Certain integrals involving multivariate Mittag Leffler function

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R E S E A R C H

Open Access

Certain integrals involving multivariate

Mittag-Lefﬂer function

D.L. Suthar

1*

_{, Hafte Amsalu}

1

_{and Kahsay Godifey}

1

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

College of Natural Sciences, Wollo University, Dessie, Ethiopia

Abstract

The objective of this article is to present several new integral equalities involving the multivariate Mittag-Leﬄer functions which are associated with the Laguerre

polynomials. To emphasize our main results, we also consider some important special cases. The main results of our paper are quite general in nature and yield a very large number of integral equalities involving polynomials occurring in problems of mathematical analysis and mathematical physics.

MSC: Primary 26A33; 33B15; 33C05; secondary 33C20; 44A10; 44A20

Keywords: Multivariate Mittag-Leﬄer function; Laguerre polynomials; Finite integrals

1 Introduction and preliminaries

The function deﬁned by the series representation

Eξ(z) =

∞

n=0 zn

Γ(ξn+ 1) (ξ> 0,z∈C) (1)

and its generalization

Eξ,ν(z) = ∞

n=0 zn

Γ(ξn+ν) (ξ> 0,ν> 0,z∈C) (2)

were introduced and studied by Agarwal [1], Mittag-Leﬄer [2,3], Humburt [4], Humbert and Agrawal [5], and Wiman [6,7], whereCis the set of complex numbers. The main prop-erties of these functions are given in the book by Erdélyi et al. [8, Sect. 18.1], and a more extensive and detailed account on Mittag-Leﬄer functions is presented in Dzherbashyan [9, Chap. 2]. In particular, the functions (1) and (2) are entire functions of orderρ= 1/ξ

and typeσ= 1; see, for example, [9, p. 118]. For a detailed account of various properties, generalizations, and applications of these functions, the reader may refer to an excellent work of Dzherbashyan [9], Kilbas and Saigo [10–13], Gorenflo and Mainardi [14], Goren-flo, Luchko and Rogosin [15], and Gorenflo, Kilbas and Rogosin [16].

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The series representation of a generalization of (2) was introduced by Prabhaker [17] as:

Eδ ξ,ν(z) =

∞

n=0

(δ)n

Γ(ξn+ν)n!z

n_, ₍₃₎

whereξ,ν,δ∈C((ξ) > 0). It is entire function of order [(ξ)]–1_{(see [}₁₇_{, p. 7]) and (}_δ₎ n

denotes the Pochhammer symbol deﬁned as:

(δ)n=

Γ(δ+n)

Γ(δ) = ⎧ ⎨ ⎩

1, n= 0,λ∈C/{0},

δ(δ+ 1)· · ·(δ+n– 1), n∈C;δ∈C. (4)

Srivastava and Tomoviski [18] studied and generalized the Mittag-Leﬄer-type function

Eδ ξ,ν(z) as

Eδ_ξ;_;κ_ν(z) =

∞

n=0

(δ)κnzn

Γ(ξn+ν)n!, (5) whereδ,κ,ξ,ν,z∈C;(δ) > 0,(κ) > 0,(ξ) > 0.

A generalization of (5) was initiated by Salim and Faraj [19] as follows:

Eδξ,,κν,,ρε(z) = ∞

n=0

(δ)ρn

Γ(ξn+ν)(κ)εn zn

n!, (6)

whereδ,ξ,ν,κ,z∈C;min((δ),(ξ),(ν),(κ) > 0);ρ,ε> 0,ρ≤ (ξ) +ε. Further, a multivariate generalization of Mittag-Leﬄer function Eδi;κi;ρi

ξi;ν;εi (·), which is a

generalization of (6), was studied by Gujar et al. [20] in the following form:

Eδi;κi;ρi

ξi;ν;εi(z1, . . . ,zr) = ∞

m1,...,mr=0

(δ1)ρ1m1· · ·(δr)ρrmr(z1)m1· · ·(zr)mr

Γ(r_i₌₁ξimi+ν)(κ1)ε1m1· · ·(κr)εrmr

, (7)

where δi,ξi,ν,κi,zi∈ C;min1≤i≤r((δi),(ξi),(ν),(κi)) > 0 and ρi,εi > 0;ρi <(ξi) +

εi(i= 1, 2, . . . ,r).

For a more detailed account of various properties, generalizations, and applications in terms of fractions of this function, the reader may refer to Mishra et al. [21], Purohit et al. [22], Saxena et al. [23] and Suthar et al. [24–26].

The polynomialL(nμ,τ)(x) was deﬁned by Prabhaker and Suman [27] as:

L(_nμ,τ)(x) =Γ(μn+τ+ 1)

Γ(n+ 1)

n

r=0

(–n)rxr

r!Γ(μr+τ+ 1), (8)

whereμ∈C+,τ∈C_–1+ ;n∈N. Ifμ= 1, then (8) reduces to

L(1,τ) n (x) =

Γ(n+τ+ 1)

Γ(n+ 1)

∞

r=0

(–n)rxr r!Γ(r+τ+ 1)=L

τ

n(x), (9)

whereLτ

(3)

In the sequel, the Konhauser polynomials of the second kind were deﬁned by Srivastava [29] as:

Zτ n[x;r] =

Γ(rn+τ+ 1)

Γ(n+ 1)

n

j=0

(–1)j

n j

xrj

r!Γ(rj+τ+ 1), (10)

whereτ∈C+

–1,n∈Nandr∈Z.

