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.
Highlights
Zrxξr where m,n,α ,β μ1 ,τ1 ,μ2 ,τ2 is given by Theorem 8 If α , β , η ∈ C+; μ1, μ2, τ1, τ2 ∈ C+–1, τ1, τ2 ∈ C+–1; n, m ∈ N, the following integral equality holds: z tv–1L(nμ1,τ1) β , ηt Lμm1,τ1 α , ηt Eξδii;;νκ;iε;ρi i z1tξ1 ,
Introduction and preliminariesThe function defined by the series representation ∞ znEξ (z) = Γ (ξ n + 1) (ξ > 0, z ∈ C) (1)n=0 and its generalizationEξ,ν(z) = Γ (ξ n + ν)(ξ > 0, ν > 0, z ∈ C) (2)n=0 were introduced and studied by Agarwal [1], Mittag-Leffler [2, 3], Humburt [4], Humbert and Agrawal [5], and Wiman [6, 7], where C is the set of complex numbers
The main properties 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
Summary
Zrxξr where m,n,α ,β μ1 ,τ1 ,μ2 ,τ2 is given by Theorem 8 If α , β , η ∈ C+; μ1, μ2, τ1, τ2 ∈ C+–1, τ1, τ2 ∈ C+–1; n, m ∈ N, the following integral equality holds: z tv–1L(nμ1,τ1) β , ηt Lμm1,τ1 α , ηt Eξδii;;νκ;iε;ρi i z1tξ1 , .
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