Abstract:
The principle aim of this research article is to investigate the properties of k-fractional integration introduced and defined by Mubeen and Habibullah [1],and secondly to solve the integral equation of the form
, for k > 0, β > 0, y > 0, 0 < x < t < ∞, where is the confluent k-hypergeometric functions, by using k-fractional integration.

Abstract:
Here we aim at establishing certain new fractional integral inequalities involving the Gauss hypergeometric function for synchronous functions which are related to the Chebyshev functional. Several special cases as fractional integral inequalities involving Saigo, Erdélyi-Kober, and Riemann-Liouville type fractional integral operators are presented in the concluding section. Further, we also consider their relevance with other related known results. 1. Introduction Fractional integral inequalities have many applications; the most useful ones are in establishing uniqueness of solutions in fractional boundary value problems and in fractional partial differential equations. Further, they also provide upper and lower bounds to the solutions of the above equations. These considerations have led various researchers in the field of integral inequalities to explore certain extensions and generalizations by involving fractional calculus operators. One may, for instance, refer to such type of works in the book [1] and the papers [2–11]. In a recent paper, Purohit and Raina [9] investigated certain Chebyshev type [12] integral inequalities involving the Saigo fractional integral operators and also established the -extensions of the main results. The aim of this paper is to establish certain generalized integral inequalities for synchronous functions that are related to the Chebyshev functional using the fractional hypergeometric operator, introduced by Curiel and Galu [13]. Results due to Purohit and Raina [9] and Belarbi and Dahmani [2] follow as special cases of our results. In the sequel, we use the following definitions and related details. Definition 1. Two functions and are said to be synchronous on , if for any . Definition 2. A real-valued function is said to be in the space , if there exists a real number such that , where . Definition 3. Let , , and ; then a generalized fractional integral (in terms of the Gauss hypergeometric function) of order for a real-valued continuous function is defined by [13] (see also [14]): where the function appearing as a kernel for the operator (2) is the Gaussian hypergeometric function defined by and is the Pochhammer symbol: The object of the present investigation is to obtain certain Chebyshev type integral inequalities involving the generalized fractional integral operators [13] which involves in the kernel, the Gauss hypergeometric function (defined above). The concluding section gives some special cases of the main results. 2. Main Results Our results in this section are based on the preliminary assertions giving

Abstract:
The object of this paper is to solve a fractional integro-differential equation involving a generalized Lauricella confluent hypergeometric function in several complex variables and the free term contains a continuous function f(τ). The method is based on certain properties of fractional calculus and the classical Laplace transform. A Cauchy-type problem involving the Caputo fractional derivatives and a generalized Volterra integral equation are also considered. Several special cases are mentioned. A number of results given recently by various authors follow as particular cases of formulas established here.

Abstract:
In previous paper I construct an approximative solution of the power series expansion in closed forms of Grand Confluent Hypergeometric (GCH) function only up to one term of A_n's [4]. And I obtain normalized constant and orthogonal relation of GCH function. In this paper I will apply three term recurrence formula [3] to the power series expansion in closed forms of GCH function (infinite series and polynomial) including all higher terms of A_n's. In general most of well-known special function with two recursive coefficients only has one eigenvalue for the polynomial case. However this new function with three recursive coefficients has infinite eigenvalues that make B_n's term terminated at specific value of index n because of three term recurrence formula [3]. This paper is 9th out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 6 for all the papers in the series. Previous paper in series deals with generating functions of Lame polynomial in the Weierstrass's form [28]. The next paper in the series describes the integral formalism and the generating function of GCH function [30].

Abstract:
Several expansions of the solutions of the double-confluent Heun equation in terms of the Kummer confluent hypergeometric functions are presented. Three different sets of these functions are examined. Discussing the expansions without pre-factor, it is shown that two of these functions provide expansions the coefficients of which obey three-term recurrence relations, while for the third confluent hypergeometric function the corresponding recurrence relation generally involves five-terms. This relation is reduced to a three-term one only in the case when the double-confluent Heun equation degenerates to the confluent hypergeometric equation. The conditions for obtaining finite sum solutions via termination of the expansions are discussed. The possibility of constructing expansions of different structure using certain equations related to the double-confluent Heun equation is discussed. An example of such expansion derived using the equation obeyed by a function involving the derivative of a solution of the double-confluent Heun equation is presented. In this way, expansions governed by three- or more term recurrence relations for expansion coefficients can be constructed. An expansion with coefficients obeying a seven-term recurrence relation is presented. This relation is reduced to a five-term one if the additional singularity of the equation obeyed by the considered auxiliary function coincides with a singularity of the double-confluent Heun equation.

Abstract:
Biconfluent Heun (BCH) function, a confluent form of Heun function, is the special case of Grand Confluent Hypergeometric (GCH) function: this has a regular singularity at x=0, and an irregular singularity at infinity of rank 2. In this paper I apply three term recurrence formula (3TRF) [arXiv:1303.0806] to the integral formalism of GCH function including all higher terms of A_n's and the generating function for the GCH polynomial which makes B_n term terminated. I show how to transform power series expansion in closed forms of GCH equation to its integral representation analytically. This paper is 10th out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 6 for all the papers in the series. The previous paper in the series describes the power series expansion in closed forms of GCH equation and its asymtotic behaviours. [arXiv:1303.0813]

Abstract:
In this paper, we establish new formulas for the product of parabolic cylinder functions with different parameters. These formulas entail the Laplace transform of Kummer's confluent hypergeometric functions. We show then that these new identities yield Nicholson-type integrals for the product of two parabolic cylinder functions generalizing thus some known ones. Besides, we use the new integral representations to derive other series expansions for products of parabolic cylinder functions.

Abstract:
It is known that the radial equation of the massless fields with spin around Kerr black holes cannot be solved by special functions. Recently, the analytic solution was obtained by use of the expansion in terms of the special functions and various astrophysical application have been discussed. It was pointed out that the coefficients of the expansion by the confluent hypergeometric functions are identical to those of the expansion by the hypergeometric functions. We explain the reason of this fact by using the integral equations of the radial equation. It is shown that the kernel of the equation can be written by the product of confluent hypergeometric functions. The integral equaton transforms the expansion in terms of the confluent hypergeometric functions to that of the hypergeometric functions and vice versa,which explains the reason why the expansion coefficients are universal.

Abstract:
We give a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F_1 equation (equivalent to the Bessel equation), the Gegenbauer equation and the Hermite equation. In particular, we discuss recurrence relations of their solutions, their integral representations and discrete symmetries.

Abstract:
We study the monodromies at infinity of confluent A-hypergeometric functions introduced by Adolphson. In particular, we extend the result of the third author for non-confluent A-hypergeometric functions to the confluent case. The integral representation by rapid decay homology cycles will play a central role in the proof.