# Generalization of Euler’ Integral of the First Kind

## DOI:

https://doi.org/10.20535/1810-0546.2016.4.77167## Keywords:

Generalization of Euler’ integral of the I-kind, r-generalized beta function, Hypergeometric function, Macdonald’ function, Whittaker’ function## Abstract

**Background.** The new generalization of Euler’ integral of the I-kind (beta-functions) is considered, its main properties are investigated. Such distributions have a special place among the special functions due to their widespread use in many areas of applied mathematics.

**Objective.** The aim of the paper is to study the generalization of the new r-generalized beta-function and its application to the calculation of the new integrals.

**Methods.** To obtain results the general methods of the theory of special functions have been used.

**Results.** The article deals with new generalization of Euler’ integral of the I-kind. For the corresponding r-generalized beta functions were obtained important functional relations and differentiation formulas. For a wide application in the theory of integral and differential equations are important theorems on the connection of new beta functions with classical hypergeometric functions, Macdonald’ and Whittaker’ functions.

**Conclusions.**Considered in the article new generalization of Euler’ integral of the I-kind opens up opportunities for the use of Euler’ integrals in the theory of special functions, in the application of mathematical and physical problems. In the future we plan to use r-generalized beta functions to solve the new problems of the theory of probability, mathematical statistics, the theory of integral equations, etc.

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