The Gauss product and Raabe’s integral for k-gamma functions

We obtain an extension of the famous Gauss product formula to the case of k-gamma functions. The Sándor–Tóth short product formula [16] is also attended to these functions. An asymptotic formula and Raabe’s integral analogue are also considered.

ON SOLVABILITY OF NONLOCAL PROBLEM FOR THIRD-ORDER EQUATION

In this paper nonlocal problem with integral conditions for partial differential equation of the third order is considered. The existence of a unique classical solution is proved in rectangular domain. The proof is carried out by the method of auxiliary problems. At first the problem for a new function for partial differential equation of the first order is considered. Then the solvability of integral analogue of Goursat problem for hyperbolic equation of the second order is proved by equivalent reduction of the problem to the Volterra integral equation of the second kind.

An identity involving integration with respect to variable order of fractional derivative

An identity is presented, in which an expression involving Riemann–Liouville fractional derivative is integrated with respect to the derivative’s order. As a by-product an integral analogue of an identity for a sum of binomial coefficients is derived.

A Local Fractional Integral Inequality on Fractal Space Analogous to Anderson’s Inequality

Anderson's inequality (Anderson, 1958) as well as its improved version given by Fink (2003) is known to provide interesting examples of integral inequalities. In this paper, we establish local fractional integral analogue of Anderson's inequality on fractal space under some suitable conditions. Moreover, we also show that the local fractional integral inequality on fractal space, which we have proved in this paper, is a new generalization of the classical Anderson's inequality.

An Integral Analogue of Theta Functions and Gauss Sums in Ramanujan's Lost Notebook

One page in Ramanujan's lost notebook is devoted to claims about a certain integral with two parameters. One claim gives an inversion formula for the integral that is similar to the transformation formula for theta functions. Other claims are remindful of Gauss sums. In this paper we prove all the claims made by Ramanujan about this integral.

On two new multidimensional integral inequalities of the Hilbert type

The main aim of this paper is to establish two new multidimensional integral inequalities similar to the integral analogue of the well known Hilbert's inequality by using elementary analysis.