

(163) (164)

Special values By (163) and [GR, (II.4)]: Knqtm (1; p, N ; q) = 1,

Knqtm (q −N ; p, N ; q) = (pq; q)n .

(165)

By (164) and (165) we have the self-duality Knqtm (q x−N ; p, N ; q) Knqtm (q −N ; p, N ; q)

=

Kxqtm (q n−N ; p, N ; q) Kxqtm (q −N ; p, N ; q)

(n, x ∈ {0, 1, . . . , N }).

(166)

By (165) and (166) we have also qtm −x (q ; p, N ; q) = (pq N ; q −1 )x KN

Limit for q → 1 to Krawtchouk

(x ∈ {0, 1, . . . , N }).

(167)

(see (14.14.14) and Section 9.11):

lim Knqtm (1 + (1 − q)x; p, N ; q) = Kn (x; p−1 , N ),

(168)

lim Knqtm (q −x ; p, N ; q) = Kn (x; p−1 , N ).

(169)

q→1

q→1

Quantum q −1 -Krawtchouk

By (163), (165), (6) and (172) (see also p.496, second formula):  −n −1  Knqtm (y; p, N ; q −1 ) 1 q ,y −N = ; q, pyq (170) 2 φ1 (pq −1 ; q −1 )n q −N Knqtm (q N ; p, N ; q −1 ) = KnAff (q −N y; p−1 , N ; q).

(171)

Rewrite (171) as  qtm Km (1 + (1 − q −1 )qx; p−1 , N ; q −1 ) = ((pq)−1 ; q −1 )n KnAff 1 + (1 − q)q −N

1−q N 1−q


− x ; p, N ; q .

In view of (168) and (177) this tends to (71) as q → 1. The orthogonality relation (14.14.2) holds with positive weights for q > 1 if p > q −1 . History The origin of the name of the quantum q-Krawtchouk polynomials is by their interpretation as matrix elements of irreducible corepresentations of (the quantized function algebra of) the quantum group SUq (2) considered with respect to its quantum subgroup U (1). The orthogonality relation and dual orthogonality relation of these polynomials are an expression of the unitarity of these corepresentations. See for instance [343, Section 6]. 30

14.16 Affine q-Krawtchouk q-Hypergeometric representation

For n = 0, 1, . . . , N (see (14.16.1)):  −n −N −1  1 q ,q y Aff −1 Kn (y; p, N ; q) = −1 −1 −1 2 φ1 ; q, p y (p q ; q )n q −N  −n  q , y, 0 = 3 φ2 ; q, q . q −N , pq

Self-duality

(172) (173)

By (173): KnAff (q −x ; p, N ; q) = KxAff (q −n ; p, N ; q)

(n, x ∈ {0, 1, . . . , N }).

(174)

Special values By (172) and [GR, (II.4)]: KnAff (1; p, N ; q) = 1,

KnAff (q −N ; p, N ; q) =

1 ((pq)−1 ; q −1 )n

.

(175)

By (175) and (174) we have also Aff −x KN (q ; p, N ; q) =

Limit for q → 1 to Krawtchouk

1 ((pq)−1 ; q −1 )x

.

(176)

(see (14.16.14) and Section 9.11):

lim KnAff (1 + (1 − q)x; p, N ; q) = Kn (x; 1 − p, N ),

(177)

lim KnAff (q −x ; p, N ; q) = Kn (x; 1 − p, N ).

(178)

q→1

q→1

A relation between quantum and affine q-Krawtchouk By (163), (172), (175) and (174) we have for x ∈ {0, 1, . . . , N }: KxAff (q −n ; p, N ; q) KxAff (q −N ; p, N ; q) K Aff (q −x ; p, N ; q) = nAff −x . KN (q ; p, N ; q)

qtm −x −1 −N −1 ;p q , N ; q) = KN −n (q

(179) (180)

Formula (179) is given in [K3, formula after (12)] and [K12, (59)]. In view of (169) and (178) formula (180) has (72) as a limit case for q → 1. Affine q −1 -Krawtchouk

By (172), (175), (6) and (163) (see also p.505, first formula):  −n −N  KnAff (y; p, N ; q −1 ) q ,q y −1 n+1 = 2 φ1 ; q, p q (181) q −N KnAff (q N ; p, N ; q −1 ) = Knqtm (q −N y; p−1 , N ; q).

(182)

Formula (182) is equivalent to (171). Just as for (171), it tends after suitable substitutions to (71) as q → 1. The orthogonality relation (14.16.2) holds with positive weights for q > 1 if 0 < p < q −N . 31

History The affine q-Krawtchouk polynomials were considered by Delsarte [161, Theorem 11], [K8, (16)] in connection with certain association schemes. He called these polynomials generalized Krawtchouk polynomials. (Note that the 2 φ2 in [K8, (16)] is in fact a 3 φ2 with one upper parameter equal to 0.) Next Dunkl [186, Definition 2.6, Section 5.1] reformulated this as an interpretation as spherical functions on certain Chevalley groups. He called these polynomials q-Kratchouk polynomials. The current name affine q-Krawtchouk polynomials was introduced by Stanton [488, (4.13)]. He chose this name because, in [488, pp. 115–116] the polynomials arise in connection with an affine action of a group G on a space X. Here X  is the set of (v − n) × n A 0 matrices over GF(q). Let G be the group of block matrices , where A ∈ GLn (q), SA B   A 0 B ∈ GLv−n (q) and S ∈ X. Then G acts on X by · T = BT A−1 + S. SA B

14.17 Dual q-Krawtchouk Symmetry Kn (x; c, N | q) = cn Kn (c−1 x; c−1 , N | q).

(183)

This follows from (14.17.1) combined with [GR, (III.11)]. In particular, Kn (x; −1, N | q) = (−1)n Kn (−x; −1, N | q).

