MPEMathematical Problems in Engineering1563-51471024-123XHindawi10.1155/2020/85967368596736Research ArticleCertain Generating Relations Involving the Generalized Multi-Index Bessel–Maitland FunctionJainShilpi1https://orcid.org/0000-0001-8202-6578NietoJuan J.2SinghGurmej34ChoiJunesang5RodinoLuigi1Department of MathematicsPoornima College of EngineeringJaipur 302022India2Instituto de MatematicasUniversidade de Santiago de Compostela15782 Santiago de CompostelaSpainusc.es3Department of MathematicsMata Sahib Kaur Girls CollegeTalwandi SaboBathinda 151103India4Department of MathematicsSinghania UniversityPacheri BariJhunjhunuIndiasinghaniauniversity.co.in5Department of MathematicsDongguk UniversityGyeongju 38066Republic of Koreadongguk.edu202025820202020290420200408202025820202020Copyright © 2020 Shilpi Jain et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 function Jνjm,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.

European Fund for Regional Development (FEDER)MTM2016-75140-PXunta de GaliciaED431C 2019/02Science and Engineering Research BoardMTR/2017/000194
1. Introduction and Preliminaries

Here and elsewhere, let , , +, , and 0 be the sets of complex numbers, real numbers, positive real numbers, positive integers, and nonpositive integers, respectively.

The Bessel–Maitland function Jνλz is defined as (see Marichev )(1)Jνλz=r=0zrΓλr+ν+1r!,λ+,z.

Pathak  gave the following more generalized form of generalized Bessel–Maitland function (1):(2)Jν,qλ,γz=r=0γqrΓλr+ν+1zrr!,(3)λ,ν,γ,λ0,ν1,γ0,q0,1.

Remark 1.

Even though Pathak excluded q=0 in (2), the case q=0 yields (1).

If q=1, γ=1, ν is replaced by ν1, and z is replaced by z in (2), then generalized Bessel–Maitland function reduces to the Mittag–Leffler function which was studied by Wiman  as follows:(4)Jν1,1λ,1z=Eλ,νz,λ>0,ν>0.

If ν is replaced by ν1 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  as follows:(5)Jν1,qλ,γz=Eλ,νγ,qz,(6)λ>0,ν>0,γ>0;q0,1.

Jain and Agarwal  generalized Bessel–Maitland function Jνλz (1) as follows:(7)Jν,μλz=r=01rz/2ν+2μ+2rΓλr+ν+μ+1Γμ+r+1,(8)λ+,ν,μ,z\,0.

Choi and Agarwal  investigated the following generalized multi-index Bessel function:(9)Jνjm,qλjm,γz=r=0γqrj=1mΓλjr+νj+1zrr!,where m and λj, νj, γ, q, and zj=1,,m such that(10)j=1mλj>max0,q1,νj>1,γ>0,q0,1.

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  (see also ).

Pohlen  introduced the Hadamard product (or the convolution) fg of two analytic functions f and g as follows:(11)fgzn=0anbnzn=gfz,z<R,where RRf·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=0anzn,z<Rf,gz=n=0bnzn,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., ).

The k-th derivative of the function fp=pλnξλ,ξ,n is easily found to be given in terms of gamma function as follows:(13)fkp=1kpλnξkΓλ+nξ+kΓλ+nξ,k0.

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  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.

Let m and λj, νj, γ, q, and zj=1,,m such that(14)j=1mλj>max0,q1,νj>1,γ>0,q0,1.

Also, let t<1. Then,(15)1+tσJνjm,qλjm,γz1+t=k=01kσkJνjm,qλjm,γzF11σ+k;σ;ztkk!.

Proof.

We replace 1+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=sσJνjm,qλjm,γzs=r=0γqrj=1mΓλjr+νj+1zrr!sσr.

