Additions to the formula lists in Hypergeometric orthogonal polynomials and their q-analogues by Koekoek, Lesky and Swarttouw

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1 Additions to the formula lists in Hypergeometric orthogonal polynomials and their q-analogues by Koekoek, Lesky and Swarttouw Tom H. Koornwinder June 9, 205 Abstract This report gives a rather arbitrary choice of formulas for q-hypergeometric orthogonal polynomials which the author missed while consulting Chapters 9 and 4 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 4 in the book [KLS]: R. Koekoek, P. A. Lesky and R. F. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, Springer-Verlag, 200. 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 98-7, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 998; 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.

2 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,,..., N}. Throughout assume 0 < q <. 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 p n 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

3 Contents Introduction Conventions Generalities 9. 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. Gegenbauer / Ultraspherical Chebyshev 9.9 Pseudo Jacobi or Routh-Romanovski 9.0 Meixner 9. Krawtchouk 9.2 Laguerre 9.4 Charlier 9.5 Hermite 4. Askey-Wilson 4.2 q-racah 4.3 Continuous dual q-hahn 4.4 Continuous q-hahn 4.5 Big q-jacobi 4.7 Dual q-hahn 4.8 Al-Salam-Chihara 4.9 q-meixner-pollaczek 4.0 Continuous q-jacobi 4.0. Continuous q-ultraspherical / Rogers 4. Big q-laguerre 4.2 Little q-jacobi 4.4 Quantum q-krawtchouk 4.6 Affine q-krawtchouk 4.7 Dual q-krawtchouk 4.20 Little q-laguerre / Wall 4.2 q-laguerre 4.27 Stieltjes-Wigert 4.28 Discrete q-hermite I 4.29 Discrete q-hermite II Standard references References from [KLS] Other references 3

4 Generalities Criteria for uniqueness of orthogonality measure According to Shohat & Tamarkin [K28, p.50] orthonormal polynomials p n have a unique orthogonality measure up to positive constant factor if for some z C we have p n z 2 =. n=0 Also see Shohat & Tamarkin [K28, p.59], monic orthogonal polynomials p n with three-term recurrence relation xp n x = p n+ x + B n p n x + C n p n x C n necessarily positive have a unique orthogonality measure if C n /2 =. 2 n= Furthermore, if orthogonal polynomials have an orthogonality measure with bounded support, then this is unique see Chihara [46]. Even orthogonality measure If {p n } is a system of orthogonal polynomials with respect to an even orthogonality measure which satisfies the three-term recurrence relation then xp n x = A n p n+ x + C n p n x p 2n 0 p 2n 2 0 = C 2n A 2n. 3 Appell s bivariate hypergeometric function F 4 This is defined by F 4 a, b; c, c a m+n b m+n ; x, y := c m c n m! n! xm y n x 2 + y 2 <, 4 m,n=0 see [HTF, 5.79, 5.744] or [DLMF, 6.3.4]. There is the reduction formula x F 4 a, b; b, b; x y, y a, + a b = x a y a 2F ; xy, x y b see [HTF, 5.07]. When combined with the quadratic transformation [HTF, 2.34] here a b should be replaced by a b +, see also [DLMF, 5.8.5], this yields x F 4 a, b; b, b; x y, y x y x y a = 2 2F a, 2 a + 4xy ; + xy b + xy 2. This can be rewritten as F 4 a, b; b, b; x, y = x y a 2F 2 a, 2 a + 4xy ; b x y 2. 5 Note that, if x, y 0 and x 2 + y 2 <, then x y > 0 and 0 4xy x y 2 <. 4

5 q-hypergeometric series of base q By [GR, Exercise.4i]: a,..., a r a rφ s ; q, z = s+ φ,... a r, 0,..., 0 s b,... b s b,..., ; q, qa... a r z b s b... b s 6 for r s +, a,..., a r, b,..., b s 0. In the non-terminating case, for 0 < q <, there is convergence if z < b... b s /qa... a r. A transformation of a terminating 2 φ By [GR, Exercise.5i] we have q n, b q 2φ ; q, z = bz/cq; q n, c/b, 0 n 3 φ 2 c c, cq/bz ; q, q. 7 Very-well-poised q-hypergeometric series The notation of [GR, 2..] will be followed: r+w r a ; a 4, a 5,..., a r+ ; q, z := r+ φ r a, qa 2, qa 2, a 4,..., a r+ a 2, a ; q, z. 8 2, qa /a 4,..., qa /a r+ Theta function The notation of [GR,.2.] will be followed: θx; q := x, q/x; q, θx,..., x m ; q := θx ; q... θx m ; q Wilson Symmetry The Wilson polynomial W n y; a, b, c, d is symmetric in a, b, c, d. This follows from the orthogonality relation 9..2 together with the value of its coefficient of y n given in 9..5b. Alternatively, combine 9.. with [AAR, Theorem 3..]. As a consequence, it is sufficient to give generating function Then the generating functions 9..3, 9..4 will follow by symmetry in the parameters. Hypergeometric representation In addition to 9.. we have see [53, 2.2]: W n x 2 ; a, b, c, d = a ix nb ix n c ix n d ix n 2ix n 7 F 6 2ix n, ix 2n +, a + ix, b + ix, c + ix, d + ix, n ix 2n, n a + ix, n b + ix, n c + ix, n d + ix; The symmetry in a, b, c, d is clear from 0. Special value. 0 W n a 2 ; a, b, c, d = a + b n a + c n a + d n, and similarly for arguments b 2, c 2 and d 2 by symmetry of W n in a, b, c, d. 5

6 Uniqueness of orthogonality measure Under the assumptions on a, b, c, d for 9..2 or 9..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..2 or 9..3 as h n δ m,n. Observe from 9..2 and that W n a 2 ; a, b, c, d 2 h n = On 4Re a as n. Therefore holds, from which the uniqueness of the orthogonality measure follows. By a similar, but necessarily more complicated argument Ismail et al. [28, 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.96 Racah polynomials are expressed in terms of Wilson polynomials. This can be equivalently written as R n xx N + δ; α, β, N, δ = W n x + 2 δ N2 ; 2 δ N, α + 2 δ N, β + 2 δ + N +, 2 δ + N. 2 α + n β + δ + n N n 9.3 Continuous dual Hahn Symmetry The continuous dual Hahn polynomial S n y; a, b, c is symmetric in a, b, c. This follows from the orthogonality relation together with the value of its coefficient of y n given in 9.3.5b. Alternatively, combine 9.3. with [AAR, Corollary 3.3.5]. As a consequence, it is sufficient to give generating function Then the generating functions 9.3.3, will follow by symmetry in the parameters. Special value S n a 2 ; a, b, c = a + b n a + c n, 3 and similarly for arguments b 2 and c 2 by symmetry of S n in a, b, c. Uniqueness of orthogonality measure Under the assumptions on a, b, c for or 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 or as h n δ m,n. Observe from and 3 that S n a 2 ; a, b, c 2 h n = On 2Re a as n. Therefore holds, from which the uniqueness of the orthogonality measure follows. 6

