Integral Transforms Involving the Product of Humbert and Bessel Functions and Its Application

Home , Appell series

AIMS Mathematics, 5(2): 1260–1274. DOI:10.3934/math.2020086 Received: 10 October 2019 Accepted: 06 January 2020 http://www.aimspress.com/journal/Math Published: 19 January 2020

Research article Integral transforms involving the product of Humbert and Bessel functions and its application

A. Belafhal1, N. Nossir1, L. Dalil-Essakali1 and T. Usman2,∗

1 LPNAMME, Laser Physics Group, Department of Physics, Faculty of Sciences, Choua¨ıb Doukkali University, P. B 20, 24000 El Jadida, Morocco 2 Department of Mathematics, School of Basic and Applied Sciences, Lingaya’s Vidyapeeth, Faridabad-121002, Haryana, India

* Correspondence: Email: [email protected]; Tel: +917417803673.

Abstract: In this paper, we develop some integral transforms involving a product of Humbert and Bessel functions with a weight e−γx2 . These integral transforms will be evaluated in terms of hypergeometric functions. Various transformation formulae are also evaluated in terms of Appell functions to complete this study. Some special cases of the evaluated integrals yield some inﬁnite series of generalized hypergeometric and Appell functions. As application, one of our main results is investigated to give an expression of the Generalized Humbert-Gaussian beams (GHGBs) propagating through a paraxial ABCD optical system.

Keywords: integral transforms; Bessel functions; Humbert functions; Appell functions; Hypergeometric functions Mathematics Subject Classiﬁcation: 33B15, 33C10, 33C15

1. Introduction

Integral transforms involving special functions have gained considerable attention in the bibliography. In the last decade, many papers are investigated to study the integral transforms involving the product of Bessel and other special functions (see [4,8,9]). In view of this, it is worth investigating a general integral transform involving Humbert and Bessel functions with a weight e−γx2 . This integral transform is important in evaluating the expression of the generalized Humbert-Gaussian beams propagating in the space. A closed form of the considered integral will be derived. To the best of our knowledge, the results of the present contribution have not been previously published. The following deﬁnitions are essential to recall for the present investigation: 1261

The four two-variables hypergeometric functions, called conﬂuent Humbert functions [7], are deﬁned by

X∞ X∞ (a) (b) xr ys Ψ (a, b; c, d; x, y) = r+s r , (1.1) 1 (c) (d) r! s! r=0 s=0 r s X∞ X∞ (a) (b) xr ys Φ (a, b; c; x, y) = r+s r , (1.2) 1 (c) r! s! r=0 s=0 r+s X∞ X∞ (a) (b) (c) xr ys Ξ (a, b, c; d; x, y) = r s r , (1.3) 1 (d) r! s! r=0 s=0 r+s and

X∞ X∞ (a) (b) xr ys Ξ (a, b; c; x, y) = r r , (1.4) 2 (c) r! s! r=0 s=0 r+s with |x| < 1, |y| < ∞ and c, d , 0, −1, −2,... The hypergeometric function 2F1 is deﬁned by (see [1])

X∞ (α) (β) zn F (α, β; γ; z) = n n , (1.5) 2 1 (γ) n! n=0 n where (α)n is the Pochhammer symbol deﬁned by Γ(α + n) (α) = , n Γ(α) with Γ is the gamma function (see [1]). The Kummer function is deﬁned by the series

X∞ (α) zk F (α; β; z) = k . (1.6) 1 1 (β) k! k=0 k In terms of the hypergeometric function (1.5), it’s easy to obtain Humbert functions (1.1)–(1.4) (see [13]). Therefore, we can write

X∞ (a) ys Ψ (a, b; c, d; x, y) = s F (a + s, b; c; x) , (1.7) 1 (d) s! 2 1 s=0 s

X∞ (a) ys Φ (a, b; c; x, y) = s F (a + s, b; c + s; x) , (1.8) 1 (c) s! 2 1 s=0 s X∞ (b) ys Ξ (a, b, c; d; x, y) = s F (a, c; d + s; x) , (1.9) 1 (d) s! 2 1 s=0 s and X∞ 1 ys Ξ (a, b; c; x, y) = F (a, b; c + s; x) , (1.10) 2 (c) s! 2 1 s=0 s

