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

Home , Gegenbauer polynomials, 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 polynomials 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

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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 (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 polynomials 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 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 (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 orthogonality 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 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 (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 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 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)

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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 . 0 On the other hand, since a, b Γ(c) 1 − c−b−1 b−1 − −a (2.7) 2F1 x = − (1 z) z (1 xz) dz c Γ(b)Γ(c b) 0

is the integral representation of the Gauss hypergeometric function [4, 11] for Re c> Re b>0and|x|≤1, the last integral in (2.6) can be shown in terms of (2.7), and

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we ﬁnally obtain [n/2] is is − n 1 − c − n+1 (2.8) F(g)(s)=22β−1 B β + ,β− 2 k 2 2 k 2 2 (−c − d − n +1/2)kk! k=0 2k − n − 2α, β + is × F 2 2 . 2 1 2β

Remark 2.1. Since the series 2F1(·) in (2.8) does not generally terminate and also 2 ∈/ [−1, 1], the evaluation at x = 2 is rather troublesome unless we assume in (2.6) or(2.8)thatRe(β) > 0andRe(2α + n − 2k) > −1. For simplicity if we deﬁne (2.9) Kn(x; p1,p2,p3,p4) [ n/2] − n 1 − − n+1 − − x 2 k 2 p3 2 k 2k n 2p1,p2 + 2 = 2F1 2 , (−p3 − p4 − n +1/2)kk! 2p k=0 2 then it is clear in (2.8) that 22β − 1 is is (2.10) F(g)(s)= Γ β + Γ β − K (is; α, β, c, d). Γ(2β) 2 2 n Now, by substituting (2.10) in Parseval’s identity (2.4) and noting (2.5), we get ∞ 2(α+l) − 2 β+u (c,d) (v,w) (2.11) 2π (tanh x) (1 tanh x) Un (tanh x)Um (tanh x)dx −∞

1 2(α+l) − 2 β+u−1 (c,d) (v,w) =2π t (1 t ) Un (t)Um (t)dt − 1 22β+2u−2 ∞ is is is is = Γ β + Γ β − Γ u + Γ u − Γ(2β)Γ(2u) −∞ 2 2 2 2

× Kn(is; a, b, c, d)Km(is; l, u, v, w)ds. On the other hand, if in the left-hand side of (2.11) we take (2.12) c = v = α + l and d = w = β + u − 1, then according to the orthogonality relation (1.9), equation (2.11) ﬁnally reads as (2.13) 1 ∞ is is is is Γ β + Γ β − Γ u + Γ u − 2π −∞ 2 2 2 2

× Kn(is; α, β, α + l, β + u − 1)Km(−is; l, u, α + l, β + u − 1) ds Γ(2β)Γ(2u)Γ(α + l + 1 )Γ(β + u) = 2 22β+2u+2n−2Γ(α + β + l + u + 1 ) 2 n j +(1− (−1)j)(α + l) j +(1− (−1)j)(α + l)+2β +2u − 1 × δ , (j + α + β + l + u − 3 )(j + α + β + l + u − 1 ) n,m j=1 2 2

n · − 1 where j=1( )=1,β,u > 0, and α + l> 2 .

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Remark 2.2. Note that Kn(x; p1,p2,p3,p4) in (2.9) is not in general a polynomial function and it is reduced to a polynomial of degree n only when p1 =0.Thismay be a reason why we could not derive an analogue or a sieved type of continuous Hahn polynomials in relations (2.6) to (2.13). Remark 2.3. Replacing n = m =0andα = l = 0 in (2.13) gives a special case of the Barnes beta integral [18],

1 ∞ Γ(a + ix)Γ(b + ix)Γ(c − ix)Γ(d − ix)dx 2π −∞ (2.14) Γ(a + d)Γ(a + c)Γ(b + d)Γ(b + c) = , Γ(a + b + c + d) if one applies the duplication Legendre formula [6] 22z−1 1 (2.15) Γ(2z)= √ Γ(z)Γ z + . π 2 Moreover, the weight function corresponding to the orthogonality relation (2.13) can be simpliﬁed by using the Cauchy beta integral [2, 6],

∞ − 1 dt Γ(c + d 1) 1−(c+d) (2.16) c d = (a + b) , 2π −∞ (a + it) (b − it) Γ(c)Γ(d) and one of its consequences, i.e. 22−2pπΓ(2p − 1) (2.17) Γ(p + iq)Γ(p − iq)= π , 2 2qx 2p−2 − π e cos x dx 2 which results in is − is is − is (2.18) Γ β + 2 Γ β 2 Γ u + 2 Γ u 2 24−2β−2uπ2Γ(2β − 1)Γ(2u − 1) = π π . 2 sx 2β−2 2 sx 2u−2 − π e cos x dx − π e cos x dx 2 2 Therefore, if we deﬁne π 2 (2.19) W ∗(x; λ)= exθ cos2λ−2 θ dθ, − π 2 then the following theorem will eventually be derived.

