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This paper deals with some multidimensional integral operators involving the Gauss hypergeometric function in the kernel and generating the multidimensional modified fractional calculus operators introduced in . Some mapping properties, weighted inequalities, a formula of integration by parts and index laws are obtained.
M. Saigo [Math. Rep. Coll. Gen. Educ., Kyushu Univ., 11 (1978) 135-143] has defined a pair of fractional integral operators and fractional derivatives involving generalizd hypergeometric function. The aim of present paper is to define their q-analogues. First, we define a pair of q-analogues of Saigo’s fractional integral operators and establish some… (More)
Recently, many papers in the theory of univalent functions have been devoted to mapping and characterization properties of various linear integral or integro-differential operators in the class S (of normalized analytic and univalent functions in the open unit disk U), and in its subclasses (as the classes S∗ of the starlike functions and K of the convex… (More)
In the paper a convolution of the Hankel transform is constructed. The convolution is used to the calculation of an integral containing Bessel functions of the first kind.
where Jν(x) is the Bessel function of the first kind of order ν , and =z denotes the imaginary part of z. An extensive table of integral transforms involving the Bessel functions in the kernels is collected in . Since the integration in (2) is with respect to the order of the Bessel function, such a pair of integral transforms is called index… (More)
Integral transforms (Hf)(x) = ∞ 0 H m,n p,q xt f (t)dt involving Fox's H-functions as kernels are studied in the spaces L ν,r of functions f such that ∞ 0 |t ν f (t)| r dt t < ∞ (1 ≦ r < ∞, ν ∈ R). Mapping properties such as the boundedness, the representation and the range of the transforms H are given.
Evaluation of integrals Meijer G-function Bessel functions Cosine and sine functions a b s t r a c t This paper is devoted to the study of the special function Z ν ρ (z) of z ∈ C with two parameters ρ ∈ R and ν ∈ C. In a special case such a function coincides with the McDonald function. Using the representations of Z ν ρ (z) in terms of the H-function,… (More)
The paper is devoted to study the H-function defined by the Mellin-Barnes integral H p,q (z) = 1 2πi ∫ L H m,n p,q (s)z ds, where the function H p,q (s) is a certain ratio of products of Gamma functions with the argument s and the contour L is specially chosen. The conditions for the existence of Hm,n p,q (z) are discussed and explicit power and… (More)
The Volterra nonlinear integral equation
where the function m,n tp,q (S) is a certain ratio of products of the Gammafunctions with the argument s and the contour specially chosen. The conditions for the existence of m,n Hp, q (Z) are discussed and explicit power and power-logarithmic series expansions of m,n Hp, q (Z) near zero and infinity are given. The obtained results define more precisely the… (More)