7 9.4 Continuous Hahn Orthogonality relation and symmetry The orthogonality relation 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 p n x; a, b, c, d is symmetric in a, b and in c, d. This follows from the orthogonality relation together with the value of its coefficient of x n given in 9.4.4b. As a consequence, it is sufficient to give generating function Then the generating function will follow by symmetry in the parameters. Uniqueness of orthogonality measure The coefficient of p n x in behaves as On 2 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: p n x; a, a + 2, a, a + 2 = 2a n2a + 2 n 4a n P n 2a 2x; 2π. 4 See [342, 2.6] note that in [342, 2.3] the Meixner-Pollaczek polynonmials are defined different from 9.7., without a constant factor in front. For 0 < a < the continuous Hahn polynomials p n x; a, a, a, a are orthogonal on, with respect to the weight function cosh2πx cos2πa by straightforward computation from For a = 4 the two special cases coincide: Meixner-Pollaczek with weight function cosh2πx. 9.5 Hahn Special values Q n 0; α, β, N =, Q n N; α, β, N = n β + n α + n. 5 Use 9.5. and compare with 9.8. and 34. From and 3 it follows that Q 2n N; α, α, 2N = From 9.5. and [DLMF, ] it follows that 2 nn + α + n N + 2 nα + n. 6 Q N x; α, β, N = N β x α + x x = 0,,..., N. 7 7

8 Symmetries By the orthogonality relation 9.5.2: Q n N x; α, β, N Q n N; α, β, N = Q n x; β, α, N, 8 It follows from 25 and 20 that Q N n x; α, β, N Q N x; α, β, N = Q n x; N β, N α, N x = 0,,..., N. 9 Duality The Remark on p.208 gives the duality between Hahn and dual Hahn polynomials: Q n x; α, β, N = R x nn + α + β + ; α, β, N n, x {0,,... N} Dual Hahn Special values By 7 and 20 we have It follows from 5 and 20 that R n NN + γ + δ + ; γ, δ, N = N δ n γ + n. 2 R N xx + γ + δ + ; γ, δ, N = x δ + x γ + x x = 0,,..., N. 22 Symmetries Write the weight in as w x α, β, N := N! 2x + γ + δ + x + γ + δ + N+ γ + x δ + x N. 23 x Then δ + N w N x γ, δ, N = γ N N w x δ N, γ N, N. 24 Hence, by 9.6.2, R n N xn x + γ + δ + ; γ, δ, N R n NN + γ + δ + ; γ, δ, N Alternatively, 25 follows from 9.6. and [DLMF, 6.4.]. It follows from 8 and 20 that = R n xx 2N γ δ ; N δ, N γ, N. 25 R N n xx + γ + δ + ; γ, δ, N R N xx + γ + δ + ; γ, δ, N = R n xx + γ + δ + ; δ, γ, N x = 0,,..., N. 26 8

9 Re: where The generating function 9.6. can be written in a more conceptual way as x N, x + γ + t x 2F ; t = δ N γ + n ω n := n N! δ + N N ω n R n λx; γ, δ, N t n, 27 n=0 δ + N n N n, 28 i.e., the denominator on the right-hand side of By the duality between Hahn polynomials and dual Hahn polynomials see 20 the above generating function can be rewritten in terms of Hahn polynomials: where n N, n + α + t n 2F ; t = β N N! β + N N w x Q n x; α, β, N t x, 29 x=0 α + x β + N x w x :=, 30 x N x i.e., the weight occurring in the orthogonality relation for Hahn polynomials. Re: There should be a closing bracket before the equality sign. 9.7 Meixner-Pollaczek Uniqueness of orthogonality measure The coefficient of p n x in behaves as On 2 as n. Hence 2 holds, by which the orthogonality measure is unique. 9.8 Jacobi Orthogonality relation Write the right-hand side of as h n δ m,n. Then h n = n + α + β + α + n β + n, h 0 = 2α+β+ Γα + Γβ +, h 0 2n + α + β + α + β + 2 n n! Γα + β + 2 h n h 0 P α,β n 2 = n + α + β + 2n + α + β + β + n n! α + n α + β + 2 n. 3 In the numerator factor Γn + α + β + in the last line should be Γβ +. When thus corrected, can be rewritten as: P m α,β x P n α,β x x α x + β dx = h n δ m,n, β > α >, m, n < 2 α + β +, h n = n + α + β + α + n β + n, h 0 = 2α+β+ Γα + Γ α β. h 0 2n + α + β + α + β + 2 n n! Γ β 32 9

10 Symmetry Use and 9.8.5b or see [DLMF, Table 8.6.]. Special values P n α,β x = n P n β,α x. 33 P n α,β = α + n n!, P n α,β = n β + n, n! P n α,β P n α,β = n β + n. 34 α + n Use 9.8. and 33 or see [DLMF, Table 8.6.]. Generating functions Formula was first obtained by Brafman [09]. Bilateral generating functions For 0 r < and x, y [, ] we have in terms of F 4 see 4: α + β + n n! r n P n α,β x P n α,β y = α + n β + n n=0 n=0 + r α+β+ F 4 2 α + β +, r x y r + x + y 2 α + β + 2; α +, β + ; + r 2, + r 2, 35 2n + α + β + n + α + β + α + β + 2 n n! r n P n α,β x P n α,β y = α + n β + n r + r α+β+2 F 4 2 α + β + 2, r x y r + x + y 2 α + β + 3; α +, β + ; + r 2, + r Formulas 35 and 36 were first given by Bailey [9, 2., 2.3]. See Stanton [485] for a shorter proof. However, in the second line of [485, ] z and Z should be interchanged. As observed in Bailey [9, p.0], 36 follows from 35 by applying the operator r 2 α+β d dr r 2 α+β+ to both sides of 35. In view of 3, 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 C α+ 2 2n x C α+ 2 2n C α+ 2 2n+ x C α+ 2 2n+ = = P α,α 2n x P α,α 2n = P α, 2 n 2x 2 P α, 2 n, 37 P α,α 2n+ x P α,α 2n+ = x P α, 2 n 2x P α, 2 n See p.22, Remarks, last two formulas together with 34 and 49. Or see [DLMF, 8.7.3, 8.7.4]. 0

