A Note On Laurent Type Hypergeometric Generating Relations

A Note On Laurent Type Hypergeometric Generating Relations

Mohammad Idris Qureshi

, Mahvish Ali

Received 29 January 2020

Abstract

In this paper, a Laurent type hypergeometric generating relation is derived by using series rearrange- ment technique. Some special cases are obtained as generating functions of the Bessel functions of di¤erent kinds. Finally explicit forms of these Bessel functions are obtained as applications.

1 Introduction and Preliminaries

Korsch et al. [4, p.14948, Eq. (2)] have discussed the general properties of two-dimensional generalized Bessel functionsJ_n^p;q(u; v). They have given the following generating function for the two-dimensional Bessel functions:

exp u

2 t^p 1 t^p +v

2 t^q 1

t^q =

X1 n= 1

J_n^p;q(u; v)tⁿ; (1)

where(p; q)being relatively prime positive integers and u; v2R. J_n^{p; q}(u; v) =

(J^p;qn (u; v); for ⁿ 2Z

0; else; (2)

6

= 0;1.

The two-dimensional Bessel functions J_n^p;q(u; v)have the following bounds:

jJ₀^p;q(u; v)j 1 and jJ_n^p;q(u; v)j 1

p2 forn6= 0:

Miller introduced a new class of Bessel functionsJn^(p;q)(x)with generating function [5, p. 497, Eq. (19)]:

exp ix

p+q(t^p+t ^q) = X1 n= 1

J_n^(p;q)(x)tⁿ; (3)

where(p; q)being relatively prime positive integers.

Also, the two variable Bessel functionsD^n(p;m)(x; y)possess the generating function [1, p. 116, Eq. (2.15)]:

exph

xt^p y t^m

i

= X1 n= 1

Dn^(p;m)(x; y)tⁿ; 0<jtj<1; x; y2R; (4) where(p; m)being relatively prime positive integers.

Mathematics Sub ject Classi…cations: 33B10, 33B15, 33Cxx.

yDepartment of Applied Sciences, Faculty of Engineering and Technology, Jamia Millia Islamia (A Central University), New Delhi-110025, India

zDepartment of Applied Sciences, Faculty of Engineering and Technology, Jamia Millia Islamia (A Central University), New Delhi-110025, India

493

TheD^(p;m)n (x; y)is given by the following series:

Dn^(p;m)(x; y) = X1 r=0

( 1)^rx(^n+mrp )y^r

n+mr p + 1 r!

: (5)

The following special cases of theD^n(p;m)(x; y)are as follows:

D^(1;1)n

x 2;x

2 = J_n(x) (Bessel functions) (6)

D^(1;1)n

x 2; x

2 = In(x) (Modi…ed Bessel functions) (7)

D^(1;1)n (1; y) = C_n(y) (Bessel Cli¤ord or Tricomi functions) (8)

Dn^(1;m)(1; y) = W_n^m(y) (Wright functions): (9)

A natural generalization of the Gaussian hypergeometric series 2F1[ ; ; ;z], is accomplished by intro- ducing any arbitrary number of numerator and denominator parameters. Thus, the resulting series

pFq

2 4 ( p);

z ( _q);

3 5=pFq

2

4 ¹; 2; : : : ; p; z

1; ₂; : : : ; _q; 3 5=

X1 n=0

( 1)n( 2)n: : :( p)n

( ₁)n( ₂)n: : :( _q)n

zⁿ

n! (10)

is known as the generalized hypergeometric series, or simply, the generalized hypergeometric function [10].

Here pand q are positive integers or zero and we assume that the variable z, the numerator parameters

1; ₂; : : : ; _p and the denominator parameters ₁; ₂; : : : ; _q take on complex values, provided that

j 6= 0; 1; 2; : : : ; j= 1;2; : : : ; q:

In contracted notation, the sequence of pnumerator parameters ₁; ₂; : : : ; _p is denoted by ( _p)with similar interpretation for others throughout this paper.