It can be easily veriﬁed that

L(_nr,τ) xr= Zτ_n[x;r], (11)

Lτ

n(x) = Zτn[x; 1]. (12)

Further, the polynomial Z(nμ,τ)[x;r] is deﬁned [30] as:

Z(_nμ,τ)[x;r] =

n

j=0

(–1)j Γ(rn+τ+ 1)x

rj

j!Γ(rj+τ+ 1)Γ(μn–ηj+ 1). (13)

From Eqs. (10) and (13), we obtain

Z_nτ[x;r] = Z_n(1,τ)[x;r]. (14)

Ifμ∈N, then Eq. (12) can be written in the following form:

Z(μ,τ) n [x;r] =

Γ(rn+τ+ 1)

Γ(μn+ 1)

n

j=0

(–1)j (–μn)μjx

rj

j!Γ(rj+τ+ 1)(–1)(μ–1)j. (15)

The set of polynomialsL(nμ,τ)[ϑ;x] is deﬁned [30] as:

L(_nμ,τ)[ϑ;x] =

n

j=0

(–1)j Γ(rn+τ+ 1)x

j

j!Γ(rj+τ+ 1)Γ(ϑn–ϑj+ 1), (16)

whereμ,ϑ∈C+,τ∈C+_–1,n∈N. 2 Main integral equalities

Throughout this paper, we assume thatδi,ξi,v,κi,α∈C;min1≤i≤r((δi),(ξi),(v),(κi)) >

0, (ρi,εi) > 0 andρi<(ξi) +εi;i= 1, 2, . . . ,r.

Theorem 1 The following integral equality holds:

1

Γ(α) 1

0

uv–1_{(1 –}_u₎α–1_Eδi;κi;ρi ξi;v;εi z1u

ξ1_{, . . . ,}_z

ruξr

du=Eδi;κi;ρi

ξi;v+α;εi(z1, . . . ,zr). (17)

Proof Applying Eq. (7) to the left-hand side of Eq. (17), we obtain

=

∞

m1,...,mr=0

(4)

× 1

Γ(α) 1

0 uν+

r

i=1miξi–1_{(1 –}_u₎α–1_du

=

∞

m1,...,mr=0

(δ1)ρ1m1· · ·(δr)ρrmr(z1)m1· · ·(zr)mrB(α,ν+ r

i=1ξimi)

Γ(α)Γ(ν+r_i₌₁ξimi)(κ1)ε1m1· · ·(κr)εrmr

=

∞

m1,...,mr=0

Γ(ν+α+r_i₌₁ξimi)(κ1)ε1m1· · ·(κr)εrmr =Eδi;κi;ρi

ξi;ν+α;εi(z1, . . . ,zr).

This completes the proof of Theorem1.

1

Γ(α) x

t

(x–s)α–1(s–t)v–1Eδi;κi;ρi

ξi;ν;εi z1(s–t)

ξ1_{, . . . ,}_zr₍_s_–_t₎ξr_ds

= (x–t)α+v–1_Eδi;κi;ρi

ξi;v+α;εi z1(x–t)

ξ1_{, . . . ,}_zr₍_x_–_t₎ξr_. ₍₁₈₎

Proof Using Eq. (6) in the left-hand side of Eq. (18), we obtain

1

Γ(α) x

t

(x–s)α–1₍_s_–_t₎v–1

×

∞

m1,...,mr=0

(δ1)ρ1m1· · ·(δr)ρrmr(z1(s–t)

ξ1₎m1· · ·₍_zr₍_s_–_t₎ξr₎mr

Γ(v+r_i₌₁ξimi)(κ1)ε1m1· · ·(κr)εrmr

ds.

Changing the variablestou=s–t

x–t, we obtain

=

∞

m1,...,mr=0

(δ1)ρ1m1· · ·(δr)ρrmr(z1)

m1· · ·₍_z

r)mr

Γ(ν+r_i₌₁ξimi)(κ1)ε1m1· · ·(κr)εrmr

1

Γ(α) ×

1

0

x–t–u(x–t)α–1 t+u(x–t) –tv+

r

i=1ξimi–1₍_x_–_t₎_du

=

∞

m1,...,mr=0

(δ1)ρ1m1· · ·(δr)ρrmr(z1(x–t)

ξ1₎m1· · ·₍_zr₍_x_–_t₎ξr₎mr

Γ(ν+r_i₌₁ξimi)(κ1)ε1m1· · ·(κr)εrmr ×(x–t)α+v–1

Γ(α) 1

0

(1 –u)α–1(u)v+ri=1ξimi–1_du

=

∞

m1,...,mr=0

(δ1)ρ1m1· · ·(δr)ρrmr(z1(x–t)

ξ1₎m1· · ·₍_zr₍_x_–_t₎ξr₎mr (x–t)–α–v+1_Γ₍_v₊_α₊r

i=1ξimi)(κ1)ε1m1· · ·(κr)εrmr ,

from which, after little simpliﬁcation, we easily arrive at the required Eq. (18).

Theorem 3 If ωi,τ,∈C;min1≤i≤r((ωi),(τ)) > 0, then the following integral equality holds:

x

0

tτ–1(x–t)v–1Eδiξi;;vκi;ε;1i z1(x–t)

ξ1_{, . . . ,}_zr₍_x_–_t₎ξr

×Eωi;κi;1 ξi;τ;εi z1(t)

ξ1_{, . . . ,}_z

r(t)ξr

dt=xτ+v–1_E(ωi+δi);κi;1 ξi;v+τ;εi z1x

ξ1_{, . . . ,}_z

rxξr

(5)

