相关论文: Analytical valuation of some non-elementary integr…
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…
The non-elementary integrals $\text{Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\beta\ge1,\alpha\le\beta+1$ and $\text{Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx, \beta\ge1,…
A formula for the non-elementary integral $\int e^{\lambda x^\alpha} dx$ where $\alpha$ is real and greater or equal two, is obtained in terms of the confluent hypergeometric function $_1F_1$. This result is verified by directly evaluating…
The non-elementary integrals $\mbox{Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\beta\ge1,\alpha>\beta+1$ and $\mbox{Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx, \beta\ge1,…
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…
We give a brief review of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's…
In numerous instances, the generalized exponential distribution can be used as an alternative to the most widely used non-regular family of distributions: Weibull, gamma, lognormal with three-parameters when analyzing lifetime or any skewed…
New asymptotic approximations of the non-central $t$ distribution are given, a generalization of the Student's $t$ distribution. Using new integral representations, we give new asymptotic expansions for large values of the noncentrality…
Let $\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ be the class of normalized analytic functions $f$ defined in the region $|z|<1$ and satisfying \begin{align*} {\rm Re\,}…
Hyperbolic hypergeometric integrals are defined as Barnes-type integrals of products of hyperbolic gamma functions. Their reduction to ordinary hypergeometric functions is well known. We study in detail their degeneration to complex…
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…
Two-term asymptotic formulae for the probability distribution functions for the smallest eigenvalue of the Jacobi $ \beta $-Ensembles are derived for matrices of large size in the r\'egime where $ \beta > 0 $ is arbitrary and one of the…
Motivated mainly by certain interesting recent extensions of the Gamma, Beta and hypergeometric functions, we introduce here new extensions of the Beta function, hypergeometric and confluent hypergeometric functions. We systematically…
We present a general non-parametric statistical inference theory for integrals of quantiles without assuming any specific sampling design or dependence structure. Technical considerations are accompanied by examples and discussions,…
In this paper, we introduce a family of robust estimates for the parametric and nonparametric components under a generalized partially linear model, where the data are modeled by $y_i|(\mathbf{x}_i,t_i)\sim F(\cdot,\mu_i)$ with…
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…
In order to better fit real-world datasets, studying asymmetric distribution is of great interest. In this work, we derive several mathematical properties of a general class of asymmetric distributions with positive support which shows up…
We introduce iterated beta integrals, a new class of iterated integrals on the universal abelian covering of the punctured projective line that unifies hyperlogarithms and classical beta integrals while preserving their fundamental…
Equilibrium statistical physics is considered from the point of view of statistical estimation theory. This involves the notions of statistical model, of estimators, and of exponential family. A useful property of the latter is the…
In the present context, superintegrability is a property of certain probability density functions coming from matrix models, which relates to the average over a distinguished basis of symmetric functions, typically the Jack or Macdonald…