Related papers: New indefinite integrals of Heun functions
First derivatives of the Whittaker function $\mathrm{M}_{\kappa ,\mu }\left(x\right) $ with respect to the parameters are calculated. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of…
We consider the derivatives of Horn hypergeometric functions of any number variables with respect to their parameters. The derivative of the function in $n$ variables is expressed as a Horn hypergeometric series of $n+1$ infinite summations…
The paper classifies algebraic transformations of Gauss hypergeometric functions with the local exponent differences $(1/2,1/4,1/4)$, $(1/2,1/3,1/6)$ and $(1/3,1/3,1/3)$. These form a special class of algebraic transformations of Gauss…
Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and…
In this paper we consider the confluent Heun equation, which is a linear differential equation of second order with three singular points --- two of them are regular and the third one is irregular of rank 1. The purpose of the work is to…
In this paper, we first introduce certain forms of extended incomplete Pochhammer symbols which are then used to define families of extended incomplete generalized hypergeometric functions. For these functions, we investigate various…
The indefinite integral $$ \int x^\alpha e^{\eta x^\beta}\,_pF_q (a_1, a_2, \cdot\cdot\cdot a_p; b_1, b_2, \cdot\cdot\cdot, b_q; \lambda x^{\gamma})dx, $$ where $\alpha, \eta, \beta, \lambda, \gamma\ne0$ are real or complex constants and…
In this paper, we study some extended hypergeometric functions from matrix point of view. We have given the integral representations of these matrix functions. Finally, we obtain some generating function relations using fractional…
Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a…
In this paper, we give explicit evaluation for some infinite series involving generalized (alternating) harmonic numbers. In addition, some formulas for generalized (alternating) harmonic numbers will also be derived.
We obtain new inequalities for certain hypergeometric functions. Using these inequalities, we deduce estimates for the hyperbolic metric and the induced distance function on a certain canonical hyperbolic plane domain.
We provide a set of diagonals of simple rational functions of three and four variables that are squares of Heun functions. These Heun functions obtained through creative telescoping, turn out to be either pullbacked $_2F_1$ hypergeometric…
The theoretical computing of special values assumed by the hypergeometric functions has a high interest not only on its own, but also in sight of the remarkable implications to both pure Mathematics and Mathematical Physics. Accordingly, in…
This paper is devoted for the study of a new generalization of Struve function type. In this paper , We establish four new integral formulas involving the Galue type Struve function, which are express in term of the generalized (Wright)…
The confluent Heun equation is one of 4 confluent forms of Heun's differential equation in which is the Fuchsian equation of second order with four regular singularities. A confluent Heun function is applicable to diverse areas such as…
We introduce two new types of Dehn functions of group presentations which seem more suitable (than the standard Dehn function) for infinite group presentations and prove the fundamental equivalence between the solvability of the word…
In this work we establish new forms of Heun-to-Heun transformations and Heun-to-Hypergeometric transformations. The transformations are realised by changing the independent variable in a non-linear way. Using these we also point out some…
We present a simple systematic algorithm for construction of expansions of the solutions of ordinary differential equations with rational coefficients in terms of mathematical functions having indefinite integral representation. The…
In calculus, an indefinite integral of a function $f$ is a differentiable function $F$ whose derivative is equal to $f$. In present paper, we generalize this notion of the indefinite integral from the ring of real functions to any ring. The…
We show that many integrals containing products of confluent hypergeometric functions follow directly from one single integral that has a very simple formula in terms of Appell's double series F_2. We present some techniques for computing…