Abstract
In this article, we investigate the so-called Inayat integral operator T_{p,q}^{m,n}, p,q,m,nin mathbb{Z}, 1leq mleq q, 0leq nleq p , on classes of generalized integrable functions. We make use of the Mellin-type convolution product and produce a concurrent convolution product, which, taken together, establishes the Inayat integral convolution theorem. The Inayat convolution theorem and a class of delta sequences were derived and employed for constructing sequence spaces of Boehmians. Moreover, by the aid of the concept of quotients of sequences, we present a generalization of the Inayat integral operator in the context of Boehmians. Various results related to the generalized integral operator and its inversion formula are also derived.
Highlights
Special functions are a generalization of the more familiar elementary functions and include, among many others, gamma functions, zeta functions, Bessel functions, Legendre functions, Laguerre functions, hypergeometric functions, and Hermite polynomials
The special functions of mathematical physics and chemistry are mostly obtained in the solution of differential equations, which were already met in some elementary analysis, series solutions of the harmonic oscillator and atomic one-electron problems
Under the assumption that only sectors having Hpm,q,n(st) = O(est) at infinity with w > 0 and s > 0 and the sectors having the asymptotic expansion of the H -function of algebraic order at infinity with ws > 0 should be considered, the Inayat integral operator Tpm,q,n is defined by [1]
Summary
Special functions are a generalization of the more familiar elementary functions and include, among many others, gamma functions, zeta functions, Bessel functions, Legendre functions, Laguerre functions, hypergeometric functions, and Hermite polynomials. Q) are complex numbers, in terms of Mellin–Barnes type contour integrals, the Inayat Hpm,q,n-function is defined as [1]. Under the assumption that only sectors having Hpm,q,n(st) = O(est) at infinity with w > 0 and s > 0 and the sectors having the asymptotic expansion of the H -function of algebraic order at infinity with ws > 0 should be considered, the Inayat integral operator Tpm,q,n is defined by [1]. On the bases of Eq (12) and Eq (13), the convolution theorem of the Inayat integral operator can be presented as follows. We make use of the convolution theorem and the delta sequences, we already obtained, to derive the deterministic axioms necessary for defining the spaces of extension of the Inayat integral operator. A sequence of Boehmians (xn) in βL∗A is said to be -convergent to a Boehmian x in βL∗A , xn → x, if there exists a (ωn) ∈ such that (xn –x)×ωn ∈ βL∗A , ∀n ∈ N, and (xn –x)×ωn → 0 as n → ∞ in βL∗A
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