(184)

14.20 Little q-Laguerre / Wall Notation

Here the little q-Laguerre polynomial is denoted by pn (x; a; q) instead of pn (x; a | q).

Re: (14.20.11) The right-hand side of this generating function converges for |xt| < 1. We can rewrite the left-hand side by use of the transformation     0, 0 1 − ; q, z = ; q, cz . 2 φ1 0 φ1 c (z; q)∞ c Then we obtain:  (t; q)∞ 2 φ1

 X 1 ∞ 0, 0 (−1)n q 2 n(n−1) ; q, xt = pn (x; a; q) tn aq (q; q)n

(|xt| < 1).

(185)

n=0

Expansion of xn Divide both sides of (185) by (t; q)∞ . Then coefficients of the same power of t on both sides must be equal. We obtain: xn = (a; q)n

n X (q −n ; q)k k=0

(q; q)k 32

q nk pk (x; a; q).

(186)

Quadratic transformations 1

Little q-Laguerre polynomials pn (x; a; q) with a = q ± 2 are related to discrete q-Hermite I polynomials hn (x; q): pn (x2 ; q −1 ; q 2 ) = xpn (x2 ; q; q 2 ) =

(−1)n q −n(n−1) h2n (x; q), (q; q 2 )n

(187)

(−1)n q −n(n−1) h2n+1 (x; q). (q 3 ; q 2 )n

(188)

14.21 q-Laguerre Notation

(α)

Here the q-Laguerre polynomial is denoted by Lαn (x; q) instead of Ln (x; q).

Orthogonality relation (14.21.2) can be rewritten with simplified right-hand side: Z ∞ xα dx = hn δm,n Lαm (x; q) Lαn (x; q) (−x; q)∞ 0

(α > −1)

(189)

with

hn (q −α ; q)∞ π (q α+1 ; q)n , h = − . (190) = 0 n h0 (q; q)n q (q; q)∞ sin(πα) The expression for h0 (which is Askey’s q-gamma evaluation [K1, (4.2)]) should be interpreted by continuity in α for α ∈ Z≥0 . Explicitly we can write 1

hn = q − 2 α(α+1) (q; q)α log(q −1 )

(α ∈ Z≥0 ).

(191)

Expansion of xn 1

xn = q − 2 n(n+2α+1) (q α+1 ; q)n

n X (q −n ; q)k k α q Lk (x; q). (q α+1 ; q)k

(192)

k=0

This follows from (186) by the equality given in the Remark at the end of §14.20. Alternatively, it can be derived in the same way as (186) from the generating function (14.21.14). Quadratic transformations q-Laguerre polynomials Lαn (x; q) with α = ± 21 are related to discrete q-Hermite II polynomials e hn (x; q): 2

L−1/2 (x2 ; q 2 ) n

(−1)n q 2n −n e = h2n (x; q), (q 2 ; q 2 )n

2 2 xL1/2 n (x ; q )

(−1)n q 2n +n e = h2n+1 (x; q). (q 2 ; q 2 )n

(193)

2

(194)

These follows from (187) and (188), respectively, by applying the equalities given in the Remarks at the end of §14.20 and §14.28. 33

14.27 Stieltjes-Wigert An alternative weight function The formula on top of p.547 should be corrected as 1 γ w(x) = √ x− 2 exp(−γ 2 ln2 x), π

x > 0,

with γ 2 = −

1 . 2 ln q

(195) 1

For w the weight function given in [Sz, §2.7] the right-hand side of (195) equals const. w(q − 2 x). See also [DLMF, §18.27(vi)].

14.28 Discrete q-Hermite I History Discrete q Hermite I polynomials (not yet with this name) first occurred in Hahn [261], see there p.29, case V and the q-weight π(x) given by the second expression on line 4 of p.30. However note that on the line on p.29 dealing with case V, one should read k 2 = q −n instead of k 2 = −q n . Then, with the indicated substitutions, [261, (4.11), (4.12)] yield constant multiples of h2n (q −1 x; q) and h2n+1 (q −1 x; q), respectively, due to the quadratic transformations (187), (188) together with (4.20.1).

14.29 Discrete q-Hermite II Basic hypergeometric representation (see (14.29.1))  −n −n+1  q ,q n 2 2 −2 e hn (x; q) = x 2 φ1 ; q , −q x . 0

(196)

Standard references [AAR]

G. E. Andrews, R. Askey and R. Roy, Special functions, Cambridge University Press, 1999.

[DLMF] NIST Handbook of Mathematical Functions, Cambridge University Press, 2010; DLMF, Digital Library of Mathematical Functions, http://dlmf.nist.gov. [GR]

G. Gasper and M. Rahman, Basic hypergeometric series, 2nd edn., Cambridge University Press, 2004.

[HTF1] A. Erd´elyi, Higher transcendental functions, Vol. 1, McGraw-Hill, 1953. [HTF2] A. Erd´elyi, Higher transcendental functions, Vol. 2, McGraw-Hill, 1953. [Ism]

M. E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Cambridge University Press, 2005; reprinted and corrected, 2009. 34

[KLS]

R. Koekoek, P. A. Lesky and R. F. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, Springer-Verlag, 2010.

[Sz]

G. Szeg˝ o, Orthogonal polynomials, Colloquium Publications 23, American Mathematical Society, Fourth Edition, 1975.

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[46]

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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]

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