11 Differentiation formulas Each differentiation formula is given in two equivalent forms. d x α P n α,β x = n + α x α P n α,β+ x, dx x d 39 dx α P n α,β x = n + α P n α,β+ x. d dx + x d dx + β + x β P n α,β x x = n + β P α+,β x. P α,β n = n + β + x β P n α+,β x, Formulas 39 and 40 follow from [DLMF, 5.5.4, 5.5.6] together with They also follow from each other by 33. Generalized Gegenbauer polynomials S α,β 2m α,β x := const. P m 2x 2, S α,β n These are defined by 2m+ 40 x := const. x P α,β+ m 2x 2 4 in the notation of [46, p.56] see also [K5], while [K9, Section.5.2] has C n λ,µ x = const. S λ 2,µ 2 n x. For α, β > we have the orthogonality relation S m α,β x S n α,β x x 2β+ x 2 α dx = 0 m n. 42 For β = α generalized Gegenbauer polynomials are limit cases of continuous q-ultraspherical polynomials, see 59. If we define the Dunkl operator T µ by T µ fx := f x + µ and if we choose the constants in 4 as fx f x x S α,β 2m x = α + β + m P m α,β 2x 2, S α,β 2m+ β + x = α + β + m+ x P m α,β+ 2x 2 m β + m+ 44 then see [K6,.6] T β+ S n α,β = 2α + β + S α+,β n Formula 45 with 44 substituted gives rise to two differentiation formulas involving Jacobi polynomials which are equivalent to and 40. Composition of 45 with itself gives T 2 S β+ n α,β = 4α + β + α + β + 2 S α+2,β n 2, 2 which is equivalent to the composition of and 40: d 2 dx 2 + 2β + d P n α,β 2x 2 = 4n + α + β + n + β P α+2,β n 2x x dx Formula 46 was also given in [322, 2.4]. 43

12 9.8. Gegenbauer / Ultraspherical Notation Here the Gegenbauer polynomial is denoted by C λ n instead of C λ n. Orthogonality relation Write the right-hand side of as h n δ m,n. Then h n h 0 = λ 2λ n λ + n n!, h 0 = π 2 Γλ + 2, Γλ + h n h 0 C λ n 2 = λ λ + n n! 2λ n. 47 Hypergeometric representation Beside we have also C λ nx = n/2 l=0 See [DLMF, 8.5.0]. l λ n l l! n 2l! 2xn 2l = 2x n λ n n! 2 2F n, 2 n + 2 λ n ; x Special value Use or see [DLMF, Table 8.6.]. Expression in terms of Jacobi Cnx λ Cn λ = P λ n 2,λ 2 x P λ 2,λ 2 n C λ n = 2λ n n! Re: By iteration of recurrence relation 9.8.2: x 2 C λ nx =. 49, Cnx λ = 2λ n λ + 2 P λ 2,λ 2 n x. 50 n n + n + 2 n 2 + 2nλ + λ 4n + λn + λ + Cλ n+2x + 2n + λ n + λ + Cλ nx Bilateral generating functions n=0 n! 2λ n r n C λ nx C λ ny = + 2rxy + r 2 λ 2 2 F λ, 2 n + 2λ n + 2λ 2 4n + λn + λ λ + λ + 2 C λ n 2x. 5 ; 4r2 x 2 y 2 2rxy + r 2 2 r,, x, y [, ]. 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 2

13 r λ+ d dr rλ to both sides of 52: n=0 λ + n λ n! 2λ n r n C λ nx C λ ny = 2 2 F λ +, 2 λ + 2 λ + 2 r 2 2rxy + r 2 λ+ ; 4r2 x 2 y 2 2rxy + r 2 2 r,, x, y [, ]. 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, 8.5., 5.8.]: n Cncos λ λ k λ n k θ = k! n k! ein 2kθ = e inθ λ n n, λ 2F n! λ n ; e 2iθ 54 k=0 = λ n 2 λ n! e λ, λ 2 iλπ e in+λθ sin θ λ 2F λ n ; ie iθ 55 2 sin θ = λ n λ k λ k cosn k + λθ + 2 k λπ n! λ n k k! 2 sin θ k+λ. 56 k=0 In 55 and 56 we require that 6 π < θ < 5 6 π. Then the convergence is absolute for λ > 2 and conditional for 0 < λ 2. By [DLMF, 4.3., 4.3.2, 5.8.]]: C λ ncos θ = 2Γλ + 2 π 2 Γλ + = 2Γλ + 2 π 2 Γλ + = 2λ Γλ + 2 π 2 Γλ + = 22λ Γλ + 2 π 2 Γλ + 2λ n λ + n sin θ 2λ 2λ n λ + n sin θ 2λ Im 2λ n λ + n sin θ λ Re 2λ n λ + n k=0 λ k n + k sin 2k + n + θ 57 n + λ + k k! k=0 λ, n + e in+θ 2F n + λ + ; e2iθ e 2 iλπ e in+λθ 2F λ, λ + λ + n ; e iθ 2i sin θ λ k λ k cosn + k + λθ 2 k + λπ + λ + n k k! 2 sin θ k+λ. 58 We require that 0 < θ < π in 57 and 6 π < θ < 5 6π in 58 The convergence is absolute for λ > 2 and conditional for 0 < λ 2. For λ Z >0 the above series terminate after the term with k = λ. Formulas 57 and 58 are also given in [Sz, , ]. Fourier transform Γλ + Γλ + 2 Γ 2 See [DLMF, and 8.7.8]. C λ ny C λ n y2 λ 2 e ixy dy = i n 2 λ Γλ + x λ J λ+n x. 59 3

14 Laplace transforms 2 n! Γλ 0 H n tx t n+2λ e t2 dt = C λ nx. 60 See Nielsen [K24, p.48, 4 with p.47, and p.28, 0] 98 or Feldheim [K0, 28] Γλ Cnt λ Cn λ t2 λ 2 t x/t n+2λ+ e x2 /t 2 dt = 2 n H n x e x2 λ > 2. 6 Use Askey & Fitch [K2, 3.29] for α = ± 2 together with 33, 37, 38, 86 and 87. Addition formula see [AAR, ]] R n α,α xy + x 2 2 y 2 n k n k n + 2α + k 2 t = 2 2k α + k 2 k=0 where R α,β n x 2 k/2 R α+k,α+k n k x y 2 k/2 R α+k,α+k n k y ω α 2,α 2 k R α 2,α 2 k t, 62 x := P α,β Chebyshev n x/p α,β n, ω α,β n := xα + x β dx Rα,β n x 2 x α + x β dx. In addition to the Chebyshev polynomials T n of the first kind and U n of the second kind , T n x := P n 2, 2 P 2, 2 n U n x := n + P n x = cosnθ, x = cos θ, 63 2, 2 x = P 2, 2 n sinn + θ sin θ, x = cos θ, 64 we have Chebyshev polynomials V n of the third kind and W n of the fourth kind, V n x := P n 2, 2 x P 2, 2 n W n x := 2n + P n = cosn + 2 θ cos 2 θ, x = cos θ, 65 2, 2 x P 2, 2 n see [K22, Section.2.3]. Then there is the symmetry = sinn + 2 θ sin 2 θ, x = cos θ, 66 V n x = n W n x. 67 4

