The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.

Various best-choice problems related to the planar homogeneous Poisson process in finite or semi-infinite rectangle are studied. The analysis is largely based on properties of the onedimensional box-area process associated with the sequence of records. We prove a series of distributional identities involving exponential and uniform random variables, and resolve the Petruccelli-Porosinski-Samuels paradox on coincidence of asymptotic values in certain discretetime optimal stopping problems. 1

The relation showing that the Gru¨nwald-Letnikov and generalised Cauchy derivatives are equal is presented. This establishes a bridge between two different formulations and simultaneously between the classic integer order derivatives and the fractional ones. Starting from the generalised Cauchy derivative formula, new relations are obtained, namely a regularised version that makes the concept of pseudo-function appear naturally without the need for a rejection of any infinite part. From the regularised derivative, new formulations are deduced and specialised first for the real functions and afterwards for functions with Laplace transforms obtaining the definitions proposed by Liouville. With these tools suitable definitions of fractional linear systems are obtained. r 2006 Elsevier B.V. All rights reserved.

Abstract—This paper develops and compares the maximum a posteriori (MAP) and minimum mean-square error (MMSE) estimators for spherically contoured multivariate Laplace random vectors in additive white Gaussian noise. The MMSE estimator is expressed in closed-form using the generalized incomplete gamma function. We also find a computationally efficient yet accurate approximation for the MMSE estimator. In addition, this paper develops an expression for the MSE for any estimator of spherically contoured multivariate Laplace random vectors in additive white Gaussian noise (AWGN), the development of which again depends on the generalized incomplete gamma function. The estimators are motivated and tested on the problem of wavelet-based image denoising. Index Terms—Denoising, estimation, Laplace distribution.

Two representations of the extended gamma functions Γ 2,0 0,2 [(b,x)] are proved. These representations are exploited to find a transformation relation between two Fox’s H-functions. These results are used to solve Fox’s H-function in terms of Meijer’s G-function for certain values of the parameters. A closed form representation of the kernel of the Bessel type integral transform is also proved. 1.

Abstract. This document describes the new gamma and beta functions in GAMS. 1.

This paper develops and compares the MAP and MMSE estimators for spherically-contoured multivariate Laplace random vectors in additive white Gaussian noise. The MMSE estimator is expressed in closed-form using the generalized incomplete gamma function. We also find a computationally efficient yet accurate approximation for the MMSE estimator. In addition, this paper develops an expression for the mean square error MSE for any estimator of spherically-contoured multivariate Laplace random vectors in AWGN, the development of which again depends on the generalized incomplete gamma function. The estimators are motivated and tested on the problem of wavelet-based image denoising.

Abstract—We present new Bayesian estimators for spherically-contoured Bessel K form (BKF) random vectors in additive white Gaussian noise (AWGN). The derivations are an extension of existing results for the scalar BKF and multivariate Laplace (MLAP) densities. MAP and MMSE estimators are derived. We show that the MMSE estimator can be written in exact form in terms of the generalized incomplete Gamma function. Computationally efficient approximations are given. We compare the proposed exact and approximate MMSE estimators with recent results using the BKF density, both in terms of the shrinkage rules and the associated mean-square error. Index Terms—Bayesian estimation, Bessel K form density, MAP estimator, MMSE estimator, wavelet denoising.

Abstract. An overview is presented on statistical techniques for the analytic evaluation of integrals for non-resonant, non-resonant depleted, non-resonant cut-off, non-resonant sccreened, and resonant thermonuclear reaction rates. The techniques are based on statistical distribution theory and the theory of Meijer’s G-function and Fox’s H-function. The implementation of Meijer’s G-function in Mathematica constituts an additional utility for analytic manipulations and numerical computation of thermonuclear reaction rate integrals. Recent results in the astrophysical literature related to the use of analytic thermonuclear reaction rates are incorporated. 1 1

Abstract. The Maxwell-Boltzmannian approach to nuclear reaction rate theory is extended to cover Tsallis statistics (Tsallis, 1988) and more general cases of distribution functions. An analytical study of respective thermonuclear functions is being conducted with the help of statistical techniques. The pathway model, recently introduced by Mathai (2005), is utilized for thermonuclear functions and closed-form representations are obtained in terms of H-functions and G-functions. Maxwell-Boltzmannian thermonuclear functions become particular cases of the extended thermonuclear functions. A brief review on the development of the theory of analytic representations of nuclear reaction rates is given. 1