AIMS Mathematics Volume 5, Issue 2, 1260–1274. 1262 with |x| < 1, |y| < ∞ and c, d , 0, −1, −2,... The four Appell series [2], which are double hypergeometric series, are deﬁned by

X∞ X∞ (a) (b) (b0) xm yn F a, b, b0; c; x, y = m+n m n , (1.11) 1 (c) m! n! m=0 n=0 m+n with max{|x|, |y|} < 1;

X∞ X∞ (a) (b) (b0) xm yn F a, b, b0; c, c0; x, y = m+n m n , (1.12) 2 (c) (c0) m! n! m=0 n=0 m n with |x| + |y| < 1;

X∞ X∞ (a) (a0) (b) (b0) xm yn F a, a0, b, b0; c; x, y = m n m n , (1.13) 3 (c) m! n! m=0 n=0 m+n with max{|x|, |y|} < 1; and X∞ X∞ (a) (b) xm yn F a, b; c, c0; x, y = m+n m+n , (1.14) 4 (c) (c0) m! n! m=0 n=0 m n √ p with |x| + |y| < 1. These expressions, with c and c0 are neither zero nor a negative integer, can be expressed in terms of 2F1 as follows (see [13])

X∞ (a) (b) xm F a, b, b0; c; x, y = m m F a + m, b0; c + m; y , (1.15) 1 (c) m! 2 1 m=0 m X∞ (a) (b) xm F a, b, b0; c, c0; x, y = m m F a + m, b0; c0; y , (1.16) 2 (c) m! 2 1 m=0 m X∞ (a) (b) xm F a, a0, b, b0; c; x, y = m m F a0, b0; c + m; y , (1.17) 3 (c) m! 2 1 m=0 m and

X∞ (a) (b) xm F a, b; c; c0; x, y = m m F a + m, b + m; c0; y . (1.18) 4 (c) m! 2 1 m=0 m

2. Main results

2.1. Evaluation of the integral Im In this section, we now evaluate the following integral transform containing Humbert functions F (= Ψ1, Φ1, Ξ1 or Ξ2):

Z ∞ 2m+d −γt2 2 Im(F ) = t e Jd−1(βt)F (y, xt )dt, (2.1) 0

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<(2m + 2d) > 0 and <(γ) > 0. In (2.1), we replaced x and y with y and xt2 respectively in (1.1), (1.2), (1.3) and (1.4). Theorem 2.1. The following transformations hold true:

∞ x q 2 ! X (m + d)q ( γ ) −β I (Ψ ) = A (a) F (a + q, b; d; y) F m + d + q; d; , (2.2) m 1 q (c0) q! 2 1 1 1 4γ q=0 q

∞ x q 2 ! X (a)q(m + d)q ( γ ) −β I (Φ ) = A F (a + q, b; d + q; y) F m + d + q; d; , (2.3) m 1 (d) q! 2 1 1 1 4γ q=0 q

∞ x q 2 ! X (b)q(m + d)q ( γ ) −β I (Ξ ) = A F a, d; c0 + q; y F m + d + q; d; , (2.4) m 1 (c0) q! 2 1 1 1 4γ q=0 q and

∞ x q 2 ! X (m + d)q ( γ ) −β I (Ξ ) = A F (a, b; d + q; y) F m + d + q; d; , (2.5) m 2 (d) q! 2 1 1 1 4γ q=0 q d A 1 β where, = βγm 2γ (d)m. Proof. The proof of integral transform in (2.2) is as follows: By substituting (1.7) in (2.1), we obtain

∞ q X (a)q x I (Ψ ) = A F (a + q, b; d; y) I , (2.6) m 1 (c0) q! 2 1 q q=0 q where Z ∞ µ −γt2 Iq = t e Jd−1(βt)dt, (2.7) 0 with µ = 2m + 2q + d. (2.8) By means of the identity [6]