15 The names of Chebyshev polynomials of the third and fourth kind and the notation V n x are due to Gautschi [K]. The notation W n x was first used by Mason [K2]. Names and notations for Chebyshev polynomials of the third and fourth kind are interchanged in [AAR, Remark 2.5.3] and [DLMF, Table 8.3.]. 9.9 Pseudo Jacobi or Routh-Romanovski In this section in [KLS] the pseudo Jacobi polynomial P n x; ν, N in 9.9. is considered for N Z 0 and n = 0,,..., n. However, we can more generally take 2 < N R so here I overrule my convention formulated in the beginning of this paper, N 0 integer such that N 2 N 0 < N + 2, and n = 0,,..., N 0 see [382, 5, case A.4]. The orthogonality relation is valid for m, n = 0,,..., N 0. History These polynomials were first obtained by Routh [K27] in 885, and later, independently, by Romanovski [463] in 929. Limit relation: See also 42. Pseudo big q-jacobi Pseudo Jacobi References See also [Ism, 20.], [5], [384], [K7], [K20], [K25]. 9.0 Meixner History In 934 Meixner [406] see. and case IV on pp. 0, and 2 gave the orthogonality measure for the polynomials P n given by the generating function where e xut ft = n=0 P n x tn n!, e ut = βt α β, αt ft = βt αt k 2 βα β k 2 αα β k 2 < 0; α > β > 0 or α < β < 0. Then P n can be expressed as a Meixner polynomial: P n x = k 2 αβ n β n M n x + k 2α α β, k 2 αβ, βα. In 938 Gottlieb [K5, 2] introduces polynomials l n of Laguerre type which turn out to be special Meixner polynomials: l n x = e nλ M n x;, e λ. Uniqueness of orthogonality measure The coefficient of p n x in behaves as On 2 as n. Hence 2 holds, by which the orthogonality measure is unique. 5

16 9. Krawtchouk Special values By 9.. and the binomial formula: The self-duality p.240, Remarks, first formula combined with 68 yields: K n 0; p, N =, K n N; p, N = p n. 68 K n x; p, N = K x n; p, N n, x {0,,..., N} 69 K N x; p, N = p x x {0,,..., N}. 70 Symmetry By the orthogonality relation 9..2: By 7 and 69 we have also K N n x; p, N K N x; p, N and, by 72, 7 and 68, K N n N x; p, N = A particular case of 7 is: Hence From 9..: Quadratic transformations K n N x; p, N K n N; p, N = K n x; p, N. 7 = K n x; p, N n, x {0,,..., N}, 72 p n+x N K n x; p, N n, x {0,,..., N}. 73 p K n N x; 2, N = n K n x; 2, N. 74 K 2m+ N; 2, 2N = K 2m N; 2, 2N = 2 m N m K 2m x + N; 2, 2N = 2 m N + 2 m R m x 2 ; 2, 2, N, 77 K 2m+ x + N; 2, 2N = 3 2 m N N + 2 m K 2m x + N + ; 2, 2N + = K 2m+ x + N + ; 2, 2N + = 2 m N 2 m 3 2 m N 2 m+ x R m x 2 ; 2, 2, N, 78 R m xx + ; 2, 2, N, 79 x + 2 R mxx + ; 2, 2, N, 80 where R m is a dual Hahn polynomial For the proofs use 9.6.2, 9..2, and

17 Generating functions N N K m x; p, NK n x; q, Nz x x p z + pz m q z + qz = p q x=0 n + z N m n K m n; p z + pzq z + qz, N. z 8 This follows immediately from Rosengren [K26, 3.5], which goes back to Meixner [K23]. 9.2 Laguerre Notation Here the Laguerre polynomial is denoted by L α n instead of L α n. Hypergeometric representation L α nx = α + n n! = xn n! = xn n! n F α + ; x n, n α 2F 0 ; x C n n + α; x, 84 where C n in 84 is a Charlier polynomial. Formula 82 is 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 [46, 2.7.0]. Uniqueness of orthogonality measure The coefficient of p n x in behaves as On 2 as n. Hence 2 holds, by which the orthogonality measure is unique. Special value Use 9.2. or see [DLMF, 8.6.]. L α n0 = α + n n!. 85 Quadratic transformations H 2n x = n 2 2n n! L /2 n x 2, 86 H 2n+ x = n 2 2n+ n! x L /2 n x See p.244, Remarks, last two formulas. Or see [DLMF, 8.7.9, ]. 7

18 Fourier transform see [DLMF, ]. Γα + 0 L α ny L α n0 e y y α e ixy dy = i n y n, 88 iy + n+α+ Differentiation formulas d dx xα L α nx = n + α x α Ln α x, Each differentiation formula is given in two equivalent forms. x d dx + α L α nx = n + α L α n x. 89 d dx e x L α nx = e x L α+ n x, d dx L α nx = L α+ n x. 90 Formulas 89 and 90 follow from [DLMF, 3.3.8, ] together with Generalized Hermite polynomials by See [46, p.56], [K9, Section.5.]. These are defined H µ 2m x := const. Lµ 2 m x 2, H µ 2m+ x := const. x Lµ+ 2 m x 2. 9 Then for µ > 2 we have orthogonality relation H µ mx H µ nx x 2µ e x2 dx = 0 m n. 92 Let the Dunkl operator T µ be defined by 43. If we choose the constants in 9 as H µ 2m x = m 2m! µ + 2 m L µ 2 m x 2, H µ 2m+ x = m 2m +! µ + 2 x L µ+ 2 m x 2 93 m+ then see [K6,.6] T µ H µ n = 2n H µ n. 94 Formula 94 with 93 substituted gives rise to two differentiation formulas involving Laguerre polynomials which are equivalent to and 89. Composition of 94 with itself gives T 2 µh µ n = 4nn H µ n 2, which is equivalent to the composition of and 89: d 2 dx 2 + 2α + x d L α dx nx 2 = 4n + α L α n x

19 9.4 Charlier Hypergeometric representation n, x C n x; a = 2 F 0 ; a = x n n a n F x n + ; a = n! a n Lx n n a, 98 where L α nx is a Laguerre polynomial. Formula 96 is Then 97 follows by reversal of the summation. Finally 98 follows by 97 and It is also the Remark on p.249 of [KLS], and it was earlier given in [46, 2.7.0]. Uniqueness of orthogonality measure The coefficient of p n x in behaves as On as n. Hence 2 holds, by which the orthogonality measure is unique. 9.5 Hermite Uniqueness of orthogonality measure The coefficient of p n x in behaves as On as n. Hence 2 holds, by which the orthogonality measure is unique. Fourier transforms 2π H n y e 2 y2 e ixy dy = i n H n x e 2 x2, 99 see [AAR, 6..5 and Exercise 6.]. see [DLMF, ]. see [AAR, 6..4]. 4. Askey-Wilson π i n 2 π H n y e y2 e ixy dy = i n x n e 4 x2, 00 y n e 4 y2 e ixy dy = H n x e x2, 0 Symmetry The Askey-Wilson polynomials p n x; a, b, c, d q are symmetric in a, b, c, d. This follows from the orthogonality relation 4..2 together with the value of its coefficient of x n given in 4..5b. Alternatively, combine 4.. with [GR, III.5]. As a consequence, it is sufficient to give generating function Then the generating functions 4..4, 4..5 will follow by symmetry in the parameters. 9

