Related papers: On incomplete Gamma and Beta integrals
If the prime numbers are pseudo-randomly distributed, then analogy with quantum systems suggests that counting primes might be modeled by a non-homogeneous Poisson process. Consequently, postulating underlying gamma statistics, more-or-less…
Lower and upper bounds $B_a(x)$ on the incomplete gamma function $\Gamma(a,x)$ are given for all real $a$ and all real $x>0$. These bounds $B_a(x)$ are exact in the sense that $B_a(x)\underset{x\downarrow0}\sim\Gamma(a,x)$ and…
We generalize the well-known mean value inequality of subharmonic functions for a slightly more general function class. We also apply this generalized mean value inequality to weighted boundary behavior and nonintegrability questions of…
In this paper, we present some new inequalities for the gamma function. The main tools are the multiple-correction method developed in our previous works, and a generalized Mortici's lemma.
In this paper, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and develop some transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have…
Some remarkable properties of the beta distribution are based on relations involving independence between beta random variables such that a parameter of one among them is the sum of the parameters of an other (see (1.1) et (1.2) below).…
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…
An integral over the interval $(0,\pi)$ is given for the cumulative distribution function of a sum of independent gamma random variables with different scale and shape parameters. The cumulative distribution function of a positive definite…
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…
Integral means are important class of bivariate means. In this paper we prove the very general algorithm for calculation of coefficients in asymptotic expansion of integral mean. It is based on explicit solving the equation of the form…
In this very short note we will derive an inequality for a class of entire functions including all the confluent basic hypergeometric series and an inequality for a class of meromorphic functions including theta functions.
In this work, generalized hypergeometric functions for bicomplex argument is introduced and its convergence criteria is derived. Furthermore, integral representation of this function has been established. Moreover, quadratic transformation,…
Partition function of beta-gamma systems on the orbifolds C^2/Z_N and C^3/Z_M x Z_N are obtained as the invariant part of that on the respective affine spaces, by lifting the geometric action of the orbifold group to the fields.…
In this paper, we find new integral representations for the generalized Hermite linear functional in the real line and the complex plane. As an application, new integral representations for the Euler Gamma function are given.
The gamma distribution is a useful model for small area prediction of a skewed response variable. We study the use of the gamma distribution for small area prediction. We emphasize a model, called the gamma-gamma model, in which the area…
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.
A new uniform asymptotic expansion for the incomplete gamma function $\Gamma(a,z)$ valid for large values of $z$ was given by the author in {\it J. Comput. Appl. Math.} {\bf 148} (2002) 323--339. This expansion contains a complementary…
Motivated by the substantial development of the special functions, we contribute to establish some rigorous results on the general series identities with bounded sequences and hypergeometric functions with different arguments, which are…
In this paper, additional properties of the lower gamma functions and the error functions are introduced and proven. In particular, we prove interesting relations between the error functions and Laplace transform.
The main object of this paper is to present a new generalized beta function which defined by three parametres Mittag-Leffler function. We also introduce new generalizations of hypergeometric and confluent hypergeometric functions with the…