Supposing that none of numerator parameters is zero or a negative integer and for _j 6= 0; 1; 2; : : :;

j= 1; 2; : : : ; q, we note that the_pF_q series de…ned by equation (10):

(i) converges forjzj<1, ifp q, (ii) converges forjzj<1, ifp=q+ 1 and (iii) diverges for allz,z6= 0, ifp > q+ 1.

A multivariable hypergeometric function provides an interesting and useful uni…cation of the generalized hypergeometric functionpFq of one variable (withpnumerator andqdenominator parameters).

The following generalization of the hypergeometric function in several variables has been given by Srivas- tava and Daoust [11], which is referred to in the literature as the generalized Lauricella function of several variables:

F_C:D^A:B₀⁰_;D^;B_00(n)⁰⁰⁽ⁿ⁾ 0

@[(a) : ⁰; ⁰⁰⁽ⁿ⁾] : [(b⁰) : ⁰]; [(b⁰⁰) : ⁰⁰]; ; [(b⁽ⁿ⁾) : ⁽ⁿ⁾];

z1; z2; ; zn

: [(d⁰) : ⁰]; [(d⁰⁰) : ⁰⁰]; ; [(d⁽ⁿ⁾) : ⁽ⁿ⁾];

1 A

=

X1 m1;m2;:::;mn=0

(m1; m2; :::; mn) z₁^m1 m₁!

z^m₂² m₂!

z_n^mn

m_n!; (11)

where

(m1; m2; :::; mn)

: = QA

j=1(aj)_m

1 0 j+m2 00

j+ +mn (n) j

QB⁰

j=1(b⁰_j)_m1 ⁰_jQB⁰⁰

j=1(b⁰⁰_j)_m2 ⁰⁰_j ::: QB⁽ⁿ⁾ j=1(b⁽ⁿ⁾_j )_m

n (n)

QC j

j=1(cj)_m

1 0 j+m2 00

j+ +mn (n) j

QD⁰

j=1(d⁰_j)_m1 ⁰_jQD⁰⁰

j=1(d⁰⁰_j)_m2 ⁰⁰_j ::: QD⁽ⁿ⁾ j=1(d⁽ⁿ⁾_j )_m

n (n) j

and the coe¢ cients

(k)

j ; j= 1;2; :::; A; ^(k)_j ; j= 1;2; :::; B^(k); ^(k)_j ; j= 1;2; :::; C; ^(k)_j ; j= 1;2; :::; D^(k);

for all k2 f1;2; :::; ng are real and positive, (a)abbreviates the array of Aparameters a1; a2; :::; aA, (b^(k)) abbreviates the array ofB^(k) parametersb^(k)_j ; j= 1;2; :::; B^(k); for allk2 f1;2; :::; ng with similar interpre- tations for(c)and(d^(k)); k= 1;2; :::; n; et cetera.

A signi…cant progress has been made in the study of generalized Bessel functions. Notably, fractional integral operators involving generalized Bessel functions are obtained in [6]. In [3], di¤erential subordinations and superordinations for generalized Bessel functions are given. In [9], the inclusion properties of new subclasses of analytic functions are established by using the generalized Bessel functions of the …rst kind. The computation of image formulas of generalized fractional hypergeometric operators, involving the product of multivariable Srivastava polynomial and multi-index Bessel function is a recent investigation, see for example [2].

Recently, Qureshiet al. [7,8] have obtained explicit expressions of some hybrid special functions related to the Bessel and Tricomi functions. In this article, our main motive is to derive a Laurent type hypergeometric generating relation. In Section 2, a general series identity is derived. Using the general series identity, a Laurent type hypergeometric generating relation is obtained. In Section 3, some special cases of the obtained results are presented in terms of the generating relations of the Bessel functions.

