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Two Classes of Special Functions Using Fourier Transforms of Generalized Ultraspherical and Generalized Hermite Polynomials

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Two Classes of Special Functions Using Fourier Transforms of Generalized Ultraspherical and Generalized Hermite Polynomials PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 6, June 2012, Pages 2053–2063 S 0002-9939(2011)11063-3 Article electronically published on October 14, 2011 TWO CLASSES OF SPECIAL FUNCTIONS USING FOURIER TRANSFORMS OF GENERALIZED ULTRASPHERICAL AND GENERALIZED HERMITE POLYNOMIALS MOHAMMAD MASJED-JAMEI AND WOLFRAM KOEPF (Communicated by Walter Van Assche) Abstract. Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. Motivated by the paper [H. T. Koelink, On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc., 124 (1996) 887- 898], in this paper we introduce two new classes of orthogonal functions, which are respectively Fourier transforms of the generalized ultraspherical polynomi- als and generalized Hermite polynomials, and then obtain their orthogonality relations using Parseval’s identity. 1. Introduction It is known that the generalized ultraspherical (Gegenbauer) polynomials with weight |x|2a(1−x2)b on [−1, 1] and the generalized Hermite polynomials with weight 2 |x|2ae−x on (−∞, ∞) are two basic examples of sieved orthogonal polynomials [1, 8]. The date of these polynomials goes back to 1984, when Al-Salam, Allaway and Askey [1] observed that some interesting polynomials arise as limiting cases of the continuous q-ultraspherical polynomials introduced by L.J. Rogers [16]. They then named them “sieved ultraspherical polynomials of the first and second kind” and showed that the polynomials of the first kind satisfy a three-term recurrence relation as Cλ (x; k)=2xCλ(x; k) − Cλ (x; k) k = n, (1.1) n+1 n n−1 λ λ − λ (m +2λ)Cmk+1(x; k)=2x(m + λ)Cmk(x; k) mCmk−1(x; k) m>0, λ λ for C0 (x; k)=1,C1 (x; k)=x(k>1), while the polynomials of the second kind satisfy the relation ⎧ ⎪ λ λ − λ ⎨Bn+1(x; k)=2xBn(x; k) Bn−1(x; k) k = n +1, (1.2) mBλ (x; k)=2x(m + λ)Bλ (x; k) ⎩⎪ mk mk−1 − λ (m +2λ)Bmk−2(x; k) m>0, λ λ in which B0 (x; k)=1,B1 (x; k)=2x (k>1). Received by the editors August 17, 2010 and, in revised form, January 10, 2011 and Febru- ary 10, 2011. 2010 Mathematics Subject Classification. Primary 33C45, 42A38. This research was supported in part by a grant from IPM, No. 89330021, and by a grant from “Bonyade Mellie Nokhbegan”, No. PM/1184. c 2011 By the authors 2053 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2054 MOHAMMAD MASJED-JAMEI AND WOLFRAM KOEPF λ Al-Salam, Allaway and Askey also derived orthogonality relations of Cn (x; k)and λ Bn(x; k) from the orthogonality relation of the continuous q-ultraspherical polyno- mials and showed that the first and second kind of sieved ultraspherical polynomials are respectively orthogonal on [−1, 1] with respect to the weight functions 2 α−1/2 2α (1.3) W1(x, k, α)=(1− x ) | Uk−1(x) | , 2 α+1/2 2α (1.4) W2(x, k, α)=(1− x ) | Uk−1(x) | , where k ∈ N is a fixed integer and Uk(x) denotes the second kind of Chebyshev polynomials [5, 9]. Later on, Askey in [3] generalized the above-mentioned results and introduced two further classes of sieved polynomials as limiting cases of the q-Wilson polynomi- als that are respectively orthogonal on [−1, 1] with respect to the weight functions 2 α−1/2 2α λ (1.5) W1(x, k, α, λ)=(1− x ) | Uk−1(x) | | Tk(x) | and 2 α+1/2 2α λ (1.6) W2(x, k, α, λ)=(1− x ) | Uk−1(x) | | Tk(x) | , where Tk(x) denotes the first kind of Chebyshev polynomials [9, 17]. 2α 2 b In this paper, we deal with the special case W1(x, 1,b+1/2, 2α)=|x| (1 − x ) on [−1, 1] corresponding to the well-known generalized ultraspherical polynomials (GUP), which were first investigated by Chihara in detail [5]. Chihara could ob- tain the general properties of GUP via a direct relation between them and Jacobi orthogonal polynomials [9]. By referring to [14], GUP has the explicit hypergeometric representation −[n/2], −a +1/2 − [(n +1)/2] 1 (1.7) U (a,b)(x)=xn F , n 2 1 −a − b − n +1/2 x2 where 2F1(·) is a special case of the generalized hypergeometric function [4, 11] of order (p, q)as ∞ k a1 a2 ...ap (a1)k(a2)k ...