Z ∞ ν ν+µ+1 2 ! µ −γt2 1 β Γ( 2 ) ν + µ + 1 −β t e Jν(βt)dt = ν+µ+1 1F1 ; ν + 1; (2.9) 0 2 2 ν!γ 2 2 4γ (<(µ + ν) > −1 , <(γ) > 0), and using (1.6), (2.7) can be written as

− ! 1 βd 1 1 Γ(m + q + d) −β2 I = F m + q + d; d; . (2.10) q 2 2 (d − 1)! γm+q+d 1 1 4γ Substituting (2.10) into (2.6), the assertion (2.1) of Theorem 2.1 directly follows.

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Note that if m = 0 and c = d, one ﬁnds the undermentioned integral: Z ∞ c −γt2 0 2 Im=0(Ψ1) = t e Jc−1(βt)Ψ1 a, b; c, c ; y, xt dt 0 !c ∞ x q 2 ! (2.11) 1 β X (a)q(c)q ( γ ) −β = F (a + q, b; c; y) F c + q; c; . β 2γ (c0) q! 2 1 1 1 4γ q=0 q Proof. The proof of integral transform (2.3) is as follows: 2 2 Considering F (y, xt ) = Φ1 a, b; d; y, xt given by (1.8), (2.1) becomes in this case

∞ q X (a)q x I (Φ ) = F (a + q, b; d + q; y) I , (2.12) m 1 (d) q! 2 1 q q=0 q where Iq is given by (2.7). By using the identity (2.9), (2.12) can be written as !d ! 1 β Γ(m + q + d) −β2 I (Φ ) = F m + q + d; d; . (2.13) m 1 β 2γ Γ(d)γm+q 1 1 4γ

Substituting this last equation in the expression of Im(Φ1), yields the assertion (2.3) of Theorem 2.1. Note here that for m = 0 and c = d,(2.12) becomes Z ∞ c −γt2 2 Im=0(Φ1) = t e Jc−1(βt) Φ1 a, b; c; y, xt dt 0 !c ∞ x q 2 ! (2.14) 1 β X ( γ ) −β = (a) F (a + q, b; c + q; y) F c + q; c; . β 2γ q q! 2 1 1 1 4γ q=0

Proof. The proof of the integral transform (2.4) is as follows: Following the same procedure as above, we use the deﬁnition of Ξ1 given by (1.9) and replace 2 2 0 2 F (y, xt ) by F (y, xt ) = Ξ1 a, b, c; c ; y, xt , then the considered integral Im(Ξ1) can be written as

∞ q X (b)q x I (Ξ ) = F a, d; c0 + q; y I , (2.15) m 1 (c0) q! 2 1 q q=0 q where Iq is given by (2.7). Now, by using (2.9) and substituting (2.7) in (2.15), (2.4) is proved. By putting m = 0 and c = d,(2.15) can be written as Z ∞ c −γt2 0 2 Im=0(Ξ1) = t e Jc−1(βt) Ξ1 a, b, c; c ; y, xt dt 0 !c ∞ x q 2 ! (2.16) 1 β X (b)q(c)q ( γ ) −β = F a, c; c0 + q; y F c + q; c; . β 2γ (c0) q! 2 1 1 1 4γ q=0 q

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Proof. The proof of the integral transform (2.5) is as follows: 2 2 Taking F (y, xt ) = Ξ2 a, b; d; y, xt given by (1.10), (2.1) becomes

X∞ 1 xq I (Ξ ) = F (a, b; d + q; y) I . (2.17) m 2 (d) q! 2 1 q q=0 q

Finally, by introducing the expression of Iq given by (2.10), one ﬁnds (2.5). This completes the proof of Theorem 2.1. For m = 0 and c = d,(2.17) becomes

!c ∞ x q 2 ! 1 β X ( γ ) −β I (Ξ ) = F (a, b; c + q; y) F c + q; c; . (2.18) m=0 2 β 2γ q! 2 1 1 1 4γ q=0

2.2. Evaluation of some series In this section, we evaluate the series obtained from the integral transforms given by Theorem 2.1 in the case of m = 0 and c = d. These series are given by (see (2.11), (2.14), (2.16) and (2.18)).