Theorem 2.4. The special function Kn(x; p1,p2,p3,p4) defined in (2.9) satisfies an orthogonality relation as (2.20) ∞ Kn(ix; α, β, α + l, β + u − 1) Km(−ix; l, u, α + l, β + u − 1) ∗ ∗ dx −∞ W (x; β) W (x; u) (2u − 1)(2β − 1) Γ(α + l + 1 )Γ(β + u) = × 2 π 22n+1 Γ(α + β + l + u + 1 ) 2 n j +(1− (−1)j)(α + l) j +(1− (−1)j)(α + l)+2β +2u − 2 × δ , (j + α + β + l + u − 3 )(j + α + β + l + u − 1 ) n,m j=1 2 2 1 − 1 ∗ · where β,u > 2 , α + l> 2 and W (x; ) is defined as (2.19).

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For instance, for β = u = 1 the relation (2.20) is reduced to

∞ 2 x − − (2.21) 2 πx Kn(ix; α, 1,r,1) Km( ix; r α, 1,r,1) dx −∞ sinh ( 2 ) Γ(r + 1 )Γ(r + 2 ) = 2 3 2n−1 3 5 π2 Γ(r + 2 + n)Γ(r + 2 + n) n j j × j +(1− (−1) )r j +(1− (−1) )r +2 δn,m , j=1 − 1 where r = α + l> 2 . The mentioned approach can similarly be applied to the generalized Hermite polynomials. For this purpose, we first define the following specific functions: − 1 2 − 1 2 2a 2 x (b) 2c 2 x (d) (2.22) u(x)=x e Hn (x)andv(x)=x e Hm (x) , for (−1)2a =(−1)2c =1. If we take the Fourier transform for e.g. u(x), then we have ∞ − − 1 2 isx 2a 2 x (b) (2.23) F(u)(s)= e x e Hn (x)dx −∞ ∞ −isx − 1 x2 2a+n = e e 2 x −∞⎛ ⎞ [ n ] − n 2 (−[ n ]) (−[ n ] − b + ( 1) ) × ⎝ 2 k 2 2 k (−x−2)k⎠ dx k! k=0

[ n ] − n 2 (−[ n ]) (−[ n ] − b + ( 1) ) = 2 k 2 2 k (−1)k k! k=0 ∞ −isx − 1 x2 2a+n−2k × e e 2 x dx . −∞ So, it remains in (2.23) to evaluate the deﬁnite integral: ∞ −isx − 1 x2 2a+n−2k (2.24) In,k(s; a)= e e 2 x dx. −∞ There are two ways to compute the above integral. First, by noting that (−1)2a = 1 in (2.24), we can directly compute In,k(s; a)forn =2m as follows: ⎛ ⎞ ∞ ∞ − j ⎝ ( isx) ⎠ 2a+2m−2k − 1 x2 (2.25) I2m,k(s; a)= x e 2 dx −∞ j! j=0 ∞ − j j j ∞ ( 1) i s j+2a+2m−2k − 1 x2 = x e 2 dx j! −∞ j=0 ∞ − r 2r ∞ ( 1) s 2r+2a+2m−2k − 1 x2 = 2 x e 2 dx (2r)! r=0 0 ∞ − r 2r ( 1) s r+a+m−k+ 1 1 = 2 2 Γ(r + a + m − k + ) . (2r)! 2 r=0

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To simplify the last summation of (2.25) in terms of a hypergeometric form we need to use (2.16) and the relations

(2.26) − k − ( 1) Γ(a) 1 2n Γ(a + k)=(a)kΓ(a), Γ(a k)= , and (2n)! = ( 2 )n 2 n!. (1 − a)k

Therefore, (2.25) becomes

(2.27) ∞ r 2r r+a+m−k+ 1 1 1 (−1) s 2 2 Γ(a + m − k + )(a + m − k + ) I (s; a)= 2 2 r 2m,k 1 2r ( )r 2 r! r=0 2 − 1 2 a+m−k+ 1 1 a + m k + 2 s =2 2 Γ(a + m − k + ) F − . 1 1 1 2 2 2

This computational method can similarly be applied for I2m+1,k(s; a). We should − 2 ∞ j+2a+2m+1−2k x just note in this turn that −∞ x e 2 dx =0foranyj =0, 2, 4,.... After some computations we obtain

(2.28) − 3 2 a+m−k+ 3 3 a + m k + 2 s I (s; a)=(−is)2 2 Γ(a + m − k + ) F − . 2m+1,k 1 1 3 2 2 2