2 Main Results

In this section, we derive a general series identity in the form of the following lemma:

Lemma 1 Let f ¹(`)g, f <sup>2</sup>(`)g, f ³(`)g and f <sup>4</sup>(`)g; ` 2 f1;2;3; g are four bounded sequences of arbitrary complex numbers and i(0)6= 0 (i= 1;2;3;4). Then

X1 i;j;k;`=0

1(i) 2(j) 3(k) 4(`)

( t^p)ⁱ ( t^q)^j _tr

k t^s

`

i! j!k! `!

= X1 n= 1

n p

1 +ⁿ_p X1 j;k;`=0

1

n qj+rk+s`

p ²(j) ₃(k) ₄(`)

1 1 +ⁿ_p

r

pk+^s_p` _p^qj

q p

j

j!

r p

k

k!

s p

`

`! tⁿ; (12)

where(p; r),(p; s),(q; r),(q; s)being relatively prime positive integers and each of the multiple series involved is absolutely convergent.

Proof. Suppose the l.h.s. of equation (12) is denoted by . Then, we have

= X1 i;j;k;`=0

1(i) 2(j) 3(k) 4(`)

i j k `

i!j!k!`! t<sup>pi+qj rk s`</sup>: (13)

Puttingpi+qj rk s`=nori=<sup>n qj+rk+s`</sup>_p in equation (13), we get

= X1 n= 1

X1 j;k;`=0

1

n qj+rk+s`

p ²(j) ₃(k) ₄(`)

(<sup>n qj+rk+s`</sup>p ) <sup>j k</sup> ` n qj+rk+s`

p + 1 j! k!`!

tⁿ: (14) On simpli…cation, we get Lemma1.

Theorem 2 The following Laurent type hypergeometric generating relation holds true:

AFB

2

4 (a_A) ; t^p (b_B) ;

3 5_CFD

2

4 (c_C) ; t^q (d_D) ;

3 5_GFH

2

4 (gG) ;

t^r

(hH) ; 3 5_VFW

2

4 (v_V) ;

t^s

(w_W) ; 3 5

= X1 n= 1

QA m=1

(a_m)ⁿ_p QB m=1

(bm)ⁿ_p

n p

1 +ⁿ_p

F_B+1:D;H;W^A:C;G;V 0 BB

@

h

(a_A) +ⁿ_p : ^q_p;^r_p;^s_pi

: [(c_C) : 1] ; h

(b_B) +ⁿ_p : ^q_p;^r_p;^s_pi h

1 + ⁿ_p : ^q_p;^r_p;_p^si

: [(d_D) : 1] ; [(g_G) : 1] ; [(v_V) : 1] ;

q

p; ^rp; ^sp [(h_H) : 1] ; [(w_W) : 1] ;

1

A tⁿ; t6= 0; (15)

where(p; r),(p; s),(q; r),(q; s)being relatively prime positive integers and each of the multiple series involved is absolutely convergent.

Proof. Taking

1(i) = QA m=1

(am)i

QB m=1

(b_m)_i

; ₂(j) = QC m=1

(cm)j

QD m=1

(d_m)_j

; ₃(k) = QG m=1

(gm)k

QH m=1

(h_m)_k

; ₄(`) = QV m=1

(vm)` WQ

m=1

(w_m)_`

;

in general series identity (12), applying some algebraic properties of Pochhammer symbols and after simpli-

…cation, we obtain:

AFB

2

4 (aA) ; t^p (bB) ;

3 5_CFD

2

4 (cC) ; t^q (dD) ;

3 5_GFH

2 4

(g_G) ;

t^r

(h_H) ; 3 5_VFW

2

4 (vV) ;

t^s

(wW) ; 3 5

= X1 n= 1

n p

1 +ⁿ_p X1 j;k;`=0

QA m=1

(am)^{n qj+rk+s`}

p

QB m=1

(bm)^{n qj+rk+s`}

p

QC m=1

(cm)j

QD m=1

(dm)j

QG m=1

(gm)k

QH m=1

(hm)k

QV m=1

(vm)`

QW m=1

(wm)`

1 1 +ⁿ_{p q}

pj+^r_pk+^s_p`

q p

j

j!