∞ q X (a)q(c)q x S = F (a + q, b; c; y) F (c + q; c; z) , (2.19) 1 (c0) q! 2 1 1 1 q=0 q

X∞ xq S = (a) F (a + q, b; c + q; y) F (c + q; c; z) , (2.20) 2 q q! 2 1 1 1 q=0

∞ q X (b)q(c)q x S = F a, c; c0 + q; y F (c + q; c; z) , (2.21) 3 (c0) q! 2 1 1 1 q=0 q and

X∞ xq S = F (a, b; c + q; y) F (c + q; c; z) . (2.22) 4 q! 2 1 1 1 q=0

Theorem 2.2. The following result holds true:

y ez X∞ (a) (b) ( )n x xz S = n n 1−x Φ a + n, c0 − c; c0; , (2.23) 1 (1 − x)a (c) n! 1 x − 1 1 − x n=0 n

X∞ (a) Yn = G n F a + n, c0 − c, b; c0 + n; X, Z (2.24) (c0) n! 2 n=0 n X∞ zn = F a, c + n, b; c0, c; x, y . (2.25) n! 2 n=0

− Y G e X x xz y We note that in (2.24), = (1−X)−a ,X = x−1 ,Y = 1−x and Z = 1−x .

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Proof of Eq. (2.23): We start from (2.19), which, by using the identities (see [10, 12])   (−1)k(n)! ≤ ≤  (n−k)! , 0 k n (−n)k = , (2.26)  0, k > n

(λ)m+n = (λ)m(λ + m)n, (2.27) and

X∞ Xn X∞ X∞ B(k, n) = B(k, n + k), (2.28) n=0 k=0 n=0 k=0 can be rearranged as

X∞ (a) (b) yn S = ez n n G , (2.29) 1 (c) n! n n=0 n where

∞ ∞ s q X X (a + n)q s(c)q s (xz) x G = + + . (2.30) n (c) (c0) s! q! s=0 q=0 s s+q With the help of (1.5) and (2.27), this last expression can be written as

X∞ (a + n) (xz)s G = s F a + n + s, c + s; c0 + s; x . (2.31) n (c0) s! 2 1 s=0 s By using the Euler’s transformation [11] z F (a, b; c; z) = (1 − z)−a F a, c − b; c; , (2.32) 2 1 2 1 z − 1 (2.29) becomes

z ∞ y n ∞ ∞ 0 x p xz s e X (a) (b) ( ) X X (a + n)s p(c − c)p ( ) ( ) S = n n 1−x + x−1 1−x , (2.33) 1 (1 − x)a (c) n! (c0) p! s! n=0 n s=0 p=0 s+p 0 − 0 x xz where, the double summation in this last expression is Φ1 a + n, c c; c ; x−1 , 1−x given by (1.2). Therefore, the result in (2.23) is proved. Proof of Eq. (2.24): To prove (2.24), we start from (2.23) by puting x xz y X = , Y = and Z = , (2.34) x − 1 1 − x 1 − x so that, the summation S1 can be written as

−Y ∞ ∞ 0 ∞ p m n e X X X (c − c)p X (a)n m p(b)n X Y Z S = + + , (2.35) 1 (1 − X)−a (c0) (c) p! m! n! m=0 p=0 m+p n=0 n

AIMS Mathematics Volume 5, Issue 2, 1260–1274. 1267 which, by using the identity

(a)m+p+n = (a)n(a + n)m+p = (a)m+p(a + m + p)n, (2.36) becomes −Y ∞ ∞ 0 p m e X X X (a)m p(c − c)p X Y S = + F (a + m + p, b; c; Z) . (2.37) 1 (1 − X)−a (c0) p! m! 2 1 m=0 p=0 m+p After some simpliﬁcations and by using (1.15), and the elementary identities

