# Certain Generating Relations Involving the Generalized Multi-Index Bessel–Maitland Function

Authors: Shilpi Jain; Juan J. Nieto; Gurmej Singh; Junesang Choi
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Generating relations involving the special functions have already proved their important role in mathematics and other fields of sciences. In this paper, we aim to provide some presumably new generating relations in connection with the generalized multi-index Bessel–Maitland functionJνjm,qλjm,γ.. The main results presented here, being very general, can yield a number of particular or equivalent identities, some of which are explicitly demonstrated.


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## Body

## 1. Introduction and Preliminaries

Here and elsewhere, letℂ, ℝ, ℝ+, ℕ, and ℤ0− be the sets of complex numbers, real numbers, positive real numbers, positive integers, and nonpositive integers, respectively.The Bessel–Maitland functionJνλz is defined as (see Marichev [1])(1)Jνλz=∑r=0∞−zrΓλr+ν+1r!,λ∈ℝ+,z∈ℂ.Pathak [2] gave the following more generalized form of generalized Bessel–Maitland function (1):(2)Jν,qλ,γz=∑r=0∞γqrΓλr+ν+1−zrr!,(3)λ,ν,γ∈ℂ,ℜλ≥0,ℜν≥−1,ℜγ≥0,q∈0,1∪ℕ.Remark 1.
Even though Pathak excludedq=0 in (2), the case q=0 yields (1).
Ifq=1, γ=1, ν is replaced by ν−1, and z is replaced by −z in (2), then generalized Bessel–Maitland function reduces to the Mittag–Leffler function which was studied by Wiman [3] as follows:(4)Jν−1,1λ,1−z=Eλ,νz,ℜλ>0,ℜν>0.
Ifν is replaced by ν−1 and z is replaced by −z in (2), then the generalized Bessel–Maitland function reduces to the well-known generalized Mittag–Leffler function Eλ,νγ,qz which was introduced by Shukla and Prajapati [4] as follows:(5)Jν−1,qλ,γ−z=Eλ,νγ,qz,(6)ℜλ>0,ℜν>0,ℜγ>0;q∈0,1∪ℕ.
Jain and Agarwal [5] generalized Bessel–Maitland function Jνλz (1) as follows:(7)Jν,μλz=∑r=0∞−1rz/2ν+2μ+2rΓλr+ν+μ+1Γμ+r+1,(8)λ∈ℝ+,ν,μ∈ℂ,z∈ℂ\−∞,0.
Choi and Agarwal [6] investigated the following generalized multi-index Bessel function:(9)Jνjm,qλjm,γz=∑r=0∞γqr∏j=1mΓλjr+νj+1−zrr!,where m∈ℕ and λj, νj, γ, q, and z∈ℂj=1,…,m such that(10)∑j=1mℜλj>max0,ℜq−1,ℜνj>−1,ℜγ>0,q∈0,1∪ℕ.Remark 2.
It is easily found that generalized multi-index Bessel–Maitland function (9) is equivalent to the generalized multi-index Mittag–Leffler function defined and studied by Saxena and Nishimoto [7] (see also [8]).
Pohlen [9] introduced the Hadamard product (or the convolution) f∗g of two analytic functions f and g as follows:(11)f∗gz≔∑n=0∞anbnzn=g∗fz,z<R,where R≥Rf·Rg. Here, fz and gz are analytic at z=0 whose Maclaurin series with their respective radii of convergence Rf and Rg are(12)fz=∑n=0∞anzn,z<Rf,gz=∑n=0∞bnzn,z<Rg.
The concept of the Hadamard product has turned out to be useful, particularly, in factorizing a newborn function, which is usually expressed as a Maclaurin series, into two known functions (see, e.g., [10–13]).
Thek-th derivative of the function fp=p−λ−nξλ,ξ∈ℂ,n∈ℕ is easily found to be given in terms of gamma function as follows:(13)fkp=−1kp−λ−nξ−kΓλ+nξ+kΓλ+nξ,k∈ℕ0.
Generating functions have been widely used in exploring certain properties and formulas involving sequences and polynomials in a wide range of research subjects. Many researchers have developed a remarkably large number of generating functions associated with a variety of special functions. For some works on this subject, one may refer, for example, to an extensive monograph [14–25] and the literature cited therein. In this search, we aim to provide some presumably new generating relations in connection with generalized multi-index Bessel–Maitland function (9). The main results developed here, being very general, can be reduced to produce a large number of presumably new and potentially useful generating relations for other known functions, some of which are demonstrated.