=

∞

m1,...,mr=0 ∞

l1,...,lr=0

(δ1)ρ1m1· · ·(δr)ρrmr(ω1)ρ1l1· · ·(ωr)ρrlr

Γ(ν+r_i₌₁ξimi)Γ(τ+r_i₌₁ξili)(κ1)ε1m1· · ·(κr)εrmr ×(z1)m1+l1· · ·(zr)mr+lr

(κ1)ε1l1· · ·(κr)εrlr x

0 tτ+

r

i=1liξi–1₍_x_–_t₎v+

r

i=1miξi–1_dt

=

∞

m1,...,mr=0

∞

l1,...,lr=0

xτ+v–1₍_δ

1)ρ1m1· · ·(δr)ρrmr(ω1)ρ1l1· · ·(ωr)ρrlr

Γ(ν+r_i₌₁ξimi)Γ(τ+_ir₌₁ξili)(κ1)ε1m1· · ·(κr)εrmr

×(z1xξ1)m1+l1· · ·(zrxξr)mr+lr

(κ1)ε1l1· · ·(κr)εrlr B

τ+

r

i=1 liξi,v+

r

i=1 miξi

=xτ+v–1 ∞

m1,...,mr=0

∞

l1,...,lr=0

(δ1)ρ1m1· · ·(δr)ρrmr(ω1)ρ1l1· · ·(ωr)ρrlr

Γ(τ+v+r_i₌₁(mi+li)ξi)(κ1)ε1m1· · ·(κr)εrmr ×(z1xξ1)m1+l1· · ·(zrxξr)mr+lr

(κ1)ε1l1· · ·(κr)εrlr

=xτ+v–1 ∞

m1,...,mr=0

∞

l1,...,lr=0

m1 l1

· · ·

mr lr

(δ1)(m1–l1)ρ1· · ·(δr)(mr–lr)ρr

Γ(τ+v+r_i₌₁(mi)ξi)

×(ω1)l1ρ1· · ·(ωr)lrρr(z1xξ1)m1· · ·(zrxξr)mr

(κ1)ε1m1· · ·(κr)εrmr

. (20)

Substitutingρ1=· · ·=ρr= 1 in Eq. (20) reduces it to

x

0

tτ–1(x–t)v–1Eδi;κi;1

ξi;v;εi z1(x–t)

ξ1_{, . . . ,}_zr₍_x_–_t₎ξr

ξ1_{, . . . ,}_zr₍_t₎ξr_du

=xτ+v–1 ∞

m1,...,mr=0

∞

l1,...,lr=0

m1 l1

· · ·

mr lr

(δ1)(m1–l1)· · ·(δr)(mr–lr)

Γ(τ+v+r_i₌₁(mi)ξi)

×(ω1)l1· · ·(ωr)lr(z1x

ξ1₎m1· · ·₍_z

rxξr)mr

(κ1)ε1m1· · ·(κr)εrmr

. (21)

Now applying the formula (α+β)m=

∞

n=0 m n

(α)n(β)m–n, Eq. (21) is reduced to the

fol-lowing form:

=xτ+v–1 ∞

m1,...,mr=0

∞

l1,...,lr=0

(ω1+δ1)m1· · ·(ωr+δr)mr(z1x

ξ1₎m1· · ·₍_z

rxξr)mr

Γ(τ+v+r_i₌₁(mi)ξi)(κ1)ε1m1· · ·(κr)εrmr =xτ+v–1_E(ωi+δi);κi;1

ξi;v+τ;εi z1x ξ1_{, . . . ,}_z

rxξr

.

z

0

tv–1Eδi;κi;ρi ξi;ν;εi z1t

ξ1_{, . . . ,}_zrtξr

dt=zvEδi;κi;ρi ξi;ν+1;εi z1z

ξ1_{, . . . ,}_zrzξr

(6)

=

∞

m1,...,mr=0

(δ1)ρ1m1· · ·(δr)ρrmr(z1)

m1· · ·₍_zr₎mr

Γ(v+r_i₌₁ξimi)(κ1)ε1m1· · ·(κr)εrmr z

0

tv+ri=1miξi–1_dt

=

∞

m1,...,mr=0

(δ1)ρ1m1· · ·(δr)ρrmr(z1)

m1· · ·₍_zr₎mr_zv+ri=1miξi

(ν+r_i₌₁miξi)Γ(v+

r

i=1ξimi)(κ1)ε1m1· · ·(κr)εrmr =zν

∞

m1,...,mr=0

(δ1)ρ1m1· · ·(δr)ρrmr(z1z

ξ1₎m1· · ·₍_zrzξr₎mr

Γ(v+ 1 +r_i₌₁ξimi)(κ1)ε1m1· · ·(κr)εrmr =zvEδi;κi;ρi

ξi;ν+1;εi z1z

ξ1_{, . . . ,}_zrzξr_.

This completes the proof of Theorem4.

Remark2.1 Upon settingκ1=· · ·=κr=ε1=· · ·=εr= 1 andr= 1, Eqs. (17), (18), (19),

and (22) reduce to a result given by Shukla and Prajapati [31, Eqs. (2.4.1), (2.4.2), (2.4.3), (2.4.4)].

Remark 2.2 By setting the parameters in Eqs. (17), (18), (19), and (22), we obtain the known results established by Khan [32, Eqs. (2.4.4), (2.4.5), (2.4.6), (2.4.7)].