(ap)k x (1.8) pFq x = , b b ...b (b1)k(b2)k ...(bq)k k! 1 2 q k=0 k−1 and (r)k = i=0 (r + i) denotes the Pochhammer symbol in (1.8). Also, the or- thogonality relation of the polynomials (1.7) is explicitly in the form [14] (1.9) 1 | |2a − 2 b (a,b) (a,b) x (1 x ) Un (x)Um (x)dx −1 ⎛ ⎞ Γ(a +1/2)Γ(b +1)n (j +(1− (−1)j )a)(j +(1− (−1)j )a +2b) = ⎝ ⎠ δ Γ(a + b +3/2) (2j +2a +2b − 1)(2j +2a +2b +1) n,m j=1 1 =Γ a + 1 Γ a + b − 1 Γ [ n ]+1 Γ(b + +[n ]) 2 2 2 2 2 − − n − − n Γ(a +1+[n ]+ 1 ( 1) )Γ(a + b + 1 +[n ]+ 1 ( 1) ) × 2 2 2 2 2 δ , 3 − 1 1 n,m Γ(a +1)Γ(a + b + 2 )Γ(a + b + n 2 )Γ(a + b + n + 2 ) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use TWO CLASSES OF FUNCTIONS USING FOURIER TRANSFORMS 2055 0(n = m), provided that a +1/2 > 0, a + b − 1/2 > 0,δn,m = and Γ(z)= 1(n = m), ∞ z−1 −x 0 x e dx(Re(z) > 0) denotes the well-known Gamma function satisfying the fundamental recurrence relation Γ(z +1)=zΓ(z). Similarly, the generalized Hermite polynomials (GHP) were first introduced by Szeg˝o in [17] giving an explicit second order differential equation. These polynomi- als can be characterized by using a direct relationship between them and Laguerre orthogonal polynomials [9]. By referring to [14], the hypergeometric representation of GHP is explicitly in the form −[n/2], −[n/2] − a +(−1)n/2 1 (1.10) H(a)(x)=xn F − . n 2 0 − x2 Moreover, the orthogonality relation of these polynomials takes the form ∞ | |2a −x2 (a) (a) (1.11) x e Hn (x)Hm (x)dx −∞ 1 n 1 = (1 − (−1)i)a + i Γ(a + )δ 2n 2 n,m i=1 1 n +1 n =Γ a + +[ ] Γ [ ]+1 δ (a>−1/2). 2 2 2 n,m But, it is known that some orthogonal polynomial systems are mapped onto each other by the Fourier transform or other integral transforms such as the Mellin and Hankel transforms [7]. The best-known examples of this type are the Hermite 2 functions, i.e. the Hermite polynomials Hn(x) multiplied by exp(−x /2), which are eigenfunctions of the Fourier transform; see also [12, 13, 15] in this regard. Especially in [10], Koelink showed that the Jacobi and continuous Hahn polyno- mials can be mapped onto each other in such a way, and the orthogonality relations for the continuous Hahn polynomials then follow from the orthogonality relations of the Jacobi polynomials and using the Parseval identity. Motivated by Koelink’s paper, in this section we apply this approach for GUP and GHP to introduce two new classes of orthogonal functions by using Fourier transforms and the Parseval identity. 2. Fourier transforms of GUP and GHP and their orthogonality relations To derive the Fourier transforms of GUP and GHP defined in (1.7) and (1.10), we will need the Beta integral having various definitions as 1 1 λ1−1 λ2−1 2λ1−1 2 λ2−1 (2.1) B(λ1; λ2)= x (1 − x) dx = x (1 − x ) dx − 0 1 ∞ λ −1 π/2 x 1 − − = dx =2 sin(2λ1 1) x cos(2λ2 1) x dx λ1+λ2 0 (1 + x) 0 Γ(λ1)Γ(λ2) = = B(λ2; λ1). Γ(λ1 + λ2) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2056 MOHAMMAD MASJED-JAMEI AND WOLFRAM KOEPF The Fourier transform of a function, say g(x), is defined as [7] ∞ (2.2) F(s)=F(g)(s)= e−isxg(x)dx, −∞ and for the inverse transform one has the formula 1 ∞ (2.3) g(x)= eisxF(s)ds. 2π −∞ For g, h ∈ L2(R), the Parseval identity related to Fourier theory is given by [7]: ∞ 1 ∞ (2.4) g(x)h(x)dx = F(g)(s)F(h)(s)ds. −∞ 2π −∞ By noting (1.9) and (2.4) and the fact that |x|2a = x2a ⇔ (−1)2a =1,wenow define the following specific functions: g(x) = (tanh x)2α(1 − tanh2 x)βU (c,d)(tanh x)s.t.(−1)2α =1, (2.5) n 2l 2 u (v,w) 2l h(x) = (tanh x) (1 − tanh x) Um (tanh x)s.t.(−1) =1, in terms of GUP, to which we will apply the Fourier transform. Clearly for both of the above functions the Fourier transform exists. For instance, for the function g(x) defined in (2.5) we get ∞ −isx 2α − 2 β (c,d) (2.6) F(g)(s)= e (tanh x) (1 tanh x) Un (tanh x)dx −∞ 1 − is is − 2 − 2 2α − 2 β 1 (c,d) = (1 + t) (1 t) t (1 t ) Un (t)dt − 1 1 − − is is − − 2β 1 − 2 2 − 2α β 1 − β 1 (c,d) − =2 (1 z) z (1 2z) z (1 z) Un (1 2z)dz 0 1 2β−1 β−1− is β−1+ is 2α+n =2 (1 − z) 2 z 2 (1 − 2z) 0 ⎛ ⎞ [ n/2] − n 1 − c − n+1 × ⎝ 2 k 2 2 k · 1 ⎠ 2k dz (−c − d − n +1/2)kk! (1 − 2z) k=0 [ n/2] − n 1 − c − n+1 =22β−1 2 k 2 2 k (−c − d − n +1/2)kk! k=0 1 β−1− is β−1+ is 2α+n−2k × (1 − z) 2 z 2 (1 − 2z) dz .
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