# Results Concerning the Analysis of Multi-Index Whittaker Function

Authors: Nabiullah Khan; Saddam Husain; Talha Usman; Serkan Araci
Journal: Journal of Mathematics
 (2022)

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## Abstract

A variety of functions, their extensions, and variants have been extensively investigated, mainly due to their potential applications in diverse research areas. In this paper, we aim to introduce a new extension of Whittaker function in terms of multi-index confluent hypergeometric function of first kind. We discuss multifarious properties of newly defined multi-index Whittaker function such as integral representation, integral transform (i.e., Mellin transform and Hankel transform), and derivative formula. The results presented here, being very general, are pointed out to reduce to yield some known or new formulas and identities for relatively functions.


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## Body

## 1. Introduction

Generalized and multivariable forms of the special functions of mathematical physics have witnessed a significant evolution during recent years. In particular, the special functions of more than one variable provided new means of analysis for the solution of large classes of partial differential equations often encountered in physical problems. Most of the special functions of mathematical physics and their generalization have been suggested by physical problems. In mathematics, the Whittaker function is a solution of Whittaker equation, which is a modified form of confluent hypergeometric function of first kind, and it has various applications in multifarious area such as mathematical physics and many research areas, which are studied by various mathematicians (see [1–4]). Recently, many authors give an extension and generalization of several special functions such as beta function, gamma function, hypergeometric function, confluent hypergeometric function, and Whittaker function (see [2, 3, 5–12]). Ghayasuddin et al. [7] defined a new type of confluent hypergeometric function by using extended beta function in terms of multi-index Mittag–Leffler function. Inspired by the abovementioned work, in this paper, we introduced a new extension of Whittaker function in terms of multi-index Mittag–Leffler function by using an extended confluent hypergeometric function and studied their various properties such as integral transform, integral representation, and derivative formula of it. We remember below the following basic definition and extension of special function.The classical beta functionBu,v is defined as (see [13])(1)Bu,v=∫01tu−11−tv−1dt=ΓuΓvΓu+v,where(2)ℜu>0,ℜv>0.The classical Gauss hypergeometric functionFu,v;w;ω and confluent hypergeometric function Φv;w;ω are defined as (see [14])(3)Fu,v;w;ω=1Bv,w−v∫01tv−11−tw−v−11−ωt−udt,arg1−ω<π;ℜw>ℜv>0,(4)Φv;w;ω=1Bv,w−v∫01tv−11−tw−v−1expωtdt,ℜw>ℜv>0.By using the series expansion of1−ωt−u and expωt in (3) and (4), respectively, the hypergeometric and confluent hypergeometric functions are written in terms of beta function as(5)Fu,v;w;ω=∑n=0∞unBv+n,w−vBv,w−vωnn!,ω<1,ℜw>ℜv>0,Φv;w;ω=∑n=0∞Bv+n,w−vBv,w−vωnn!,ℜw>ℜv>0.In 1997, Aslam Chaudhary et al. [5] give an extension of beta function defined as(6)Bρu,v=∫01tu−11−tv−1exp−ρt1−tdt,where(7)ℜρ>0,ℜu>0,ℜv>0.Remark 1.