Lemma 2.1([33, Eq. 2.13]) Ifμ1,μ2,α,β∈C+,τ1,τ2∈C+–1 and n,m∈N,then the fol-lowing formula holds:

L(μ1,τ1)

n β,x

Lμ1,τ1

m α,x

=

n+m

h=0 h

r=0

Γ(μ1n+τ1+ 1)Γ(μ2m+τ2+ 1)

Γ(h–r+ 1)Γ(α(m–h+r) + 1)

× (–x)h

Γ(r+ 1)Γ(β(n–r) + 1)Γ(μ1r+τ1+ 1)Γ(μ2(h–r) +τ2+ 1)

. (23)

Theorem 5 Ifα,β,η∈C+,μ1,μ2,τ1,τ2∈C+–1 and n,m∈N,then the following integral equality holds:

1

Γ(α) 1

0

uv–1(1 –u)α–1L(μ1,τ1)

n β,η(1 –u)

L(μ2,τ2)

m α,η(1 –u)

×Eδi;κi;ρi ξi;v;εi z1u

ξ1_{, . . . ,}_z

ruξr

du

=

n+m

h=0

(η)hmμ1,n,τ,α1,μ,β2,τ2E

δi;κi;ρi

ξi;v+h+α;εi(z1, . . . ,zr), (24)

where

m,n,α,β μ1,τ1,μ2,τ2=

h

r=0

h r

(–1)h₍_α₎ h

Γ(h+ 1)Γ(α(m–h+r) + 1)Γ(β(n–r) + 1) × Γ(μ1n+τ1+ 1)Γ(μ2m+τ2+ 1)

Γ(μ1r+τ1+ 1)Γ(μ2(h–r) +τ2+ 1)

(7)

Proof Using Eqs. (6) and (23) in the left-hand side of Eq. (24), we obtain

I= 1

Γ(α) 1

0

uv–1(1 –u)α–1

n+m

h=0 h

r=0

Γ(μ1n+τ1+ 1)Γ(μ2m+τ2+ 1)

Γ(h–r+ 1)Γ(α(m–h+r) + 1)

× (–η(1 –u))h

Γ(μ1r+τ1+ 1)Γ(μ2(h–r) +τ2+ 1)Γ(r+ 1)Γ(β(n–r) + 1)

×

∞

m1,...,mr=0

(δ1)ρ1m1· · ·(δr)ρrmr(z1u

ξ1₎m1· · ·₍_zruξr₎mr

Γ(v+r_i₌₁ξimi)(κ1)ε1m1· · ·(κr)εrmr

du

= 1

Γ(α)

n+m

h=0 h

r=0

Γ(μ1n+τ1+ 1)Γ(μ2m+τ2+ 1)

Γ(h–r+ 1)Γ(α(m–h+r) + 1)Γ(r+ 1)Γ(β(n–r) + 1)

× (–η)h

Γ(μ1r+τ1+ 1)Γ(μ2(h–r) +τ2+ 1) ∞

m1,...,mr=0

(δ1)ρ1m1· · ·(δr)ρrmr

Γ(v+r_i₌₁ξimi) × (z1)m1· · ·(zr)mr

(κ1)ε1m1· · ·(κr)εrmr 1

0

uv+ri=1ξimi–1_{(1 –}_u₎α+h–1_du

= 1

Γ(α)

n+m

h=0 h

r=0

Γ(μ1n+τ1+ 1)Γ(μ2m+τ2+ 1)

Γ(h–r+ 1)Γ(α(m–h+r) + 1)Γ(r+ 1)Γ(β(n–r) + 1)

× (–η)hΓ(α+h)

Γ(μ1r+τ1+ 1)Γ(μ2(h–r) +τ2+ 1)

×

∞

m1,...,mr=0

Γ(v+α+h+r_i₌₁ξimi)(κ1)ε1m1· · ·(κr)εrmr

= 1

Γ(α)

n+m

h=0 h

r=0

Γ(μ1n+τ1+ 1)Γ(μ2m+τ2+ 1)

Γ(h–r+ 1)Γ(α(m–h+r) + 1)Γ(r+ 1)Γ(β(n–r) + 1)

× (–η)hΓ(α+h)

Γ(μ1r+τ1+ 1)Γ(μ2(h–r) +τ2+ 1) Eδi;κi;ρi

ξi;v+h+α;εi(z1, . . . ,zr). (26)

Using the fact that _nx=(–1)_n_!n(–x)n, where (–x)n= (–1)n(x–n+ 1)n, on the right-hand side

of Eq. (26), yields

=

n+m

h=0 h

r=0

h r

Γ(μ1n+τ1+ 1)Γ(μ2m+τ2+ 1)

Γ(h+ 1)Γ(α(m–h+r) + 1)Γ(β(n–r) + 1)

× (–η)h(α)h

Γ(μ1r+τ1+ 1)Γ(μ2(h–r) +τ2+ 1)

Eδiξi;;κiv+;hρi+α;εi(z1, . . . ,zr),

from which, after little rearrangement, we easily arrive at the required Eq. (24).

Theorem 6 Ifα,β,η∈C+;μ1,μ2,τ1,τ2∈C+–1 and n,m∈N,then the following integral equality holds:

1

Γ(α) x

t

(x–s)α–1(s–t)ν–1L(μ1,τ1)

n β,η(x–s)

L(μ2,τ2)

m α,η(x–s)

(8)

×Eξδii;;vκi;ε;ρii z1(s–t)

ξ1_{, . . . ,}_zr₍_s_–_t₎ξr_ds_{= (}_x_–_t₎α+ν–1

n+m

h=0

(η)h ×(x–t)hmμ1,n,τ,α1,μ,β2,τ2E

δi;κi;ρi

ξi;v+h+α;εi z1(x–t)

ξ1_{, . . . ,}_zr₍_x_–_t₎ξr_, ₍₂₇₎

wherem,n,α,β

μ1,τ1,μ2,τ2 is given by Eq. (25).

