1. Introduction and PreliminariesHere and elsewhere, let ℂ, 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 function Jνλz 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 excluded q=0 in (2), the case q=0 yields (1).

If q=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.

If ν 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∈ℂ j=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]).

The k-th derivative of the function fp=p−λ−nξ λ,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 RelationsWe give two generating relations involving generalized multi-index Bessel–Maitland function (9) asserted by the following theorems.

Theorem 1.Let m∈ℕ and λj, νj, γ, q, and z∈ℂ j=1,…,m such that(14)∑j=1mℜλj>max0,ℜq−1, ℜνj>−1, ℜγ>0, q∈0,1∪ℕ.

Also, let t<1. Then,(15)1+t−σJνjm,qλjm,γz1+t=∑k=0∞−1kσkJνjm,qλjm,γz∗F11σ+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=s−σJνjm,qλjm,γzs=∑r=0∞γqr∏j=1mΓλjr+νj+1−zrr!s−σ−r.

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=−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.

Expanding gs+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, setting s=1 yields desired result (15).

Theorem 2.Let m∈ℕ and λj, νj, γ, q, and z∈ℂ j=1,…,m such that(22)∑j=1mℜλj>max0,ℜq−1, ℜνj>−1, ℜγ>0, q∈0,1∪ℕ.

Also, let t<1. Then,(23)∑k=0∞γ+k−1kJνjm,qλjm,γ+kztk=1−t−γJνjm,qλjm,γz1−tq.

Proof.Let J 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 RemarksHere, 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 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.Let m∈ℕ and λj, νj, γ, q, and z∈ℂ j=1,…,m such that(28)∑j=1mℜλj>max0,ℜq−1, ℜνj>0, ℜγ>0, q∈0,1∪ℕ.

Also, let t<1. Then,(29)1+t−σEλj,νjmγ,qz1+t=∑k=0∞−1kσkEλj,νjmγ,qz∗F11σ+k;σ;−ztkk!.

Corollary 2.Let m∈ℕ and λj, νj, γ, q, and z∈ℂ j=1,…,m such that(30)∑j=1mℜλj>max0,ℜq−1, ℜνj>0, ℜγ>0, q∈0,1∪ℕ.

Also, let t<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.Let σ, 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.Let σ, 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.Let σ, 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.Let σ, 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.