## 2. Generating Relations

We give two generating relations involving generalized multi-index Bessel–Maitland function (9) asserted by the following theorems.Theorem 1.
Letm∈ℕ and λj, νj, γ, q, and z∈ℂj=1,…,m such that(14)∑j=1mℜλj>max0,ℜq−1,ℜνj>−1,ℜγ>0,q∈0,1∪ℕ.
Also, lett<1. Then,(15)1+t−σJνjm,qλjm,γz1+t=∑k=0∞−1kσkJνjm,qλjm,γz∗F11σ+k;σ;−ztkk!.Proof.
We replace1+t by s in the left-hand side of (15) and denote the resulting expression by gs. Then, using form (9), on expanding the function in series, gives(16)gs=s−σJνjm,qλjm,γzs=∑r=0∞γqr∏j=1mΓλjr+νj+1−zrr!s−σ−r.
Differentiatingk times both sides of (16) with respect to s with the aid of (13) (term-by-term differentiation can be verified under the given conditions), we find(17)gks=−1ks−σ−k∑r=0∞γqr∏j=1mΓλjr+νj+1Γσ+r+kΓσ+r−zsr1r!,which is simplified to yield(18)gks=−1ks−σ−kσk∑r=0∞γqr∏j=1mΓλjr+νj+1σ+krσr−zsr1r!.
Decomposing series (18) into Hadamard product (11), we obtain(19)gks=−1ks−σ−kσkJνjm,qλjm,γzs∗F11σ+k;σ;−zs.
Expandinggs+t as the Taylor series gives(20)gs+t=∑k=0∞tkk!gks.
Combining (16), (19), and (20), we obtain(21)s+t−σJνjm,qλjm,γzs+t=∑k=0∞−tks−σ−kk!σkJνjm,qλjm,γzs∗F11σ+k;σ;−zs.
Finally, settings=1 yields desired result (15).Theorem 2.
Letm∈ℕ and λj, νj, γ, q, and z∈ℂj=1,…,m such that(22)∑j=1mℜλj>max0,ℜq−1,ℜνj>−1,ℜγ>0,q∈0,1∪ℕ.
Also, lett<1. Then,(23)∑k=0∞γ+k−1kJνjm,qλjm,γ+kztk=1−t−γJνjm,qλjm,γz1−tq.Proof.
LetJ be the left-hand side of (23). Using (9), on expanding the function in series, gives(24)J=∑k=0∞γ+k−1k∑r=0∞γ+kqr∏j=1mΓλjr+νj+1−zrr!tk.
Interchanging the order of summations in (24) and using the known identity (see, e.g., [26, p. 5])(25)γk=Γγ+1k!Γγ−k+1,k∈ℕ0,γ∈ℂ,we have(26)J=∑r=0∞γqr∏j=1mΓλjr+νj+1∑k=0∞γ+qr+k−1ktk−zrr!.
Using the generalized binomial expansion, we find that the inner sum in (26) gives(27)∑k=0∞γ+qr+k−1ktk=1−t−γ+qr,t<1.
Finally, interpreting (26) with the help of (27) yields desired result (23).


## 3. Further Remarks

Here, we choose to give some equivalent identities and particular cases of the results in Theorems1 and 2. As noted in Remark 2, setting νj by νj−1 and z by −z in (15) and (23) gives two corresponding generating relations involving the generalized multi-index Mittag–Leffler function Eλj,νjmγ,qz, which are asserted, respectively, in Corollaries 1 and 2.Corollary 1.
Letm∈ℕ and λj, νj, γ, q, and z∈ℂj=1,…,m such that(28)∑j=1mℜλj>max0,ℜq−1,ℜνj>0,ℜγ>0,q∈0,1∪ℕ.
Also, lett<1. Then,(29)1+t−σEλj,νjmγ,qz1+t=∑k=0∞−1kσkEλj,νjmγ,qz∗F11σ+k;σ;−ztkk!.Corollary 2.
Letm∈ℕ and λj, νj, γ, q, and z∈ℂj=1,…,m such that(30)∑j=1mℜλj>max0,ℜq−1,ℜνj>0,ℜγ>0,q∈0,1∪ℕ.
Also, lett<1. Then,(31)∑k=0∞γ+k−1kEλj,νjmγ+k,qztk=1−t−γEλj,νjmγ,qz1−tq.The particular cases of (15), (23), (29), and (31) when m=1 give the following generating relations, stated, respectively, in Corollaries 3–6.Corollary 3.
Letσ, λ, ν, γ, and z∈ℂ such that ℜλ>0, ℜν≥−1, ℜγ>0, and q∈0,1∪ℕ. Also, let t<1. Then,(32)1+t−σJν,qλ,γz1+t=∑k=0∞−1kσkJν,qλ,γz∗F11σ+k;σ;−ztkk!.Corollary 4.
Letσ, λ, ν, γ, and z∈ℂ such that ℜλ>0, ℜν≥−1, ℜγ>0, and q∈0,1∪ℕ. Also, let t<1. Then,(33)∑k=0∞γ+k−1kJν,qλ,γ+kztk=1−t−γJν,qλ,γz1−tq.Corollary 5.
Letσ, λ, ν, γ, and z∈ℂ such that ℜλ>0, ℜν≥0, ℜγ>0, and q∈0,1∪ℕ. Also, let t<1. Then,(34)1+t−σEλ,νγ,qz1+t=∑k=0∞−1kσkEλ,νγ,qz∗F11σ+k;σ;−ztkk!.Corollary 6.
Letσ, λ, ν, γ, and z∈ℂ such that ℜλ>0, ℜν≥0, ℜγ>0, and q∈0,1∪ℕ. Also, let t<1. Then,(35)∑k=0∞γ+k−1kEλ,νγ+k,qztk=1−t−γEλ,νγ,qz1−tq.



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