r p

k

k!

s p

`

`! tⁿ

= X1 n= 1

QA m=1

(a_m)ⁿ

p

QB m=1

(bm)ⁿ_p

n p

1 +ⁿ_p X1 j;k;`=0

QA m=1

a_m+ⁿ_{p qj+rk+s`}

p

QB m=1

bm+ⁿ_{p qj+rk+s`}

p

QC m=1

(c_m)_j QD m=1

(dm)j

QG m=1

(g_m)_k QH m=1

(hm)k

QV m=1

(v_m)_`

WQ

m=1

(wm)`

1 1 +ⁿ_{p q}

pj+^r_pk+^s_p`

q p

j

j!

r p

k

k!

s p

`

`! tⁿ: (16)

On using the de…nition of the Srivastava-Daoust hypergeometric functions (11) in the r.h.s. of equation (16), we obtain assertion (15).

3 Applications

I. Taking A=B =C =D =G=H =V =W = 0, = ^u₂, = ^v₂, = ^u₂,r=p, = ^v₂, s=q in Theorem2, we get

exp u

2 t^p 1 t^p +v

2 t^q 1 t^q

= X1 n= 1

u 2

n p

1 +ⁿ_p

F_1:0;0;0^0:0;0;0 0 B@

: ; ; ;

v 2

u 2

q

p; ^u₄²; ^v₂ ^u₂

q

h p

1 +ⁿ_p : ^q_p;^r_p;^s_p i

: ; ; ;

1

CA tⁿ: (17)

On comparison of equations (17) and (1), the two-dimensional Bessel functions J_n^p;q(u; v) have the following explicit representation

J_n^p;q(u; v) =

u 2

n p

1 +ⁿ_p

F_1:0;0;0^0:0;0;0 0 B@

: ; ; ;

v 2

u 2

q

p; ^u₄²; ^v₂ ^u₂

q

h p

1 +ⁿ_p : _p^q;^r_p;^s_p i

: ; ; ;

1 CA:

II. TakingA=B=C=D=G=H =V =W = 0, =x, = 0, = y,r=m, = 0in Theorem2, we get

exph

xt^p y t^m

i

= X1 n= 1

X1 k=0

( 1)^kx^n+mkp y^k

n+mk

p + 1 k!

tⁿ: (18)

On comparison of equations (18) and (4), we get the explicit representation of the two-variable Bessel functions D^(p;m)n (x; y)de…ned by equation (5). In view of equations (6)–(9), we can …nd the explicit representations of the special cases of the two-variable Bessel functionsD^n(p;m)(x; y)by suitable substi- tutions of the variable and indices in equation (18).

III. TakingA=B =C=D=G=H =V =W = 0, =_p+q^ix , = 0, =_p+q^ix , r=q, = 0in Theorem 2, we get

exp ix

p+q(t^p+t ^q) = X1 n= 1

X1 k=0

ix p+q

n

p+(^1+rp)^k

n+qk

p + 1 k!

tⁿ: (19)

On comparison of equations (19) and (3), the Bessel functions Jn^(p;q)(x) have the following explicit representation:

J_n^(p;q)(x) = X1 k=0

ix p+q

n

p+(^1+rp)^k

n+qk

p + 1 k!

:

4 Conclusion

We conclude our present investigation by observing that several other Laurent type hypergeometric gener- ating relations for the complex special functions can also be deduced in an analogous manner.

Acknowledgment. This work has been sponsored by Dr. D. S. Kothari Post Doctoral Fellowship (Award letter No. F.4-2/2006(BSR)/MA/17-18/0025) awarded to Dr. Mahvish Ali by the University Grants Commission, Government of India, New Delhi.

The authors are thankful to the Reviewer(s) for several useful comments and suggestions towards the improvement of this paper.

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