Related papers: On incomplete Gamma and Beta integrals
In this article, we define a special function called the Bigamma function. It provides a generalization of Euler's gamma function. Several algebraic properties of this new function are studied. In particular, results linking this new…
The main aim of this note, which can be viewed as a certain addendum to the paper \cite{2019}, is to propose several generalized inequalities for the ratio functions of trigonometric and hyperbolic functions. We basically follow the…
Two exact evaluation formulae for multiple rarefied elliptic beta integrals related to the simplest lens space are proved. They generalize evaluations of the type I and II elliptic beta integrals attached to the root system $C_n$. In a…
The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, $ \int x^\alpha e^{\eta x^\beta}dx, \int x^\alpha \cosh\left(\eta x^\beta\right)dx, \int x^\alpha \sinh\left(\eta x^\beta\right)dx, \int…
In this paper we evaluate integrals of products of gamma functions of Ramanujan type in terms of bilateral hypergeometric series. In cases where the bilateral hypergeometric series are summable, then we evaluate these integral as beta…
The object of this paper is to investigate the certain results involving Bateman's matrix polynomials for integral index. We obtain some properties, integral representation and recurrence relations for hypergeometric matrix function. We…
Recently, there emerges different versions of beta function and hypergeometric functions containing extra parameters. Gaining enlightenment from these ideas, we will first introduce a new extension of generalized hypergeometric function and…
We give a brief account of the key properties of elliptic hypergeometric integrals -- a relatively recently discovered top class of transcendental special functions of hypergeometric type. In particular, we describe an elliptic…
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…
The beta distribution is a two-parameter family of probability distributions whose distribution function is the (regularised) incomplete beta function. In this paper, the inverse incomplete beta function is studied analytically as…
In this expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some inequalities, the complete monotonicity of several functions involving ratios of two gamma or $q$-gamma…
This paper discusses certain properties of heterogeneous hypergeometric functions with two matrix arguments. These functions are newly defined but have already appeared in statistical literature and are useful when dealing with the…
Hypergeometric functions and their generalizations play an important r\^{o}les in diverse applications. Many authors have been established generalizations of hypergeometric functions by a number ways. In this paper, we aim at establishing…
We prove a novel type of inversion formula for elliptic hypergeometric integrals associated to a pair of root systems. Using the (A,C) inversion formula to invert one of the known C-type elliptic beta integrals, we obtain a new elliptic…
The primary goal of this paper is to introduce and investigate generalized incomplete exponential functions with matrix parameters. Integral representation, differential formula, addition formula, multiplication formula, and recurrence…
This paper deals with the evaluation of some definite Euler-type integrals in terms of the Wright hypergeometric function. We obtain a theorem on the Wright hypergeometric function and then use this theorem to evaluate some definite…
This paper studies the log-convexity of the extended beta functions. As a consequence, Tur\'an-type inequalities are established.The monotonicity, log-convexity, log-concavity of extended hypergeometric functions are deduced by using the…
The hypergeometric distribution is a popular distribution, whose properties have been extensively investigated. Generating functions of this distribution, such as the probability-generating function, the moment-generating function, and the…
The gamma difference distribution is defined as the difference of two gamma distributions, with in general different shape and rate parameters. Starting with knowledge of the corresponding characteristic function, a second order linear…
The recurrence matrix relations, differentiation formulas, and analytical and fractional integral properties of incomplete gamma matrix functions $\gamma(Q, x)$ and $\Gamma(Q, x)$ are all covered in this article. The generalized incomplete…