20 Basic hypergeometric representation In addition to 4.. we have in notation 8: p n cos θ; a, b, c, d q = ae iθ, be iθ, ce iθ, de iθ ; q n e 2iθ ; q n e inθ 8 W 7 q n e 2iθ ; ae iθ, be iθ, ce iθ, de iθ, q n ; q, q 2 n /abcd. 02 This follows from 4.. by combining III.5 and III.9 in [GR]. It is also given in [53, 4.2], but be aware for some slight errors. The symmetry in a, b, c, d is evident from 02. Special value p n 2 a + a ; a, b, c, d q = a n ab, ac, ad; q n, 03 and similarly for arguments 2 b + b, 2 c + c and 2 d + d by symmetry of p n in a, b, c, d. Trivial symmetry p n x; a, b, c, d q = n p n x; a, b, c, d q. 04 Both 03 and 04 are obtained from 4... Re: 4..5 Let Then p n x := p nx; a, b, c, d q 2 n abcdq n ; q n = x n + k n x n kn = qn a + b + c + d abc + abd + acd + bcdq n 2 q abcdq 2n This follows because k n k n+ equals the coefficient 2 a + a A n + C n of p n x in Generating functions Rahman [449, 4., 4.9] gives: abcdq ; q n a n t n p n cos θ; a, b, c, d q ab, ac, ad, q; q n=0 n = abcdtq ; q abcdq 2, abcdq 2, abcd 2, abcd 2, ae iθ, ae iθ 6φ 5 t; q ab, ac, ad, abcdtq, qt ; q, q + abcdq, abt, act, adt, ae iθ, ae iθ ; q ab, ac, ad, t, ate iθ, ate iθ ; q tabcdq 2, tabcdq 2, tabcd 2, tabcd 2, ate iθ, ate iθ 6 φ 5 abt, act, adt, abcdt 2 q ; q, q, qt t <. 07 In the limit 08 the first term on the right-hand side of 07 tends to the left-hand side of 9..5, while the second term tends formally to 0. The special case ad = bc of 07 was earlier given in [236, 4., 4.6]. 20

21 Limit relations Askey-Wilson Wilson Instead of 4..2 we can keep a polynomial of degree n while the limit is approached: p n 2 lim x q2 ; q a, q b, q c, q d q q q 3n = W n x; a, b, c, d. 08 For the proof first derive the corresponding limit for the monic polynomials by comparing 4..5 with Askey-Wilson Continuous Hahn Instead of we can keep a polynomial of degree n while the limit is approached: p n cos φ x q sin φ; q a e iφ, q b e iφ, q a e iφ, q b e iφ q lim q q 2n = 2 sin φ n n! p n x; a, b, a, b 0 < φ < π. 09 Here the right-hand side has a continuous Hahn polynomial For the proof first derive the corresponding limit for the monic polynomials by comparing 4..5 with In fact, define the monic polynomial p n x := p n cos φ x q sin φ; q a e iφ, q b e iφ, q a e iφ, q b e iφ q 2 q sin φ n abcdq n. ; q n Then it follows from 4..5 that x p n x = p n+ x + qa e iφ + q a e iφ + Ãn + C n 2 q sin φ p n x + Ã n Cn q 2 sin 2 φ p n x, where Ãn and C n are as given after 4..3 with a, b, c, d replaced by q a e iφ, q b e iφ, q a e iφ, q b e iφ. Then the recurrence equation for p n x tends for q to the recurrence equation with c = a, d = b. Askey-Wilson Meixner-Pollaczek Instead of we can keep a polynomial of degree n while the limit is approached: p n cos φ x q sin φ; q λ e iφ, 0, q λ e iφ, 0 q lim q q n = n! P n λ x; π φ 0 < φ < π. 0 Here the right-hand side has a Meixner-Pollaczek polynomial For the proof first derive the corresponding limit for the monic polynomials by comparing 4..5 with In fact, define the monic polynomial p n x := p n cos φ x q sin φ; q λ e iφ, 0, q λ e iφ, 0 q 2 q sin φ n. 2

22 Then it follows from 4..5 that x p n x = p n+ x + qλ e iφ + q λ e iφ + Ãn + C n 2 q sin φ p n x + Ã n Cn q 2 sin 2 φ p n x, where Ãn and C n are as given after 4..3 with a, b, c, d replaced by q λ e iφ, 0, q λ e iφ, 0. Then the recurrence equation for p n x tends for q to the recurrence equation References See also Koornwinder [K8]. 4.2 q-racah Symmetry R n x; α, β, q N, δ q = βq, αδ q; q n αq, βδq; q n δ n R n δ x; β, α, q N, δ q. This follows from 4.2. combined with [GR, III.5]. In particular, R n x; α, β, q N, q = βq, αq; q n αq, βq; q n n R n x; β, α, q N, q, 2 and R n x; α, α, q N, q = n R n x; α, α, q N, q, 3 Trivial symmetry Clearly from 4.2.: R n x; α, β, γ, δ q = R n x; βδ, αδ, γ, δ q = R n x; γ, αβγ, α, γδα q. 4 For α = q N this shows that the three cases αq = q N or βδq = q N or γq = q N of 4.2. are not essentially different. Duality It follows from 4.2. that R n q y + γδq y+ ; q N, β, γ, δ q = R y q n + βq n N ; γ, δ, q N, β q n, y = 0,,..., N Continuous dual q-hahn The continuous dual q-hahn polynomials are the special case d = 0 of the Askey-Wilson polynomials: p n x; a, b, c q := p n x; a, b, c, 0 q. Hence all formulas in 4.3 are specializations for d = 0 of formulas in

23 4.4 Continuous q-hahn The continuous q-hahn polynomials are the special case of Askey-Wilson polynomials with parameters ae iφ, be iφ, ae iφ, be iφ : p n x; a, b, φ q := p n x; ae iφ, be iφ, ae iφ, be iφ q. In [72, 4.29] and [GR, ] who write p n x; a, b q, x = cosθ + φ and in [KLS, 4.4] who writes p n x; a, b, c, d; q, x = cosθ + φ the parameter dependence on φ is incorrectly omitted. Since all formulas in 4.4 are specializations of formulas in 4., there is no real need to give these specializations explicitly. In particular, the limit is in fact a limit from Askey-Wilson to continuous q-hahn. See also Big q-jacobi Different notation See p.442, Remarks: q P n x; a, b, c, d; q := P n qac x; a, b, ac n, q n+ ab, qac x d; q = 3 φ 2 qa, qac ; q, q. 6 d Furthermore, P n x; a, b, c, d; q = P n λx; a, b, λc, λd; q, 7 P n x; a, b, c; q = P n q c x; a, b, ac, ; q 8 Orthogonality relation equivalent to 4.5.2, see also [K9, 2.42, 2.4, 2.36, 2.35]. Let c, d > 0 and either a c/qd, /q, b d/cq, /q or a/c = b/d / R. Then where and c d P m x; a, b, c, d; qp n x; a, b, c, d; q h n h 0 = q 2 nn q 2 a 2 d c h 0 = qc n qab q 2n+ ab qx/c, qx/d; q qax/c, qbx/d; q d q x = h n δ m,n, 9 q, qb, qbc/d; q n qa, qab, qad/c; q n 20 q, d/c, qc/d, q2 ab; q qa, qb, qbc/d, qad/c; q. 2 Other hypergeometric representation and asymptotics P n x; a, b, c, d; q = qbd x; q n q n, q n b, cx q n a cd 3φ 2 ; q n qa, q n b ; q, q 22 dx = qac x n qb, cx ; q n q n, q n a, qbd x qa, qac 3φ 2 d; q n qb, q n c ; q, q n+ ac d 23 x = qac x n qb, q; q n n cx ; q n k qbd x; q k qac k q 2 kk dx k. d; q n q, qa; q n k qb, q; q k k=