(a + m)p(a)m (a + p)m = , (2.38) (a)p and

0 0 0 (c + m)p(c )m (c + p)m = 0 , (2.39) (c )p

(2.37) can be expressed in terms of the Appell function F2 as given in (2.24). This completes the proof. Proof of Eq. (2.25): We start from the expression of S1 and with the help of the identities

(c)n (c)q(c + q)n = (c)q+n, and (c + q)n = (c + n)q , (2.40) (c)q

S1 can be written in terms of 2F1 as follows

∞ n ∞ q X z X (a)q(c + n)q x S = F (a + q, b; c; y) . (2.41) 1 n! (c0) q! 2 1 n=0 q=0 q 0 If one uses (1.15), S1 can be written in terms of the second Appell function F2 (a, c + n, b; c , c; x, y). This completes the proof of (2.25) and consequently Theorem 2.2. Theorem 2.3. The following result holds true:

y ez X∞ (a) (b) ( )n x xz S = n n 1−x Φ a + n, n; c + n; , , (2.42) 2 (1 − x)a (c) n! 1 x − 1 1 − x n=0 n

y ez X∞ (a) (b) ( )n x xz = n n 1−x F a + n, −, n; c + n; , , (2.43) (1 − x)a (c) n! 2 x − 1 1 − x n=0 n X∞ zn = F a, c + n, b; c0, c; x, y . (2.44) n! 1 n=0

Proof of Eq. (2.42): To prove the result (2.42), we use the identities given by (2.26), (2.27) and (2.28) and the result [10]

q X (−q) (−z)s F (c + q; c; z) = ez F (−q; c; −z) = ez s , (2.45) 1 1 1 1 (c) s! s=0 s

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∞ ∞ ∞ q s n X X X (a)n(a + n)q s(b)n x (xz) y S = ez + , (2.46) 2 (c) (c + q + s) q! s! n! q=0 s=0 n=0 s n which can be written as

X∞ (a) (b) yn X∞ (a + n) (xz)s S = ez n n s F (a + n + s, c + s; c + n + s; x) . (2.47) 2 (c) n! (c + n) s! 2 1 n=0 n s=0 s The summation in (2.47) can be rearranged, by using the Euler’s transformation given by (2.32), as

y ez X∞ (a) (b) ( )n S = n n 1−x 2 (1 − x)a (c) n! n=0 n (2.48) X∞ X∞ (a + n) (a + n + s) (n) ( xz )s ( x )p × s p p 1−x x−1 . (c + n) (c + n + s) s! p! s=0 p=0 s p Now, by employing the following identities

(a + n)q+s = (a + n)s(a + n + s)q, (2.49) (c + n)q+s = (c + n)s(c + n + s)q,

The expression of S2 can be written as

z ∞ y n ∞ ∞ xz s x p e X (a) (b) ( ) X X (a + n)s p(n)p ( ) ( ) S = n n 1−x + 1−x x−1 . (2.50) 2 (1 − x)a (c) n! (c + n) s! p! n=0 n s=0 p=0 s+p Since, the double summation in (2.50) is the Humbert function given by (1.2), that is, x xz Φ1 a + n, n; c + n; x−1 , 1−x . Therefore, the required proof of (2.42) straightforwardly follows. Proof of Eq. (2.43): The summation in (2.50) can be rewritten as

y ez X∞ (a) (b) ( )n S = n n 1−x 2 (1 − x)a (c) n! n=0 n (2.51) X∞ (a + n) ( xz )s X∞ (a + n + s) (n) ( x )p × s 1−x p p x−1 , (c + n) s! (c + n + s) p! s=0 s p=0 p which after introducing the hypergeometric function, becomes

y ez X∞ (a) (b) ( )n S n n 1−x 2 = a (1 − x) (c)n n! n=0 (2.52) X∞ (a + n) ( xz )s x × s 1−x F a + n + s, n; c + n + s; . (c + n) s! 2 1 x − 1 s=0 s By using the expression of the second Appell function given by (1.15), (2.52) can be rearranged to yield (2.43). This completes the proof.