Ifρ=0, then extended beta function (6) is reduced to classical beta function (1).In 2004, Chaudhary et al. [6] introduced the extended hypergeometric and confluent hypergeometric functions in terms of extended beta function (6) as follows:(8)Fρu,v;w;ω=∑n=0∞unBρv+n,w−vBv,w−vωnn!,ρ≥0,ω<1,ℜw>ℜv>0,Φρv;w;ω=∑n=0∞Bρv+n,w−vBv,w−vωnn!,ρ≥0,ℜw>ℜv>0.Their integral representation is(9)Fρu,v;w;ω=1Bv,w−v∫01tv−11−tw−v−11−ωt−uexp−ρt1−tdt,ρ>0;ρ=0andarg1−ω<π;ℜw>ℜv>0,Φρv;w;ω=1Bv,w−v∫01tv−11−tw−v−1expωt−ρt1−tdt,ρ>0;ρ=0andℜw>ℜv>0.Shadab et al. [12] introduced an extension of beta function using generalized Mittag–Leffler function as follows:(10)Bαρu,v=∫01tu−11−tv−1Eα−ρt1−tdt,ℜu>0,ℜv>0,ℜρ>0;α∈ℝ0+,where Eα. is the classical Mittag–Leffler (see [15, 16]) function defined by(11)Eαω=∑n=0∞ωnΓαn+1,where(12)ω∈ℂ,α∈ℝ0+.Shadab et al. [12] expressed the extended hypergeometric and confluent hypergeometric functions in terms of extended beta function (11) as follows:(13)Fρ,αu,v;w;ω=∑n=0∞unBαρv+n,w−vBv,w−vωnn!,α∈ℝ+,ρ∈ℝ0+,ω<1,ℜw>ℜv>0,Φρ,αv;w;ω=∑n=0∞Bαρv+n,w−vBv,w−vωnn!,α∈ℝ+,ρ∈ℝ0+,ω<1,ℜw>ℜv>0,and their integral representation is(14)Fρ,αu,v;w;ω=1Bv,w−v×∫01tv−11−tw−v−11−ωt−uEα−ρt1−tdt,α∈ℝ+,ρ>0;ρ=0andarg1−ω<π;ℜw>ℜv>0,Φρ,αv;w;ω=1Bv,w−v∫01tv−11−tw−v−1eωtEα−ρt1−tdt,α∈ℝ+,ρ>0;ρ=0andℜw>ℜv>0.Ghayasuddin et al. [7] introduced an extension of beta function using multi-index Mittag–Leffler function as follows:(15)Bρa1,…,as,b1,…bsu,v=∫01tu−11−tv−1E1/aibi−ρt1−tdt,ℜu>0,ℜv>0,ℜρ>0;ai>0,bi∈ℝ,Fors = 2, if we set 1/a1=α,1/a2=0, and b1=b2=1 in (15), then we obtain the extended beta function defined by Shadab et al. [12].Multi-index Mittag–Leffler functionE1/aibi. is defined as follows (see [17]):(16)E1/aibiω=∑k=0∞ωkΓb1+k/a1…Γbs+k/as.where s>1 is an integer and a1,a2,…,as>0 and b1,b2,…,bs are arbitrary real numbers.It is easy to see that, fors=2,1/a1=α,1/a2=0, and b1=b2=1, then multi-index Mittag–Leffler function is reduced to classical Mittag–Leffler function Eαω.In 2020, Ghayasuddin et al. [7] expressed the extended hypergeometric and confluent hypergeometric functions in terms of extended beta function (15) as follows:(17)Fρa1,…,as,b1,…,bsu,v;w;ω=∑n=0∞unBρa1,…,as,b1,…,bsv+n,w−vBv,w−vωnn!,ai>0,bi∈ℝ,ρ∈ℝ0+,ω<1,ℜw>ℜv>0,Φρa1,…,as,b1,…,bsv;w;ω=∑n=0∞Bρa1,…,as,b1,…,bsv+n,w−vBv,w−vωnn!,ai>0,bi∈ℝ,ρ∈ℝ0+,ω<1,ℜw>ℜv>0,and their integral representation is(18)Fρa1,…,as,b1,…,bsu,v;w;ω=1Bv,w−v∫01tv−11−tw−v−11−ωt−uE1/aibi−ρt1−tdt,ρ>0;ρ=0andarg1−ω<π;ℜw>ℜv>0,ai>0,bi∈ℝ,(19)Φρa1,…,as,b1,…,bsv;w;ω=1Bv,w−v×∫01tv−11−tw−v−1eωtE1/aibi−ρt1−tdt,ρ>0;ρ=0andℜw>ℜv>0,ai>0,bi∈ℝ.The extension of Kummer’s relation to the generalized extended confluent hypergeometric function of the first kind is as follows:(20)Φα,βρ,η,νv,w;ω=eωΦα,βρ,η,νw−v;w;−ω.Forα=β=η=ν=1 and ρ=0, (20) is reduced to Kummer’s formula of first kind for the classical confluent hypergeometric function (see [14]).The Whittaker functionMκ,ζω in terms of confluent hypergeometric function of first kind (see [4, 18]) is defined as(21)Mκ,ζω=ωζ+1/2exp−ω2Φζ−κ+12;2ζ+1;ω,ℜζ>−12andℜζ±κ>−12.In 2013, Nagar et al. [3] generalized the Whittaker function by using extended confluent hypergeometric function Φρ which is defined as(22)Mρ,κ,ζω=ωζ+1/2exp−ω2Φρζ−κ+12;2ζ+1;ω,ℜζ>−12andℜζ±κ>−12.