Differentiating k 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=1ksσkr=0γqrj=1mΓλjr+νj+1Γσ+r+kΓσ+rzsr1r!,which is simplified to yield(18)gks=1ksσkσkr=0γqrj=1mΓλjr+νj+1σ+krσrzsr1r!.

Decomposing series (18) into Hadamard product (11), we obtain(19)gks=1ksσkσkJνjm,qλjm,γzsF11σ+k;σ;zs.

Expanding gs+t as the Taylor series gives(20)gs+t=k=0tkk!gks.

Combining (16), (19), and (20), we obtain(21)s+tσJνjm,qλjm,γzs+t=k=0tksσkk!σkJνjm,qλjm,γzsF11σ+k;σ;zs.

Finally, setting s=1 yields desired result (15).

Theorem 2.

Let m and λj, νj, γ, q, and zj=1,,m such that(22)j=1mλj>max0,q1,νj>1,γ>0,q0,1.

Also, let t<1. Then,(23)k=0γ+k1kJνjm,qλjm,γ+kztk=1tγJνjm,qλjm,γz1tq.

Proof.

Let J be the left-hand side of (23). Using (9), on expanding the function in series, gives(24)J=k=0γ+k1kr=0γ+kqrj=1mΓλjr+νj+1zrr!tk.

Interchanging the order of summations in (24) and using the known identity (see, e.g., [26, p. 5])(25)γk=Γγ+1k!Γγk+1,k0,γ,we have(26)J=r=0γqrj=1mΓλjr+νj+1k=0γ+qr+k1ktkzrr!.

Using the generalized binomial expansion, we find that the inner sum in (26) gives(27)k=0γ+qr+k1ktk=1tγ+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 Theorems 1 and 2. As noted in Remark 2, setting νj by νj1 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.

Let m and λj, νj, γ, q, and zj=1,,m such that(28)j=1mλj>max0,q1,νj>0,γ>0,q0,1.

Also, let t<1. Then,(29)1+tσEλj,νjmγ,qz1+t=k=01kσkEλj,νjmγ,qzF11σ+k;σ;ztkk!.

Corollary 2.

Let m and λj, νj, γ, q, and zj=1,,m such that(30)j=1mλj>max0,q1,νj>0,γ>0,q0,1.

Also, let t<1. Then,(31)k=0γ+k1kEλj,νjmγ+k,qztk=1tγEλj,νjmγ,qz1tq.

The particular cases of (15), (23), (29), and (31) when m=1 give the following generating relations, stated, respectively, in Corollaries 36.

Corollary 3.

Let σ, λ, ν, γ, and z such that λ>0, ν1, γ>0, and q0,1. Also, let t<1. Then,(32)1+tσJν,qλ,γz1+t=k=01kσkJν,qλ,γzF11σ+k;σ;ztkk!.

Corollary 4.

Let σ, λ, ν, γ, and z such that λ>0, ν1, γ>0, and q0,1. Also, let t<1. Then,(33)k=0γ+k1kJν,qλ,γ+kztk=1tγJν,qλ,γz1tq.

Corollary 5.

Let σ, λ, ν, γ, and z such that λ>0, ν0, γ>0, and q0,1. Also, let t<1. Then,(34)1+tσEλ,νγ,qz1+t=k=01kσkEλ,νγ,qzF11σ+k;σ;ztkk!.

Corollary 6.

Let σ, λ, ν, γ, and z such that λ>0, ν0, γ>0, and q0,1. Also, let t<1. Then,(35)k=0γ+k1kEλ,νγ+k,qztk=1tγEλ,νγ,qz1tq.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research has been partially supported by the Agencia Estatal de Investigacion (AEI) of Spain, co-financed by the European Fund for Regional Development (FEDER) corresponding to the 2014–2020 multiyear financial framework (project MTM2016-75140-P), and also supported by Xunta de Galicia (grant ED431C 2019/02). Shilpi Jain also thanks SERB (project number: MTR/2017/000194) for providing necessary facility.

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