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Proof of Eq.(2.44): By using the elementary identities

(λ)n(λ + n)q (λ + q)n = , (2.53) (λ)q and (λ)q+s (λ + q)s = , (2.54) (λ)q (2.20) can be written as ∞ n ∞ ∞ q s X z X X (a)q s(c + n)q(b)s x y S = + . (2.55) 2 n! (c) q! s! n=0 s=0 q=0 q+s With the help of the deﬁnition of the ﬁrst Appell function given by (2.55), we can easily obtain (2.44). This completes the proof of Theorem 2.3. Theorem 2.4. The following result holds true:

ez X∞ (a) (c) yn x xz S = n n Φ b, c0 + n − c; c0 + n; , , (2.56) 3 (1 − x)b (c0) n! 1 x − 1 1 − x n=0 n

ez X∞ (a) (c) yn x xz = n n F b, −, c0 + n − c; c0 + n; , , (2.57) (1 − x)b (c0) n! 1 x − 1 1 − x n=0 n X∞ zn = F b, a, c + n; c, c0; x, y . (2.58) n! 3 n=0

Proof of Eq. (2.56): By the use of (2.28), (2.53) and (2.54), S3 can be expressed as

∞ n ∞ s ∞ q X (a) (c) y X (b) (xz) X (b + s)q(c + s)q x S = ez n n s . (2.59) 3 (c0) n! (c0 + n) s! (c0 + n + s) q! n=0 n s=0 s q=0 q

By replacing the last summation in (2.59) by the hypergeometric function 2F1 and using the Euler’s transformation given by (2.32), we ﬁnd

z ∞ n ∞ ∞ 0 x p xz s e X (a) (c) y X X (b)s p(c + n − c)p ( ) ( ) S = n n + x−1 1−x . (2.60) 3 (1 − x)b (c0) n! (c0 + n) p! s! n=0 n p=0 s=0 s+p The last summation in this equation is none other than the Humbert function 0 − 0 x xz Φ1 b, c + n c; c + n; x−1 , 1−x given by (1.2). From here, the proof of (2.56) straightforwardly follows.

Proof of Eq. (2.57): We start from (2.59) which can be written, with the help of (2.32), as

ez X∞ (a) (c) yn X∞ (b) ( xz )s S = n n s 1−x 3 (1 − x)b (c0) n! (c0 + n) s! n=0 n s=0 s (2.61) x × F b + s, c0 + n − c; c0 + n + s; . 2 1 x − 1

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0 Now, by recalling (1.14), S3 can be rewritten in terms of the ﬁrst Appell function F1 (a, b, b ; c; x, y) as expressed in (2.57). This completes the proof.

Proof of Eq. (2.58): We write (2.21) of S3 as

∞ ∞ q s X X (b)q(c)q(c + q)s x z S = F a, c; c0 + q; y , (2.62) 3 (c0) (c) q! s! 2 1 q=0 s=0 q s which, with the help of the identities given by (2.27), becomes

∞ s ∞ q X z X (b)q(c + s)q x S = F a, c; c0 + q; y . (2.63) 3 s! (c0) q! 2 1 s=0 q=0 q

Recalling the expression of the third Appell function F3 given by (1.16) that yields (2.58) in terms of series of this function. This completes the proof of (2.58) and the proof of Theorem 2.4. Theorem 2.5. The following result holds true:

ez X∞ (a) (b) yn x xz S = n n Φ a, n, c + n; , , (2.64) 4 (1 − x)a (c) n! 1 x − 1 1 − x n=0 n

ez X∞ (a) (b) yn x xz = n n F a, −, n, c + n; , , (2.65) (1 − x)a (c) n! 1 x − 1 1 − x n=0 n X∞ zn = F (a, a, c + n, b; c; x, y) . (2.66) n! 3 n=0

Proof of Eq. (2.64): The use of (2.26), (2.27), (2.28) and (2.45) yields the following expression of S4