## 2. Multi-Index Whittaker Function

In this section, we give a new generalization of Whittaker function of the first kind by applying the multi-index confluent hypergeometric function (19) defined as(23)Mρ,κ,ζa1,…,as,b1,…,bsω=ωζ+1/2exp−ω2Φρa1,…,as,b1,…,bsζ−κ+12;2ζ+1;ω,ρ≥0,ℜζ>−12andℜζ±κ>−12,ai>0,bi∈ℝ.If we takeρ=0andai=⋯=as=bi=⋯=bs=1, (23) reduced to the classical Whittaker function (21).


### 2.1. Integral Representation

Here, we define integral representation of the multi-index Whittaker function by using (19) and (23) as(24)Mρ,κ,ζa1,…,as,b1,…,bsω=ωζ+1/2exp−ω/2Bζ−κ+1/2,ζ+κ+1/2×∫01tζ−κ−1/21−tζ+κ−1/2eωtE1/aibi−ρt1−tdtρ≥0,ℜζ>−12andℜζ±κ>−12,ai>0,bi∈ℝ.Substitutingt=x−a/b−a in (24), we get the multi-index Whittaker function as(25)Mρ,κ,ζa1,…,as,b1,…,bsω=b−a−2ζ+1ωζ+1/2exp−ω/2Bζ−κ+1/2,ζ+κ+1/2×∫abx−aζ−κ−1/2b−xζ+κ−1/2eωx−a/b−aE1/aibi−ρb−a2x−ab−xdx,where a and b are scalars such that b−a>0.If we takea=−1andb=1 in (23), we obtain another representation of multi-index Whittaker function:(26)Mρ,κ,ζa1,…,as,b1,…,bsω=2−2ζ+1ωζ+1/2Bζ−κ+1/2,ζ+κ+1/2×∫abx+1ζ−κ−1/21−xζ+κ−1/2eωx/2E1/aibi−4ρx+11−xdx,In (24), substitute t=x/1+x, and we get another integral representation of multi-index Whittaker function as(27)Mρ,κ,ζa1,…,as,b1,…,bsω=ωζ+1/2exp−ω/2Bζ−κ+1/2,ζ+κ+1/2×∫0∞xζ−κ−1/21+x−2ζ+1eωx/1+xE1/aibi−ρ1+x2xdx.Ifai=⋯=as=bi=⋯=bs=1 and ρ=0 in (24), (23), and (27), we obtain integral representation of classical Whittaker function.Theorem 1.
The following relation holds true:(28)Mρ,κ,ζa1,…as,b1,…bs−ω=−1ζ+1/2Mρ,−η,νa1,…,as,b1,…,bsω,ρ≥0.Proof.
Using relation (20) in (23), we have(29)Mρ,κ,ζa1,…as,b1,…bs−ω=−1ζ+1/2ωζ+1/2exp−ω2Φρa1,…,as,b1,…,bsζ−κ+12;2ζ+1;ω.
Now, writing the right-hand side of the above representation by using (23), we get the desired result.