24 Formula 22 follows from 6 by [GR, III.] and next 23 follows by series inversion [GR, Exercise.4ii]. Formulas 22 and 24 are also given in [Ism, , ]. It follows from 23 or 24 that see [298,.7] or [Ism, 8.4.3] lim n qac x n P n x; a, b, c, d; q = cx, dx ; q qac, 25 d, qa; q uniformly for x in compact subsets of C\{0}. Exclusion of the spectral points x = cq m, dq m m = 0,, 2,..., as was done in [298] and [Ism], is not necessary. However, while 25 yields 0 at these points, a more refined asymptotics at these points is given in [298] and [Ism]. For the proof of 25 use that lim n qac x n P n x; a, b, c, d; q = qb, cx ; q n qbd x qa, qac φ ; q, dx, 26 d; q n qb which can be evaluated by [GR, II.5]. Formula 26 follows formally from 23, and it follows rigorously, by dominated convergence, from 24. Symmetry see [K9, 2.5]. P n x; a, b, c, d; q P n d/qb; a, b, c, d; q = P nx; b, a, d, c; q. 27 Special values P n c/qa; a, b, c, d; q =, 28 P n d/qb; a, b, c, d; q = ad n qb, qbc/d; q n, 29 bc qa, qad/c; q n P n c; a, b, c, d; q = q ad n 2 nn+ qbc/d; q n, 30 c qad/c; q n P n d; a, b, c, d; q = q 2 nn+ a n qb; q n qa; q n. 3 Quadratic transformations see [K9, 2.48, 2.49] and 6. These express big q-jacobi polynomials P m x; a, a,, ; q in terms of little q-jacobi polynomials see 4.2. P 2n x; a, a,, ; q = p nx 2 ; q, a 2 ; q 2 p n qa 2 ; q, a 2 ; q 2, 32 P 2n+ x; a, a,, ; q = qax p nx 2 ; q, a 2 ; q 2 p n qa 2 ; q, a 2 ; q

25 Hence, by 4.2., [GR, Exercise.4ii] and 6, P n x; a, a,, ; q = qa2 ; q 2 n q qa 2 qax n n, q n+ 2φ ; q n q 2n+ a 2 ; q2, ax 2 k=0 34 = q; q [ 2 n] n qa 2 qa n k q kk qa 2 ; q 2 n k ; q n q 2 ; q 2 x n 2k. 35 k q; q n 2k q-chebyshev polynomials In 6, with c = d =, 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 because of the limit The quadratic relations 32, 33 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 32, 33, the Al-Salam-Ismail polynomials U n x; a, b q-dependence suppressed in the case a = q can be expressed as q-chebyshev polynomials of the second kind: U n x, q, b = q 3 b 2 n qn+ q P n b 2 x; q 2, q 2,, ; q. Similarly, by [K7, 5.4, 5., 5.3] and 32, 33, Cigler s q-chebyshev polynomials T n x, s, q and U n x, s, q can be expressed in terms of the q-chebyshev cases of 6: T n x, s, q = s 2 n P n qs 2 x; q 2, q 2,, ; q, U n x, s, q = q 2 s 2 n qn+ q Limit to Discrete q-hermite I P n qs 2 x; q 2, q 2,, ; q. lim a 0 a n P n x; a, a,, ; q = q n h n x; q. 36 Here h n x; q is given by For the proof of 36 use 22. 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 <. Let N be the largest nonnegative integer such that q 2N > ab/qcd. Then where z q Z z + q Z P m cx; c/b, d/a, c/a; q P n cx; c/b, d/a, c/a; q ax, bx; q cx, dx; q d q x = h n δ m,n m, n = 0,,..., N, 37 h n h 0 = n c 2 ab n q 2 nn q 2n q, qd/a, qd/b; q n qcd/ab, qc/a, qc/b; q n qcd/ab q 2n+ cd/ab 38 25

26 and h 0 = z q Z z + q Z ax, bx; q cx, dx; q d q x = qz + q, a/c, a/d, b/c, b/d; q ab/qcd; q θz /z +, cdz z + ; q θcz, dz, cz +, dz + ; q. 39 See Groenevelt & Koelink [K6, Prop. 2.2]. Formula 39 was first given by Slater [K29, 5] as an evaluation of a sum of two 2 ψ 2 series. The same formula is given in Slater [47, 7.2.6] and in [GR, Exercise 5.0], but in both cases with the same slight error, see [K6, 2nd paragraph after Lemma 2.] for correction. The theta function is given by 9. Note that P n cx; c/b, d/a, c/a; q = P n q ax; c/b, d/a, a/b, ; q. 40 In [K4] 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 lim a in q 2 nn P n q a ix; a, a,, ; q = h n x; q. 4 For the proof use 35 and 96. Note that P n q a ix; a, a,, ; q is obtained from the right-hand side of 4 by replacing a, b, c, d by ia, ia, i, i. Pseudo big q-jacobi Pseudo Jacobi lim P n iq 2 N +iν x; q N, q N, q N+iν ; q = P nx; ν, N q P n i; ν, N. 42 Here the big q-jacobi polynomial on the left-hand side equals P n cx; c/b, d/a, c/a; q with a = iq 2 N+ iν, b = iq 2 N++iν, c = iq 2 N +iν, d = iq 2 N iν. 4.7 Dual q-hahn Orthogonality relation More generally we have with positive weights in any of the following cases: i 0 < γq <, 0 < δq < ; ii 0 < γq <, δ < 0; iii γ < 0, δ > q N ; iv γ > q N, δ > q N ; v 0 < qγ <, δ = 0. This also follows by inspection of the positivity of the coefficient of p n x in Case v yields Affine q-krawtchouk in view of Symmetry R n x; γ, δ, N q = δ q N ; q n γq; q n This follows from 4.7. combined with [GR, III.]. γδq N+ n Rn γ δ q N x; δ q N, γ q N, N q

27 4.8 Al-Salam-Chihara Symmetry The Al-Salam-Chihara polynomials Q n x; a, b q are symmetric in a, b. This follows from the orthogonality relation together with the value of its coefficient of x n given in 4.8.5b. q -Al-Salam-Chihara Re: 4.8. For x Z 0 : Q n 2 aq x + a q x ;a, b q = n b n q 2 nn ab ; q n q n, q x, a 2 q x 3 φ = ab x q 2 xx+ qba ; q x a b ; q x 2φ ab ; q, q n ab q x, a 2 q x qba ; q, q n = ab x q 2 xx+ qba ; q x a b ; q x p x q n ; ba, qab ; q. 46 Formula 44 follows from the first identity in Next 45 follows from [GR, III.8]. Finally 46 gives the little q-jacobi polynomials See also [79, 3]. Orthogonality x=0 q 2x a 2 a 2, ab ; q x a 2 q, bqa ; q x ba x q x2 Q m Q n 2 aq x + a q x ; a, b q = qa 2 ; q ba q; q q, ab ; q n ab n q n2 δ m,n ab >, qb < a. 47 This follows from 46 together with 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 3.8, 3.67, Normalized recurrence relation xp n x = p n+ x + 2 a + bq n p n x + 4 q n abq n+ p n x, 48 where Q n x; a, b q = 2 n p n x. 4.9 q-meixner-pollaczek The q-meixner-pollaczek polynomials are the special case of Askey-Wilson polynomials with parameters ae iφ, 0, ae iφ, 0: P n x; a, φ q := q; q n p n x; ae iφ, 0, ae iφ, 0 q 27 x = cosθ + φ.