Theorem 7 Ifα,β,η∈C+;μ1,μ2,τ1,τ2∈C+–1,τ1,τ2∈C+–1;n,m∈N,then the following integral equality holds:

x

0

tτ–1(x–t)v–1L(μ1,τ1)

n β,ηt

Lμ1,τ1

m α,ηt

×Eδi;κi;1

ξi;v;εi z1(x–t) ξ1_{, . . . ,}_z

r(x–t)ξr

Eωi;κi;1 ξi;τ;εi z1(t)

ξ1_{, . . . ,}_z

r(t)ξr

dt

=xτ+v–1 n+m

h=0

(ηx)hμm1,n,τ,α1,μ,β2,τ2E

(ωi+δi);κi;1 ξi;v+τ+h;εi z1x

ξ1_{, . . . ,}_zrxξr_, ₍₂₈₎

wherem,n,α,β

Theorem 8 Ifα,β,η∈C+;μ1,μ2,τ1,τ2∈C+–1,τ1,τ2∈C+–1;n,m∈N,then the following integral equality holds:

z

0

tv–1L(μ1,τ1)

n β,ηt

Lμ1,τ1

m α,ηt

Eδiξi;;κiν;ε;ρii z1t

ξ1_{, . . . ,}_zrtξr_dt

=zv–1 n+m

h=0

(ηz)hm,n,α,β μ1,τ1,μ2,τ2E

δi;κi;ρi ξi;ν+h+1;εi z1z

ξ1_{, . . . ,}_z

rzξr

, (29)

wherem,n,α,β

Proof The proofs of Eqs. (27), (28), and (29) are the same as those of Eq. (25), which can

be obtained from Eqs. (18), (19), and (22).

Theorem 9 Ifα,β,η∈C+,μ1,μ2,τ1,τ2∈C+–1 and n,m∈N,then the following integral equality holds:

1

Γ(α) 1

0

uv–1(1 –u)α–1L(μ1,τ1)

n β,η(1 –u)

L(μ2,τ2)

m α,η(1 –u)

ξ1_{, . . . ,}_z

ruξr

du

=

n+m

h=0

(η)h℘_μm,n,α,β

1,τ1,μ2,τ2E

δi;κi;ρi

ξi;v+h+α;εi(z1, . . . ,zr), (30)

where

℘_μm,n,α,β

1,τ1,μ2,τ2=

Γ(μ1n+τ1+ 1)Γ(μ2m+τ2+ 1)

Γ(αm+ 1)Γ(βn+ 1) ×

h

r=0

h r

(–1)h–α(h–r)–βr₍_α₎

h(–αm)α(h–r)(–βn)βr

Γ(h+ 1)Γ(μ1r+τ1+ 1)Γ(μ2(h–r) +τ2+ 1)

(9)

Proof Using (–x)n= (–1)n(x–n+ 1)n, Eq. (26) is reduced to

=Γ(μ1n+τ1+ 1)Γ(μ2m+τ2+ 1)

Γ(αm+ 1)Γ(βn+ 1)

n+m

h=0

(η)h

h

r=0

h r

× (–1)h–α ₍_h_–_r_)–_β_r

(α)h(–αm)α(h–r)(–βn)βr

Γ(h+ 1)Γ(μ1r+τ1+ 1)Γ(μ2(h–r) +τ2+ 1) Eδi;κi;ρi

ξi;v+h+α;εi(z1, . . . ,zr)

=

n+m

h=0

(η)h℘_μm,n,α,β

1,τ1,μ2,τ2E

δi;κi;ρi

ξi;v+h+α;εi(z1, . . . ,zr),

where℘m,n,α,β

μ1,τ1,μ2,τ2is given by Eq. (31).

Theorem 10 Ifα,β,η∈C+,μ1,μ2,τ1,τ2∈C+–1and n,m∈N,then the following integral equality holds:

1

Γ(α) x

t

(x–s)α–1(s–t)ν–1L(μ1,τ1)

n β,η(x–s)

L(μ2,τ2)

m α,η(x–s)

×Eδi;κi;ρi

ξi;v;εi z1(s–t) ξ1_{, . . . ,}_z

r(s–t)ξr

ds= (x–t)α+ν–1 n+m

h=0

(η)h ×(x–t)h℘μm1,n,τ,α1,μ,β2,τ2E

δi;κi;ρi

ξi;v+h+α;εi z1(x–t)

ξ1_{, . . . ,}_zr₍_x_–_t₎ξr_, ₍₃₂₎

where℘m,n,α,β

Theorem 11 Ifα,β,η∈C+_,_μ

1,μ2,τ1,τ2∈C+–1,τ1,τ2∈C+–1;n,m∈N,then the following integral equality holds:

x

0

tτ–1₍_x_–_t₎v–1_L(μ1,τ1)

n β,ηt

Lμ1,τ1

m α,ηt

×Eδi;κi;1

ξi;v;εi z1(x–t)

ξ1_{, . . . ,}_zr₍_x_–_t₎ξr_Eωi;κi;1 ξi;τ;εi z1(t)

ξ1_{, . . . ,}_zr₍_t₎ξr_dt

=xτ+v–1 n+m

h=0

(ηx)h℘_μm₁,n_,_τ,α₁_,_μ,β₂_,_τ₂E(ωi+δi);κi;1 ξi;v+τ+h;εi z1x

ξ1_{, . . . ,}_zrxξr_, ₍₃₃₎

where℘m,n,α,β

Theorem 12 Ifα,β,η∈C+_,_μ

z

0

tv–1L(μ1,τ1)

n β,ηt

Lμ1,τ1

m α,ηt

Eδi;κi;ρi ξi;ν;εi z1t

ξ1_{, . . . ,}_z

rtξr

dt

=zv–1 n+m

h=0

(ηz)h℘_μm,n,α,β

1,τ1,μ2,τ2E

δi;κi;ρi ξi;ν+h+1;εi z1z

ξ1_{, . . . ,}_zrzξr_, ₍₃₄₎

where℘m,n,α,β

(10)

Remark2.3 Upon settingκ1=· · ·=κr=ε1=· · ·εr= 1, Eqs. (17), (18), (19), (22), (24), (27),

(28), (29), (30), (32), (33), and (34) are reduced to Eqs. (2.1), (2.3), (2.5), (2.10), (2.20), (2.27), (2.28), (2.29), (2.31), (2.35), (2.36), and (2.37) established by Agarwal et al. [33].