Thus, by combining both relations (2.27) and (2.28) and using the well-known n+1 − n 1−(−1)n identity [ 2 ] [ 2 ]= 2 we ﬁnally have a−k+ 1 +[ n+1 ] 1 n+1 (2.29) I (s; a)=2 2 2 Γ a − k + +[ ] n,k 2 2 n 1 − n+1 2 1−(−1) 2 + a k +[ 2 ] s × (−is) 2 F n − . 1 1 − (−1) 2 1 2

The second way to compute In,k(s; a) is that we respectively consider n =2m and n =2m + 1 and directly apply the cosine and sine Fourier transforms of the 2 − x 2a+n−2k function e 2 x ; see [7, Section 1.4, formula 14] and [7, Section 2.4, formula 24]. In other words, by noting that (−1)2a =1wehave

(2.30) ∞ ∞ 2a+2m−2k − 1 x2 2a+2m−2k − 1 x2 I2m,k(s; a)= cos(sx)x e 2 dx − i sin(sx)x e 2 dx −∞ −∞ ∞ 2a+2m−2k − 1 x2 =2 cos(sx)x e 2 dx 0 − 1 2 a+m−k+ 1 1 a + m k + 2 s =2 2 Γ(a + m − k + ) F − 1 1 1 2 2 2

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and

(2.31) ∞ 2a+2m+1−2k − 1 x2 I2m,+1k(s; a)= cos(sx)x e 2 dx −∞ ∞ 2a+2m+1−2k − 1 x2 − i sin(sx)x e 2 dx −∞ ∞ 2a+2m+1−2k − 1 x2 = −2i sin(sx)x e 2 dx 0 − 3 2 a+m−k+ 3 3 a + m k + 2 s =(−is)2 2 Γ(a + m − k + ) F − . 1 1 3 2 2 2

Consequently, the result (2.29) simpliﬁes (2.23) as

1 n+1 1 n+1 a+ 2 +[ 2 ] (2.32) F(u)(s)=Γ a + 2 + 2 2 n [ 2 ] − − n − n 1 − − n+1 1 ( 1) ( [ 2 ])k ( 2 b [ 2 ])k × (−is) 2 1 n+1 k ( − a − [ ])k 2 k! k=0 2 2 1 − n+1 2 2 + a k +[ 2 ] s × F n − . 1 1 − (−1) 2 1 2

On the other hand, according to Kummer’s formula [11], a c − a (2.33) F x = ex F −x , 1 1 c 1 1 c

relation (2.32) can be transformed to 1 n+1 − 1 2 1 n+1 a+ 2 +[ 2 ] 2 x (2.34) F(u)(s)=Γ a + 2 + 2 2 e n [ 2 ] − − n − n 1 − − n+1 1 ( 1) ( [ 2 ])k ( 2 b [ 2 ])k × (−is) 2 1 n+1 k ( − a − [ ])k 2 k! k=0 2 2 −a + k − [ n ] 2 1 2 × F n s . 1 1 − (−1) 2 1 2

Now, by noting (2.34), if for simplicity we deﬁne

n [ 2 ] − − n − n 1 − − n+1 1 ( 1) ( [ 2 ])k ( 2 q2 [ 2 ])k (2.35) J (x; q ,q )=(x) 2 n 1 2 1 n+1 k ( − q1 − [ ])k 2 k! k=0 2 2 −q + k − [ n ] 1 2 1 2 × F n s , 1 1 − (−1) 2 1 2

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then by referring to deﬁnitions (2.22) and applying Parseval’s identity we get ∞ 2(a+c) −x2 (b) (d) (2.36) 2π x e Hn (x) Hm (x)dx −∞

− n− − m 1 n+1 1 m+1 ( 1) ( 1) Γ(a + 2 +[ 2 ]) Γ(c + 2 +[ 2 ]) = i 2 −(a+ 1 +[ n+1 ]) −(c+ 1 +[ m+1 ]) 2 2 2 2 2 2 ∞ −s2 × e Jn(s; a, b) Jm(s; c, d)ds. −∞ Finally it is sufficient to assume in (2.36) that b = d = a + c and then refer to the orthogonality relation (1.11) to reach the following theorem. Theorem 2.5. The special function defined in (2.35) satisfies the orthogonality relation (2.37)

∞ −b−2[ n+1 ] n 1 n+1 2 π2 2 Γ([ ]+1)Γ(b + +[ ]) e−x J (x; a, b)J (x; b − a, b)dx = 2 2 2 δ n m 1 n+1 − 1 n+1 n,m −∞ Γ(a + 2 +[ 2 ])Γ(b a + 2 +[ 2 ]) − 1 − 2b where a, b > 2 and ( 1) =1.

References

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