28 In [KLS, 4.9] the parameter dependence on φ is incorrectly omitted. Since all formulas in 4.9 are specializations of formulas in 4., there is no real need to give these specializations explicitly. See also 0. There is an error in [KLS, 4.9.6, 4.9.8]. Read x = cosθ + φ instead of x = cos θ. 4.0 Continuous q-jacobi Symmetry This follows from 04 and P n α,β x q = n q 2 α βn P n β,α x q Continuous q-ultraspherical / Rogers Re: C n cos θ; β q = β2 ; q n q; q n β 2 n 4φ 3 see [GR, 7.4.3, 7.4.4]. q 2 n, βq 2 n, β 2 e iθ, β 2 e iθ β, β 2 q 4, β 2 q 4 ; q 2, q 2, 50 Special value see [63, 3.23] C n 2 β 2 + β 2 ; β q = β 2 ; q n q; q n β 2 n. 5 Re: another q-difference equation. Let C n [e iθ ; β q] := C n cos θ; β q. βz 2 z 2 C n[q 2 z; β q] + βz 2 z 2 C n[q 2 z; β q] = q 2 n + q 2 n β C n [z; β q], 52 see [35, 6.0]. Re: This can also be written as C n [q 2 z; β q] Cn [q 2 z; β q] = q 2 n β z z C n [z; qβ q]. 53 Two other shift relations follow from the previous two equations: β + C n [q 2 z; β q] = q 2 n + q 2 n βc n [z; β q] + q 2 n β z βz C n [z; qβ q], 54 β + C n [q 2 z; β q] = q 2 n + q 2 n βc n [z; β q] + q 2 n β z βzc n [z; qβ q]

29 Trigonometric representation see p.473, Remarks, first formula C n cos θ; β q = n k=0 β; q k β; q n k q; q k q; q n k e in 2kθ. 56 Limit for q see [63, pp ]. By 56 and 54 we obtain lim C 2m x; q λ q = C 2 λ+ m 2x 2 + C 2 λ+ m 2x 2, q lim C 2m+ x; q λ q = 2x C 2 λ+ m 2x 2. q By 50 and [HTF2, 0.636] this can be rewritten as lim C 2m x; q λ q = λ m q 2 λ P 2 λ, 2 λ m 2x 2, 57 m lim C 2m+ x; q λ q = 2 λ + m q 2 λ + x P 2 λ, 2 λ m 2x m By 4 the limits 57, 58 imply that lim C n x; q λ q = const. S 2 λ, 2 λ n x, 59 q where the right-hand side gives a one-parameter subclass of the generalized Gegenbauer polynomial. Note that in [K3, Section 7.] the generalized Gegenbauer polynomials are also observed as fitting in the q = Askey scheme, but the limit 59 is not observed there. 4. Big q-laguerre Symmetry The big q-laguerre polynomials P n x; a, b; q are symmetric in a, b. This follows from 4... As a consequence, it is sufficient to give generating function 4... Then the generating function 4..2 will follow by symmetry in the parameters. 4.2 Little q-jacobi Notation Here the little q-jacobi polynomial is denoted by p n x; a, b; q instead of p n x; a, b q. Special values see [K9, 2.4]. p n 0; a, b; q =, 60 p n q b ; a, b; q = qb n q 2 nn qb; q n qa; q n, 6 p n ; a, b; q = a n q 2 nn+ qb; q n qa; q n

30 4.4 Quantum q-krawtchouk q-hypergeometric representation For n = 0,,..., N see 4.4. and use 7: q Kn qtm n, y y; p, N; q = 2 φ q N ; q, pqn+ = pyq N+ ; q n 3 φ 2 q n, q N /y, 0 q N, q N n /py ; q, q Special values By 63 and [GR, II.4]: K qtm n ; p, N; q =, K qtm n q N ; p, N; q = pq; q n. 65 By 64 and 65 we have the self-duality K qtm n q x N ; p, N; q K qtm n q N ; p, N; q = K By 65 and 66 we have also qtm x q n N ; p, N; q K qtm x q N ; p, N; q n, x {0,,..., N}. 66 K qtm N q x ; p, N; q = pq N ; q x x {0,,..., N}. 67 Limit for q to Krawtchouk see and Section 9.: lim q Kqtm n + qx; p, N; q = K n x; p, N, 68 lim q Kqtm n q x ; p, N; q = K n x; p, N. 69 Quantum q -Krawtchouk By 63, 65, 6 and 72 see also p.496, second formula: Kn qtm y; p, N; q Kn qtm q N ; p, N; q = q n, y pq ; q 2φ n q N ; q, pyq N 70 Rewrite 7 as K qtm m + q qx; p, N; q = pq ; q n K Aff n = K Aff n q N y; p, N; q. 7 + qq N q N q x ; p, N; q. In view of 68 and 77 this tends to 7 as q. The orthogonality relation holds with positive weights for q > if p > q. 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 SU q 2 considered with respect to its quantum subgroup U. 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

31 4.6 Affine q-krawtchouk q-hypergeometric representation For n = 0,,..., N see 4.6.: Kn Aff q n, q N y y; p, N; q = p q ; q 2φ n q N ; q, p y 72 q n, y, 0 = 3 φ 2 q N, pq ; q, q. 73 Self-duality By 73: Special values Kn Aff q x ; p, N; q = Kx Aff q n ; p, N; q n, x {0,,..., N}. 74 By 72 and [GR, II.4]: By 75 and 74 we have also K Aff n ; p, N; q =, K Aff n q N ; p, N; q = K Aff N q x ; p, N; q = Limit for q to Krawtchouk see and Section 9.: pq ; q n. 75 pq ; q x. 76 lim q KAff n + qx; p, N; q = K n x; p, N, 77 lim q KAff n q x ; p, N; q = K n x; p, N. 78 A relation between quantum and affine q-krawtchouk By 63, 72, 75 and 74 we have for x {0,,..., N}: N n q x ; p q N, N; q = KAff x q n ; p, N; q Kx Aff q N ; p, N; q K qtm 79 = KAff n q x ; p, N; q K Aff N q x ; p, N; q. 80 Formula 79 is given in [K3, formula after 2] and [K2, 59]. In view of 69 and 78 formula 80 has 72 as a limit case for q. Affine q -Krawtchouk By 72, 75, 6 and 63 see also p.505, first formula: Kn Aff y; p, N; q q n Kn Aff q N ; p, N; q =, q N y 2φ q N ; q, p q n+ 8 = K qtm n q N y; p, N; q. 82 Formula 82 is equivalent to 7. Just as for 7, it tends after suitable substitutions to 7 as q. The orthogonality relation holds with positive weights for q > if 0 < p < q N. 3

