Two-Dimensional Generalized Gamma Function and Its Applications

In this article, we present a new two-dimensional generalization of the gamma function based on the product of the one-dimensional generalized beta function and the one-dimensional generalized gamma function. As will become clear later, one of the properties of this generalization is a formula that generalizes the famous formula establishing the connection between the classical gamma and beta functions. Next, we present the properties of this generalization, some series for the generalized beta function and the generalized two-dimensional gamma function. As a practical application of the two-dimensional generalized gamma function, we will show how it can be used to represent a fairly wide class of double integrals in the form of functional series, or in the form of the product of two one-dimensional integrals, which makes it quite easy to find their values. That is, it will be seen how, with the help of simple transformations, many types of double integrals can be reduced to a generalized two-dimensional gamma function (23), which greatly simplifies work with them, thanks to formula (24). At the end of the article, a representation of the classical and generalized hypergeometric function in the form of a two-dimensional generalized gamma function is presented.