∞ n ∞ s ∞ q X (a) (b) y X (a) (xz) X (a + s)q(c)q s x S = ez n n s + , (2.67) 4 (c) n! (c) s! (c + n) q! n=0 n s=0 s q=0 q+s or in the terms of the hypergeometric function 2F1 as

X∞ (a) (b) yn X∞ (a) (xz)s S = ez n n s F (a + s, c + s; c + n + s; x) . (2.68) 4 (c) n! (c + n) s! 2 1 n=0 n s=0 s With the help of the Euler’s transformation (see (2.32)), (2.68) can be rewritten as

z ∞ n ∞ ∞ x p xz s e X (a) (b) y X X (a)s p(n)p ( ) ( ) S = n n + x−1 1−x . (2.69) 4 (1 − x)a (c) n! (c + n) p! s! n=0 n p=0 s=0 s+p

x xz This last summation in the above equation is the Humbert function Φ1 a, n; c + n; x−1 ; 1−x . If one introduces this function, the result (2.64) is proved.

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Proof of Eq. (2.65): If we apply the Euler’s transformation on the function 2F1 which appears in (2.69), we ﬁnd

ez X∞ (a) (b) yn X∞ (a) ( xz )s x S = n n s 1−x F a + s, n; c + n + s; . (2.70) 4 (1 − x)a (c) n! (c + n) s! 2 1 x − 1 n=0 n s=0 s − x xz The last summation in (2.70) is the ﬁrst Appell function F1 a, , n, c + n; x−1 , 1−x (see (1.14)), which proves the desired result (2.65).

Proof of Eq. (2.66): The series in (2.22) can be written as

X∞ xq X∞ (c + q) zs S = (a) s F (a, b; c + q; y) . (2.71) 4 q q! (c) s! 2 1 q=0 s=0 q By using the identity given by (2.48), (2.71) can be written as

∞ s ∞ q X z X (a)q(c + s)q x S = F (a; b; c + q; y) . (2.72) 4 s! (c) q! 2 1 s=0 q=0 q

The last summation in (2.72) is the third Appell function F3 (a, a, c + s, b; c; x, y). This completes the proof of (2.66) and consequently of Theorem 2.5.

3. Applications

In this section, we will apply our main results to express the output ﬁeld resulting from the propagation of generalized Humbert Gaussian beams through a paraxial ABCD optical system. These beams are obtained by modulating generalized Humbert beams [3] with a Gaussian envelope. The output ﬁeld is given by Huygens-Fresnel integral [5]

Z ∞ Z 2π k ik Z+ D ρ2 0 0 ik Aρ02−2ρρ0cos(φ−φ0) 0 0 0 E(ρ, φ, Z) = e ( 2B ) E(ρ , φ )e 2B ( )ρ dρ dφ , (3.1) 2πiB 0 0 where k = 2π/λ is the wave number, λ is the optical wavelength and E(ρ0, φ0) is the input ﬁeld of the form 0 −ηρ0 ikρ 0 0 0 2z i(m+l)φ 2w 2 0m+l 0 E(ρ , φ ) = C1e 0 e e 0 ρ Ψ1 a, b; c, c ; x, y , (3.2) 2 0 2iα0 − k ρ 2 0 where x = α+iα0 , y = 2 0 , z0 is a constant, a = l/2 + m + 1, b = β, c = m + l + 1 and c = m + 1. 4z0(α+iα )

In (3.2), C1 is a constant, w0 is the waist width of the Gaussian part, m is the beam order and l is the topological charge. By using the development of Humbert conﬂuent Hypergeometric function in terms of Hypergeometric functions 2F1 (a, b; c; x) [14]

X∞ (a) yn ψ a, b; c; c0; x, y = n F (a + n, b; c; x) , (3.3) 1 (c0) n! 2 1 n=0 n

AIMS Mathematics Volume 5, Issue 2, 1260–1274. 1272 and by substituting (3.2) in (3.1) with the help of the identity given by (2.9) and the well-known relationship [6]