32 History The affine q-krawtchouk polynomials were considered by Delsarte [6, Theorem ], [K8, 6] in connection with certain association schemes. He called these polynomials generalized Krawtchouk polynomials. Note that the 2 φ 2 in [K8, 6] is in fact a 3 φ 2 with one upper parameter equal to 0. Next Dunkl [86, Definition 2.6, Section 5.] 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, 4.3]. He chose this name because, in [488, pp. 5 6] 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 GFq. Let G be the group of block matrices SA B A 0 B GL v n q and S X. Then G acts on X by SA B 4.7 Dual q-krawtchouk T = BT A + S., where A GL n q, Symmetry K n x; c, N q = c n K n c x; c, N q. 83 This follows from 4.7. combined with [GR, III.]. In particular, K n x;, N q = n K n x;, N q Little q-laguerre / Wall Notation Here the little q-laguerre polynomial is denoted by p n x; a; q instead of p n x; a q. Re: The right-hand side of this generating function converges for xt <. We can rewrite the left-hand side by use of the transformation Then we obtain: Expansion of x n 0, 0 2φ c ; q, z = 0, 0 t; q 2 φ ; q, xt = aq 0φ ; q, cz. z; q c n q 2 nn n=0 q; q n p n x; a; q t n xt <. 85 Divide both sides of 85 by t; q. Then coefficients of the same power of t on both sides must be equal. We obtain: x n = a; q n n k=0 q n ; q k q; q k q nk p k x; a; q

33 Quadratic transformations Little q-laguerre polynomials p n x; a; q with a = q ± 2 are related to discrete q-hermite I polynomials h n x; q: 4.2 q-laguerre Notation p n x 2 ; q ; q 2 = n q nn q; q 2 n h 2n x; q, 87 xp n x 2 ; q; q 2 = n q nn q 3 ; q 2 n h 2n+ x; q. 88 Here the q-laguerre polynomial is denoted by L α nx; q instead of L α n x; q. Orthogonality relation can be rewritten with simplified right-hand side: with 0 L α mx; q L α nx; q h n h 0 = qα+ ; q n q; q n q n, x α x; q dx = h n δ m,n α > 89 h 0 = q α ; q q; q π sinπα. 90 The expression for h 0 which is Askey s q-gamma evaluation [K, 4.2] should be interpreted by continuity in α for α Z 0. Explicitly we can write Expansion of x n h n = q 2 αα+ q; q α logq α Z 0. 9 x n = q 2 nn+2α+ q α+ ; q n n k=0 q n ; q k q α+ q k L α k x; q. 92 ; q k This follows from 86 by the equality given in the Remark at the end of Alternatively, it can be derived in the same way as 86 from the generating function Quadratic transformations q-laguerre polynomials L α nx; q with α = ± 2 hn x; q: are related to discrete q-hermite II polynomials L /2 n x 2 ; q 2 = n q 2n2 n q 2 ; q 2 h2n x; q, 93 n xl /2 n x 2 ; q 2 = n q 2n2 +n q 2 ; q 2 n h2n+ x; q. 94 These follows from 87 and 88, respectively, by applying the equalities given in the Remarks at the end of 4.20 and

34 4.27 Stieltjes-Wigert An alternative weight function The formula on top of p.547 should be corrected as wx = γ π x 2 exp γ 2 ln 2 x, x > 0, with γ 2 = 2 ln q. 95 For w the weight function given in [Sz, 2.7] the right-hand side of 95 equals const. wq 2 x. See also [DLMF, 8.27vi] Discrete q-hermite I History Discrete q Hermite I polynomials not yet with this name first occurred in Hahn [26], see there p.29, case V and the q-weight πx 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, [26, 4., 4.2] yield constant multiples of h 2n q x; q and h 2n+ q x; q, respectively, due to the quadratic transformations 87, 88 together with Discrete q-hermite II Basic hypergeometric representation see hn x; q = x n 2φ q n, q n+ 0 ; q 2, q 2 x Standard references [AAR] G. E. Andrews, R. Askey and R. Roy, Special functions, Cambridge University Press, 999. [DLMF] NIST Handbook of Mathematical Functions, Cambridge University Press, 200; DLMF, Digital Library of Mathematical Functions, [GR] G. Gasper and M. Rahman, Basic hypergeometric series, 2nd edn., Cambridge University Press, [HTF] A. Erdélyi, Higher transcendental functions, Vol., McGraw-Hill, 953. [HTF2] A. Erdélyi, Higher transcendental functions, Vol. 2, McGraw-Hill, 953. [Ism] M. E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Cambridge University Press, 2005; reprinted and corrected,

35 [KLS] [Sz] R. Koekoek, P. A. Lesky and R. F. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, Springer-Verlag, 200. G. Szegő, Orthogonal polynomials, Colloquium Publications 23, American Mathematical Society, Fourth Edition, 975. References from Koekoek, Lesky & Swarttouw [24] W. A. Al-Salam and M. E. H. Ismail, Orthogonal polynomials associated with the Rogers- Ramanujan continued fraction, Pacific J. Math , [46] R. Askey, Orthogonal polynomials and special functions, CBMS Regional Conference Series, Vol. 2, SIAM, 975. [5] R. Askey, Beta integrals and the associated orthogonal polynomials, in: Number theory, Madras 987, Lecture Notes in Mathematics 395, Springer-Verlag, 989, pp [63] R. Askey and M. E. H. Ismail, A generalization of ultraspherical polynomials, in: Studies in Pure Mathematics, Birkhäuser, 983, pp [64] R. Askey and M. E. H. Ismail, Recurrence relations, continued fractions, and orthogonal polynomials, Mem. Amer. Math. Soc , no [72] R. Askey and J. A. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc , no. 39. [79] N. M. Atakishiyev and A. U. Klimyk, On q-orthogonal polynomials, dual to little and big q-jacobi polynomials, J. Math. Anal. Appl , [9] W. N. Bailey, The generating function of Jacobi polynomials, J. London Math. Soc , 8 2. [09] F. Brafman, Generating functions of Jacobi and related polynomials, Proc. Amer. Math. Soc. 2 95, [46] T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach, 978; reprinted Dover Publications, 20. [6] Ph. Delsarte, Association schemes and t-designs in regular semilattices, J. Combin. Theory Ser. A , [86] C. F. Dunkl, An addition theorem for some q-hahn polynomials, Monatsh. Math , [234] G. Gasper and M. Rahman, Positivity of the Poisson kernel for the continuous q- ultraspherical polynomials, SIAM J. Math. Anal ,