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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 Two Classes of Special Functions Using Fourier Transforms of Generalized Ultraspherical and Generalized Hermite Polynomials Mohammad Masjed-Jamei a,c Wolfram Koepf b a Department of Mathematics, K.N.Toosi University of Technology, P.O.Box 16315-1618, Tehran, Iran, E-mail: [email protected] , [email protected] b Institute of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany, E-mail: [email protected] c School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran 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 polynomials and generalized Hermite polynomials, and then obtain their orthogonality relations using Parseval identity. Keywords. generalized ultraspherical polynomials, generalized Hermite polynomials, Fourier transform, Parseval identity, hypergeometric functions MSC(2010): 33C45, 42A38 1. Introduction It is known that the generalized ultraspherical (Gegenbauer) polynomials with weight 2 x 2a (1− x 2 )b on [−1,1] and the generalized Hermite polynomials with weight x 2a e −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 first kind of these polynomials satisfy a three term recurrence relation as λ λ λ ⎪⎧Cn+1 (x;k) = 2xCn (x;k) − Cn−1 (x;k) k ≠ 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 second kind satisfies the relation λ λ λ ⎪⎧Bn+1 (x;k) = 2x Bn (x;k) − Bn−1 (x;k) k ≠ n +1, (2) ⎨ λ λ λ ⎩⎪m Bmk (x;k) = 2x(m + λ) Bmk−1 (x;k) − (m + 2λ) Bmk−2 (x;k) m > 0, λ λ in which B0 (x;k) = 1, B1 (x;k) = 2x ( k >1). 1 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 polynomials 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 W1 (x,k,α) = (1− x ) U k−1 (x) , (3) and 2 α +1/ 2 2α W2 (x,k,α) = (1− x ) U k−1 (x) , (4) where k ∈ N is a fixed integer and U k (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 polynomials that are respectively orthogonal on [−1,1] with respect to the weight functions 2 α −1/ 2 2α λ W1 (x,k,α,λ) = (1− x ) U k −1 (x) Tk (x) , (5) and 2 α +1/ 2 2α λ W2 (x,k,α,λ) = (1− x ) U k −1 (x) Tk (x) , (6) where Tk (x) denotes the first kind of Chebyshev polynomials [9,17]. 2a 2 b In this paper, we deal with the special case W1 (x,1, b +1/ 2,2a) = 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 obtain 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 ⎞ U (a,b) (x) = x n F ⎜ ⎟ , (7) n 2 1 ⎜ 2 ⎟ ⎝ − a − b − n +1/ 2 x ⎠ where 2 F1 (.) is a special case of the generalized hypergeometric function [4,11] of order ( p,q) as ∞ k ⎛ a1 a2 ... a p ⎞ (a ) (a ) ...(a ) x F ⎜ x⎟ = 1 k 2 k p k , (8) p q ⎜ b b ... b ⎟ ∑ ⎝ 1 2 q ⎠ k =0 (b1 ) k (b2 ) k ...