Z 2π 2r r cos(θ −θ ) − 1 2 1 2 2r1r2 eilθ1 e ρ2 .dθ π −i leilθ2 J , 1 = 2 ( ) l( 2 ) (3.4) 0 ρ

The output ﬁeld can be expressed in terms of Hypergeometric functions 1F1 (a; b; x) and 2F1 (a, b; c; x) as

∞ q i(m+l)φ X (a)q(c)q x E(ρ, φ, Z) = CZe 2F1 (a + q, b; c; y) 1F1 (c + q; c; z) , (3.5) (c0) q! q=0 q

k2ρ2 where CZ is a function of the coordinate Z and z = 4γB2 . By using (2.19) of Theorem 2.2,(3.4) can be written in terms of the Humbert function Φ1 (a, b; c; x, y) as

ez E(ρ, φ, Z) =C ei(m+l)φ (3.6) Z (1 − x)a y X∞ (a) (b) ( )n x xz × n n 1−x Φ a + n, c0 − c; c0; , , (3.7) (c) n! 1 x − 1 1 − x n=0 n

0 0 or in terms of Appell function F2 (a, b, b ; c, c ; x, y) as

X∞ zn E(ρ, φ, Z) = C ei(m+l)φ F a, c + n, b; c0, c; x, y . (3.8) Z n! 2 n=0

In the following, we investigate some numerical simulations of particular cases of GHGBs called as GHGBssLGBs, GHGBsQBGBs and GHGBsWGBs which are derived from standard Laguerre-Gaussian beams, quadratic Bessel-Gaussian beams and Whittaker-Gaussian beams respectively after passing through a paraxial ABCD optical system with a spiral phase plate.

Figure1 illustrates the normalized intensity distribution as a function of the transverse coordinate for particular cases of GHGBs propagating in free space system with two values of the lowest orders beam (m=0 and m=1), where A = 1, B = z, C = 0 and D = 1. From the plots of this ﬁgure, it’s seen that the GHGBs family is characterized by a dark spot at the center with zero intensity. It’s observed that when the beam orders increase, the side lobes of GHGBssLGBs and GHGBsWGBs disappear.

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1 1 GHGBs GHGBs 0.9 WGBs 0.9 WGBs GHGBs GHGBs QBGBs QBGBs 0.8 GHGBs 0.8 GHGBs sLGBs sLGB 0.7 m=0 0.7 m=1 z=1500mm z=1500mm 0.6 0.6

0.5 0.5

0.4 0.4 Normalized intensity 0.3 Normalized intensity 0.3

0.2 0.2

0.1 0.1

0 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 ρ(mm) ρ(mm)

Figure 1. The Normalized intensity distribution versus ρ of GHGBssLGBs, GHGBsWGBs and GHGBsQBGBs as particular cases of GHGBs propagating in free space for two values of the beam orders: m = 0 and m = 1. The other parameters are given as: l = 1, the waist beam ω0 = 0.5mm, the wavelength λ = 1330nm.

4. Conclusion

In this note, we have obtained a new general formulae for the integral transforms containing the product of Humbert and Bessel functions. For each conﬂuent Appell function, we have evaluated the corresponding integral transfoms which is an inﬁnite series of generalized hypergeometric and Appell functions. An application of our main results is investigated to evaluate the output generalized Humbert-Gaussian beam propagating through a paraxial ABCD optical system.

Conﬂict of interest

The authors declare that there is no conﬂict of interest.

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8. N. U. Khan, T. Kashmin, On inﬁnite series if three variables involving Whittaker and Bessel functions, Palestine J. Math., 5 (2016), 185–190. 9. N. U. Khan, T. Usman, M. Ghayasuddin, A note on integral transforms associated with H umbert’s conﬂuent hypergeometric function, Electron. J. Math. Anal. Appl., 4 (2016), 259–265. 10. A. P. Prudnikov, Y. A. Brychkov, O. I. Marychev, Integrals and Series: Special Functions, Nauka, 1983.

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AIMS Mathematics Volume 5, Issue 2, 1260–1274.