3 Special cases

In this section, we emphasize special cases by selecting particular values of parameters. (i) Putting α=β= 1, the results in Eqs. (30), (32), (33), and (34) are reduced to the following form:

Corollary 1 Ifη∈C+,μ1,μ2,τ1,τ2∈C+–1and n,m∈N,then

1

Γ(α) 1

0

uv–1(1 –u)α–1L(μ1,τ1)

n η(1 –u)

L(μ2,τ2)

m η(1 –u)

ξ1_{, . . . ,}_z

ruξr

du

=

n+m

h=0

(η)hΓ(μ1n+τ1+ 1)Γ(μ2m+τ2+ 1)

Γ(m+ 1)Γ(n+ 1)

h

r=0

h r

× (α)h(–m)(h–r)(–n)r

Γ(h+ 1)Γ(μ1r+τ1+ 1)Γ(μ2(h–r) +τ2+ 1) Eδi;κi;ρi

ξi;v+h+α;εi(z1, . . . ,zr). (35)

Corollary 2 Ifη∈C+,μ1,μ2,τ1,τ2∈C+–1and n,m∈N,then

1

Γ(α) x

t

(x–s)α–1(s–t)ν–1L(μ1,τ1)

n η(x–s)

L(μ2,τ2)

m η(x–s)

×Eδi;κi;ρi

ξi;v;εi z1(s–t)

ξ1_{, . . . ,}_zr₍_s_–_t₎ξr_ds

= (x–t)α+ν–1

n+m

h=0

(η)h(x–t)hΓ(μ1n+τ1+ 1)Γ(μ2m+τ2+ 1)

Γ(m+ 1)Γ(n+ 1) ×

h

r=0

h r

(α)h(–m)(h–r)(–n)r

Γ(h+ 1)Γ(μ1r+τ1+ 1)Γ(μ2(h–r) +τ2+ 1)

×Eδi;κi;ρi

ξi;v+h+α;εi z1(x–t) ξ1_{, . . . ,}_z

r(x–t)ξr

. (36)

Corollary 3 Ifη∈C+_,_μ

1,μ2,τ1,τ2∈C+–1,;n,m∈N,then

x

0

n (ηt)L μ1,τ1

m (ηt)

×Eδi;κi;1

ξi;v;εi z1(x–t)

ξ1_{, . . . ,}_zr₍_t₎ξr_dt

=xτ+v–1 n+m

h=0

(ηx)hΓ(μ1n+τ1+ 1)Γ(μ2m+τ2+ 1)

Γ(m+ 1)Γ(n+ 1)

h

r=0

h r

× (α)h(–m)(h–r)(–n)r

Γ(h+ 1)Γ(μ1r+τ1+ 1)Γ(μ2(h–r) +τ2+ 1)

E(ωi+δi);κi;1 ξi;v+τ+h;εi z1x

ξ1_{, . . . ,}_z

rxξr

. (37)

Corollary 4 Ifη∈C+,μ1,μ2,τ1,τ2∈C+–1,n,m∈N,then

z

0

tv–1L(μ1,τ1)

n (ηt)L μ1,τ1

m (ηt)E δi;κi;ρi ξi;ν;εi z1t

(11)

=zv–1 n+m

h=0

(ηz)hΓ(μ1n+τ1+ 1)Γ(μ2m+τ2+ 1)

Γ(m+ 1)Γ(n+ 1)

h

r=0

h r

× (α)h(–m)(h–r)(–n)r

Γ(h+ 1)Γ(μ1r+τ1+ 1)Γ(μ2(h–r) +τ2+ 1)

E_ξiδi;_;_νκi₊;ρi_h_+1;_εi z1zξ1, . . . ,zrzξr

. (38)

(ii) Puttingμ1=μ2=α=β= 1 and using Eq. (12), the results in Eqs. (30), (32), (33),

and (34) reduced to the following form:

Corollary 5 Ifη∈C+_,_τ

1,τ2∈C+–1and n,m∈N,then

1

Γ(α) 1

0

uv–1(1 –u)α–1_L(1,τ1)

n η(1 –u); 1

L(1,τ2)

m η(1 –u); 1

ξ1_{, . . . ,}_zruξr_du

=

n+m

h=0

(η)hΓ(n+τ1+ 1)Γ(m+τ2+ 1)

Γ(m+ 1)Γ(n+ 1)

×

h

r=0

h r

(α)h(–m)(h–r)(–n)r

Γ(h+ 1)Γ(r+τ1+ 1)Γ((h–r) +τ2+ 1) Eδi;κi;ρi

ξi;v+h+α;εi(z1, . . . ,zr). (39)

Corollary 6 Ifη∈C+,τ1,τ2∈C+–1and n,m∈N,then

1

Γ(α) x

t

(x–s)α–1(s–t)ν–1L(1,τ1)

n η(x–s); 1

L(1,τ2)

m η(x–s); 1

×Eδi;κi;ρi

ξi;v;εi z1(s–t)

ξ1_{, . . . ,}_zr₍_s_–_t₎ξr_ds

= (x–t)α+ν–1 n+m

h=0

(η)h(x–t)hΓ(n+τ1+ 1)Γ(m+τ2+ 1)

Γ(m+ 1)Γ(n+ 1)

×

h

r=0

h r

(α)h(–m)(h–r)(–n)r

Γ(h+ 1)Γ(r+τ1+ 1)Γ((h–r) +τ2+ 1)