(bq ) k k! k −1 and (r) k = ∏(r + i) denotes the Pochhammer symbol in (8). i=0 Also, the orthogonality relation of the polynomials (7) is explicitly in the form [14] 2 j j 1 n 2a 2 b (a,b) (a,b) ⎛ Γ(a +1/ 2)Γ(b +1) ( j + (1− (−1) )a)( j + (1− (−1) )a + 2b)⎞ x (1− x ) U n (x)U m (x) dx = ⎜ ⎟δ n,m ∫−1 ⎜ ∏ ⎟ ⎝ Γ(a + b + 3 / 2) j=1 (2 j + 2a + 2b −1)(2 j + 2a + 2b +1) ⎠ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ n ⎞ ⎛ 1 n ⎞ ⎛ n 1− (−1) n ⎞ ⎛ 1 n 1− (−1) n ⎞ Γ⎜a + ⎟Γ⎜a + b − ⎟Γ⎜[ ]+1⎟Γ⎜b + +[ ]⎟Γ⎜a +1+[ ]+ ⎟Γ⎜a + b + +[ ]+ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 2 ⎠ ⎝ 2 2 ⎠ ⎝ 2 2 2 ⎠ = δ , ⎛ 3 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ n,m Γ()a +1 Γ⎜a + b + ⎟Γ⎜a + b + n − ⎟Γ⎜a + b + n + ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ (9) ∞ ⎧0 (n ≠ m), z−1 −x provided that a +1/ 2 > 0 , a + b −1/ 2 > 0 , δ n,m = and Γ(z) = x e dx ⎨ ∫0 ⎩1 (n = m), ( 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 Szegö in [17] giving an explicit second order differential equation. These polynomials 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 ⎞ H (a) (x) = x n F ⎜ − ⎟. (10) n 2 0 ⎜ 2 ⎟ ⎝ − x ⎠ Moreover, the orthogonality relation of these polynomials takes the form ∞ n 2a 2 ⎛ 1 ⎞ 1 x e −x H (a) (x) H (a) (x) dx = ⎜ (1− (−1)i )a + i⎟Γ(a + ) δ ∫ n m ⎜ n ∏ ⎟ n,m ⎝ 2 i=1 ⎠ 2 −∞ (11) ⎛ 1 n +1 ⎞ ⎛ n ⎞ = Γ⎜a + + [ ]⎟Γ⎜[ ] +1⎟ δ n,m (a > −1/ 2). ⎝ 2 2 ⎠ ⎝ 2 ⎠ 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 functions, i.e. the 2 Hermite polynomials Hxn ( ) 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 polynomials 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 Parseval identity. 2. Fourier transforms of GUP and GHP and their orthogonality relations To derive the Fourier transform of GUP and GHP defined in (7) and (10), we will need the Beta integral having various definitions as 3 1 1 ∞ λ1 −1 λ1 −1 λ2 −1 2λ1 −1 2 λ2 −1 x B(λ1 ;λ2 ) = x (1− x) dx = x (1− x ) dx = dx ∫ ∫ ∫ (1+ x)λ1 +λ2 0 −1 0 (12) π / 2 Γ(λ )Γ(λ ) = 2 sin (2λ1 −1) x cos(2λ2 −1) x dx = 1 2 = B(λ ;λ ). ∫ 2 1 0 Γ(λ1 + λ2 ) The Fourier transform of a function, say g(x) , is defined as [7] ∞ F(s) = F(g)(s) = e −isx g(x) dx , (13) ∫−∞ and for the inverse transform one has the formula 1 ∞ g(x) = eisx F(s) ds . (14) 2π ∫−∞ For gh,()∈ L2 R , the Parseval identity related to Fourier theory is given by [7] ∞ 1 ∞ g(x) h(x) dx = F(g)(s)F(h)(s) ds . (15) ∫−∞ 2π ∫−∞ By noting (9) and (15) and this fact that x 2a = x 2a ⇔ (−1) 2a = 1, we now define the following specific functions 2α 2 β (c,d ) 2α ⎪⎧g(x) = (tanh x) (1− tanh x) U n (tanh x) s.t. (−1) = 1, (16) ⎨ 2l 2 u (v,w) 2l ⎩⎪h(x) = (tanh x) (1− tanh x) U m (tanh x) s.t. (−1) = 1, in terms of GUP to which we will apply the Fourier transform. Clearly for both above functions the Fourier transform exists. For instance, for the function g(x) defined in (16) we get ∞ F(g)(s) = e −isx (tanh x) 2α (1− tanh 2 x) β U (c,d ) (tanh x) dx ∫ n −∞ 1 is is − = (1+ t) 2 (1− t) 2 t 2α (1− t 2 ) β −1 U (c,d ) (t) dt ∫ n −1 1 is is − = 22β −1 (1− z) 2 z 2 (1− 2z) 2α z β −1 (1− z) β −1 U (c,d ) (1− 2z) dz ∫ n 0 ⎛ n 1 n +1 ⎞ 1 is is ⎜ [n / 2] (−[ ]) ( − c −[ ]) ⎟ β −1− β −1+ k k 1 = 22β −1 (1− z) 2 z 2 (1− 2z) 2α +n ⎜ 2 2 2 ⎟ dz ∫ ⎜ ∑ (−c − d − n +1/ 2) k! (1− 2z) 2k ⎟ 0 ⎜ k =0 k ⎟ ⎝ ⎠ n 1 n +1 (−[ ]) ( − c −[ ]) 1 is is [n / 2] k k ⎛ β −1− β −1+ ⎞ = 22β −1 2 2 2 ⎜ (1− z) 2 z 2 (1− 2z) 2α +n−2k dz⎟.
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