## 2.1. Integral Representation

Here, we define integral representation of the multi-index Whittaker function by using (19) and (23) as(24)Mρ,κ,ζa1,…,as,b1,…,bsω=ωζ+1/2exp−ω/2Bζ−κ+1/2,ζ+κ+1/2×∫01tζ−κ−1/21−tζ+κ−1/2eωtE1/aibi−ρt1−tdtρ≥0,ℜζ>−12andℜζ±κ>−12,ai>0,bi∈ℝ.Substitutingt=x−a/b−a in (24), we get the multi-index Whittaker function as(25)Mρ,κ,ζa1,…,as,b1,…,bsω=b−a−2ζ+1ωζ+1/2exp−ω/2Bζ−κ+1/2,ζ+κ+1/2×∫abx−aζ−κ−1/2b−xζ+κ−1/2eωx−a/b−aE1/aibi−ρb−a2x−ab−xdx,where a and b are scalars such that b−a>0.If we takea=−1andb=1 in (23), we obtain another representation of multi-index Whittaker function:(26)Mρ,κ,ζa1,…,as,b1,…,bsω=2−2ζ+1ωζ+1/2Bζ−κ+1/2,ζ+κ+1/2×∫abx+1ζ−κ−1/21−xζ+κ−1/2eωx/2E1/aibi−4ρx+11−xdx,In (24), substitute t=x/1+x, and we get another integral representation of multi-index Whittaker function as(27)Mρ,κ,ζa1,…,as,b1,…,bsω=ωζ+1/2exp−ω/2Bζ−κ+1/2,ζ+κ+1/2×∫0∞xζ−κ−1/21+x−2ζ+1eωx/1+xE1/aibi−ρ1+x2xdx.Ifai=⋯=as=bi=⋯=bs=1 and ρ=0 in (24), (23), and (27), we obtain integral representation of classical Whittaker function.Theorem 1.
The following relation holds true:(28)Mρ,κ,ζa1,…as,b1,…bs−ω=−1ζ+1/2Mρ,−η,νa1,…,as,b1,…,bsω,ρ≥0.Proof.
Using relation (20) in (23), we have(29)Mρ,κ,ζa1,…as,b1,…bs−ω=−1ζ+1/2ωζ+1/2exp−ω2Φρa1,…,as,b1,…,bsζ−κ+12;2ζ+1;ω.
Now, writing the right-hand side of the above representation by using (23), we get the desired result.


## 3. Integral Transform of Multi-Index Whittaker Function

Theorem 2.
The following Mellin transform formula holds true:(30)∫0∞ρs−1Mρ,κ,ζa1,…,as,b1,…,bsωdρ=ΓsΓ1−s∏i=0mΓbi−s/aiBζ+s−κ+1/2,2ζ+s+1ωsBζ−κ+1/2,ζ+κ+1/2Mκ,ζ+sωℜs>0andℜζ±κ>−12,ai>0,bi∈ℝ.Proof.
Using multi-index Whittaker function (1), we obtain(31)∫0∞ρs−1Mρ,κ,ζa1,…,as,b1,…,bsωdρ=ωζ+1/2exp−ω2∫0∞ρs−1Φρa1,…,as,b1,…,bsζ−κ+12;2ζ+1;ωdρ.
Again, using multi-index confluent hypergeometric function (19), we obtain(32)=ωζ+1/2exp−ω/2Bζ−κ+1/2,ζ+κ+1/2×∫0∞ρs−1∫01tζ−κ−1/21−tζ+κ−1/2eωtE1/aibi−ρt1−tdtdρ.
Changing the order of integration, we obtain(33)=ωζ+1/2exp−ω/2Bζ−κ+1/2,ζ+κ+1/2×∫01tζ−κ−1/21−tζ+κ−1/2eωt∫0∞ρs−1E1/aibi−ρt1−tdρdt.
Substitutingu=ρ/tt−1 in integral (33), we obtain(34)=ωζ+1/2exp−ω/2Bζ−κ+1/2,ζ+κ+1/2×∫01tζ+s−κ−1/21−tζ+s+κ−1/2eωt∫0∞us−1E1/aibi−ududt.
Now, using well-known result (p. 102 of [19]) and confluent hypergeometric function (4), we obtain(35)=ωζ+1/2exp−ω/2Bζ+s−κ+1/2,2ζ+s+1Bζ−κ+1/2,ζ+κ+1/2×ϕζ+s−κ+1/2;2ζ+s+1;ωΓsΓ1−s∏i=0mΓbi−s/ai=ωζ+s+1/2exp−ω/2ϕζ+s−κ+1/2;2ζ+s+1;ωωsBζ−κ+1/2,ζ+κ+1/2×Bζ+s−κ+12,2ζ+s+1ΓsΓ1−s∏i=0mΓbi−s/ai.
By using definition of classical Whittaker function (21), we get the desired result.Theorem 3.
The following formula holds true:(36)∫0∞ωa−1e−bωMρ,κ,ζa1,…as,b1,…bscωdω=cζ+1/2Γa+ζ+1/2b+c/2a+ζ+1/2Fρa1,…,as,b1,…,bsa+ζ+12,ζ−κ+12;2ζ+1;2c2b+cρ≥0,2b>c>0,ℜa+ζ>−12,ai>0,bi∈ℝ.Proof.
Using integral representation of the multi-index Whittaker function, we obtain(37)∫0∞ωa−1e−bωMρ,κ,ζa1,…,as,b1,…,bscωdω=1Bζ−κ+1/2,ζ+κ+1/2∫0∞ωa−1e−bωcωζ+1/2exp−cω2×∫01tζ−κ−1/21−tζ+κ−1/2ecωtE1/aibi−ρt1−tdtdω.
Now, interchanging the order of integration and using the definition of gamma function, we obtain(38)∫0∞ωa−1e−bωMρ,κ,ζa1,…,as,b1,…,bscωdω=cζ+1/2Γa+ζ+1/2b+c/2a+ζ+1/2Bζ−κ+1/2,ζ+κ+1/2×∫01tζ−κ−1/21−tζ+κ−1/21−2ct2b+c−a+ζ+1/2E1/aibi−ρt1−tdtdω.
Using (18) in (38), we get the desired result (36).Corollary 1.
If we takea=c=1 in (36), we get the following special cases:(39)∫0∞e−bωMρ,κ,ζa1,…as,b1,…bsωdω=2ζ+3/2Γζ+3/22b+1ζ+3/2Fρa1,…,as,b1,…,bsζ+32,ζ−κ+12;2ζ+1;22b+c.Theorem 4.
The following Hankel transform formula holds true:(40)∫0∞ωMρ,κ,ζa1,…,as,b1,…,bsωJmaωdω=Γζ+κ+5/2a2+1/4ζ/2+5/4∑n=o∞Bρa1,…,as,b1,…,bsζ−κ+n+1/2,ζ+κ+1/2ζ+m+5/2nBζ−κ+1/2,ζ+κ+1/2a2+1/4n/2n!×Pζ+n+3/2−m14a2+1,
(41)ℜζ±κ>−12andℜζ+m>−52,ai>0,bi∈ℝ,where Pζmz is the Legendre function (see p.34. of [20]).Proof.
Using (23) and (18), expanding multi-index Whittaker function in terms of generating extended beta function and changing the order of integration and summation, we obtain(42)∫0∞zMρ,κ,ζa1,…,as,b1,…,bsωJmaωdω=∑n=o∞Bρa1,…,as,b1,…,bsζ−κ+n+1/2,ζ+κ+1/2Bζ−κ+1/2,ζ+κ+1/2n!×∫0∞ωζ+n+3/2e−ω/2Jmaωdω.
On using the known result (see p. 182 (9)of[21]),(43)∫0∞e−pttζJmatdt=Γζ+m+1r−ζ−1Pζ−mpr,where ℜζ+m>−1,r=p2+a21/2 and Pζmz is the Legendre function (see [20]).By using (43) in (42) and after some simplification, we get the desired result.