×Eδi_ξi;_;κi_v₊;ρi_h₊_α_;_εi z1(x–t)ξ1, . . . ,zr(x–t)ξr

. (40)

Corollary 7 Ifη∈C+,τ1,τ2∈C+–1,n,m∈N,then

x

0

tτ–1(x–t)v–1_L(μ1,τ1) n (ηt; 1)L

μ1,τ1

m (ηt; 1)

×Eδi;κi;1

ξi;v;εi z1(x–t)

ξ1_{, . . . ,}_zr₍_t₎ξr_dt

=xτ+v–1 n+m

h=0

(ηx)hΓ(n+τ1+ 1)Γ(m+τ2+ 1)

Γ(m+ 1)Γ(n+ 1)

h

r=0

h r

× (α)h(–m)(h–r)(–n)r

Γ(h+ 1)Γ(r+τ1+ 1)Γ((h–r) +τ2+ 1)

E_ξi(ωi_;_v+₊δi_τ₊);_hκi_;_εi;1 z1xξ1, . . . ,zrxξr

. (41)

Corollary 8 Ifη∈C+,τ1,τ2∈C+–1,n,m∈N,then

z

0

tv–1L(μ1,τ1)

n (ηt; 1)L μ1,τ1

m (ηt; 1)E δi;κi;ρi ξi;ν;εi z1t

(12)

=zv–1 n+m

h=0

(ηz)hΓ(n+τ1+ 1)Γ(m+τ2+ 1)

Γ(m+ 1)Γ(n+ 1)

h

r=0

h r

× (α)h(–m)(h–r)(–n)r

Γ(h+ 1)Γ(r+τ1+ 1)Γ((h–r) +τ2+ 1)

E_ξiδi;_;_νκi₊;ρi_h_+1;_εi z1zξ1, . . . ,zrzξr

. (42)

(iii) Puttingμ1=μ1= 0,α=β= 1, the results in Eqs. (30), (32), (33), and (34) are

re-duced to the following form:

1

Γ(α) 1

0

uv–1(1 –u)α–1 1 –η(1 –u)mEδiξi;;κiv;ε;ρii z1u

ξ1_{, . . . ,}_zruξr_du

=

m

h=0

(η)h_(–_m₎ h(α)hE

δi;κi;ρi

ξi;v+h+α;εi(z1, . . . ,zr). (43)

1

Γ(α) x

t

(x–s)α–1₍_s_–_t₎ν–1 _{1 –}_η₍_x_–_s₎m

×Eδi;κi;ρi

ξi;v;εi z1(s–t)

ξ1_{, . . . ,}_zr₍_s_–_t₎ξr_ds_{= (}_x_–_t₎α+ν–1 m

h=0

η(x–t)h ×(–m)h(α)hEξiδi;;κiv+;hρi+α;εi z1(x–t)

ξ1_{, . . . ,}_zr₍_x_–_t₎ξr_. ₍₄₄₎

x

0

tτ–1(x–t)v–1_{(1 –}_η_t₎m_Eδi;κi;1

ξi;v;εi z1(x–t)

ξ1_{, . . . ,}_zr₍_x_–_t₎ξr

ξ1_{, . . . ,}_zr₍_t₎ξr_dt

=xτ+v–1 m

h=0

(ηx)h(–m)h(α)hEξi(ωi;v++δiτ+);hκi;εi;1 z1x

ξ1_{, . . . ,}_zrxξr_. ₍₄₅₎

Corollary 12 Ifη∈C+_,_τ

1,τ2∈C+–1and n,m∈N,then

z

0

tv–1(1 –ηt)mEξδii;;νκi;ε;ρii z1t

ξ1_{, . . . ,}_zrtξr_dt

=zv–1 m

h=0

(ηz)h(–m)h(α)hE δi;κi;ρi ξi;ν+h+1;εi z1z

ξ1_{, . . . ,}_z

rzξr

. (46)

Remark3.1 If we putκ1=· · ·=κr=ε1=· · ·=εr= 1, then Eqs. (35)–(46) are reduced to

Eqs. (3.1), (3.2), (3.3), (3.4), (3.6), (3.7), (3.8), (3.9), (3.11), (3.16), (3.17), (3.18) established in Agarwal et al. [33].

(iv) Settingρi=κi=εi= 1,ξ1=· · ·=ξr= 1, the multivariate Mittag-Leﬄer function

(13)

Corollary 13 The following integral equality holds:

1

Γ(α) 1

0

uv–1(1 –u)α–1_ϕ(r) 2

δ1, . . . ,δr;v;z1uξ1, . . . ,zruξr

du

=ϕ₂(r)[δ1, . . . ,δr;v+α;z1, . . . ,zr]. (47)

Corollary 14 Then following integral equality holds:

1

Γ(α) x

t

(x–s)α–1(s–t)v–1

×ϕ₂(r)δ1, . . . ,δr;v;z1(s–t)ξ1, . . . ,zr(s–t)ξr

ds

= (x–t)α+v–1ϕ₂(r)δ1, . . . ,δr;v+α;z1(s–t)ξ1, . . . ,zr(s–t)ξr

. (48)

x

0

tτ–1(x–t)v–1ϕ₂(r)δ1, . . . ,δr;v;z1(x–t)ξ1, . . . ,zr(x–t)ξr

×ϕ₂(r)δ1, . . . ,δr;v;z1tξ1, . . . ,zrtξr

dt

=xτ+v–1_ϕ(r) 2

δ1, . . . ,δr;v+τ;z1tξ1, . . . ,zrtξr

. (49)

z

0

tv–1ϕ₂(r)δ1, . . . ,δr;v;z1tξ1, . . . ,zrtξr

dt

=zv–1ϕ₂(r)δ1, . . . ,δr;v+ 1;z1zξ1, . . . ,zrzξr

. (50)