## 4. Derivative of Multi-Index Whittaker Function

Theorem 5.
The following differential formula holds true:(44)dndωneω/2ω−ζ−1/2Mρ,κ,ζa1,…,as,b1,…,bsω=ζ−κ+1/2n2ζ+1neω/2ω−ζ−n/2−1/2Mρ,κ−n2,ζ+n2a1,…,as,b1,…,bsω.Proof.
Thenth order derivative of generalized extended confluent hypergeometric function is given by [9](45)dndωnΦρa1,…,as,b1,…,bsv;w;ω=vnwnΦρa1,…,as,b1,…,bsv+n;w+n;ω.
Now, using (23) on the left-hand side of (44), we obtain(46)dndωneω/2z−ζ−1/2ρ,κ,ζa1,…,as,b1,…,bsω=dndωnΦρa1,…,as,b1,…,bsζ−κ+12;2ζ+1;ω=ζ−κ+1/2n2ζ+1n×Φρa1,…,as,b1,…,bsζ+n2−κ−n2+12;2ζ+12+1;ω=ζ−κ+1/2n2ζ+1neω/2ω−ζ−n/2−1/2Mρ,κ−n/2,ζ+n/2a1,…,as,b1,…,bsω.
We get the desired result (44).


## 5. Conclusion

In the present paper, we introduce a multi-index Whittaker function in terms of extended confluent hypergeometric function. We have provided some important properties of Whittaker function such as integral representation, integral transform, and derivative formula. We have known that most of the special function of mathematical physics such as modified Bessel function and Laguerre and Hermite polynomial can be written in terms of Whittaker function. Therefore, extensions and generalization of the Whittaker function are playing important roles in applied mathematics and mathematical physics.



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