Corollary 17 Ifα,β,η∈C+,μ1,μ2,τ1,τ2∈C+–1and n,m∈N,then the following integral equality holds:

1

Γ(α) x

t

(x–s)α–1(s–t)ν–1L(μ1,τ1)

n β,η(x–s)

L(μ2,τ2)

m α,η(x–s)

×ϕ₂(r)δ1, . . . ,δr;v;1(s–t)ξ1, . . . ,zr(s–t)ξr

ds

= (x–t)α+ν–1 n+m

h=0

η(x–t)hm,n,α,β μ1,τ1,μ2,τ2

×ϕ₂(r)δ1, . . . ,δr;v+h+α;z1(x–t)ξ1, . . . ,zr(x–t)ξr

, (51)

wherem,n,α,β

1

Γ(α) 1

0

uv–1(1 –u)α–1_L(μ1,τ1)

n β,η(1 –u)

L(μ2,τ2)

m α,η(1 –u)

×ϕ₂(r)δ1, . . . ,δr;v;z1uξ1, . . . ,zruξr

(14)

=

n+m

h=0

(η)hmμ1,n,τ,α1,μ,β2,τ2ϕ

(r)

2 [δ1, . . . ,δr;v+α;z1, . . . ,zr], (52)

wherem,n,α,β

Corollary 19 Ifα,β,η∈C+_,_μ

x

0

n β,ηt

Lμ1,τ1

m α,ηt

×ϕ₂(r)δ1, . . . ,δr;v;z1(x–t)ξ1, . . . ,zr(x–t)ξr

×ϕ₂(r)δ1, . . . ,δr;v;z1(t)ξ1, . . . ,zr(t)ξr

dt

=xτ+v–1 n+m

h=0

(ηx)hm,n,α,β μ1,τ1,μ2,τ2ϕ

(r) 2

δ1, . . . ,δr;v+τ+h;z1xξ1, . . . ,zrxξr

, (53)

wherem,n,α,β

Corollary 20 Ifα,β,η∈C+,μ1,μ2,τ1,τ2∈C+–1,τ1,τ2∈C+–1;n,m∈N,then the following integral equality holds:

z

0

tv–1L(μ1,τ1)

n β,ηt

Lμ1,τ1

m α,ηt

ϕ₂(r)δ1, . . . ,δr;v;z1tξ1, . . . ,zrtξr

dt

=zv–1 n+m

h=0

(ηz)hμm1,n,τ,α1,μ,β2,τ2ϕ

(r) 2

δ1, . . . ,δr;ν+h+ 1;z1tξ1, . . . ,zrtξr

, (54)

wherem,n,α,β

1,μ2,τ1,τ2∈C+–1and n,m∈N,then the following integral equality holds:

1

Γ(α) 1

0

uv–1(1 –u)α–1_L(μ1,τ1)

n β,η(1 –u)

L(μ2,τ2)

m α,η(1 –u)

×ϕ₂(r)δ1, . . . ,δr;v;z1uξ1, . . . ,zruξr

du

=

n+m

h=0

(η)h_℘m,n,α,β μ1,τ1,μ2,τ2ϕ

(r)

2 [δ1, . . . ,δr;v+h+α;z1, . . . ,zr], (55)

where℘m,n,α,β

1

Γ(α) x

t

(x–s)α–1₍_s_–_t₎ν–1_L(μ1,τ1)

n β,η(x–s)

L(μ2,τ2)

m α,η(x–s)

×ϕ₂(r)δ1, . . . ,δr;v;z1(s–t)ξ1, . . . ,zr(s–t)ξr

(15)

= (x–t)α+ν–1

n+m

h=0

η(x–t)h℘_μm,n,α,β

1,τ1,μ2,τ2

×ϕ₂(r)δ1, . . . ,δr;v+h+α;z1(x–t)ξ1, . . . ,zr(x–t)ξr

, (56)

where℘m,n,α,β

x

0

n β,ηt

Lμ1,τ1

m α,ηt

×ϕ₂(r)δ1, . . . ,δr;v;z1(x–t)ξ1, . . . ,zr(x–t)ξr

×ϕ₂(r)δ1, . . . ,δr;v;z1(t)ξ1, . . . ,zr(t)ξr

dt

=xτ+v–1 n+m

h=0

(ηx)h℘m_μ,n,α,β

1,τ1,μ2,τ2ϕ

(r) 2

δ1, . . . ,δr;v+τ+h;z1xξ1, . . . ,zrxξr

, (57)

where℘m,n,α,β

z

0

tv–1L(μ1,τ1)

n β,ηt

Lμ1,τ1

m α,ηt

ϕ₂(r)δ1, . . . ,δr;v;z1tξ1, . . . ,zrtξr

dt

=zv–1 n+m

h=0

(ηz)h℘_μm,n,α,β

1,τ1,μ2,τ2ϕ

(r) 2

δ1, . . . ,δr;ν+h+ 1;z1zξ1, . . . ,zrzξr

, (58)

where℘m,n,α,β

(v) Setting ρi=κi=εi=δi=ξi=v=r= 1, the multivariate Mittag-Leﬄer function is

reduced to the exponential functionE_1;1;11;1;1(z) =E1,1(z) =exp(z); and we can ﬁnd a similar

line of results associated with exponential function from Eqs. (47)–(58).

Acknowledgements

The authors are thankful to the referee for his/her valuable remarks and comments for the improvement of the paper.

Funding

Not available.

Availability of data and materials

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally and signiﬁcantly in writing this paper. All authors read and approved the ﬁnal manuscript.

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(16)

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