On the expansion of the Kummer function in terms of incomplete Gamma functions

Asymptotic and exact series representations for the incomplete Gamma function

Using a variational approach, two new series representations for the incomplete Gamma function are derived: the first is an asymptotic series, which contains and improves over the standard asymptotic expansion; the second is a uniformly convergent series, completely analytical, which can be used to obtain arbitrarily accurate estimates of $Γ(a,x)$ for any value of $a$ or $x$. Applications of these formulas are discussed.

Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions

Starting from equations obeyed by functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four- and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples o…

Series solutions of confluent Heun equations in terms of incomplete Gamma-functions

We present a simple systematic algorithm for construction of expansions of the solutions of ordinary differential equations with rational coefficients in terms of mathematical functions having indefinite integral representation. The approach employs an auxiliary equation involving only the derivatives of a solution of the equation under consideration. Using power-series expansions of the solutions of this auxiliary equation, we construct several expansions of the four confluent Heun equations' solutions in terms of the incomplete Gamma-functions. In the cases of single- and double-confluent He…

三重球贝塞尔函数积分与指数和高斯阻尼：迈向解析的N点相关函数协方差模型

Triple-Spherical Bessel Function Integrals with Exponential and Gaussian Damping: Towards an Analytic N-Point Correlation Function Covariance Model

本文推广了递归方法，并提出了非递归方法，以解析计算带指数或高斯阻尼的三重球贝塞尔函数积分，结果用勒让德函数、伽马函数和超几何函数表示，为N点相关函数协方差建模提供工具。

Evidence of a Gamma Distribution for Prime Powers

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 standard heuristic arguments borrowed from quantum mechanics in the context of functional integration allows to derive analytic expressions of several average counting functions associated with prime numbers. The expressions are certain sums of incomplete gamma functions that are closely related to logarithmic-type integral functions --- which in turn are w…

Self-avoiding walk on the complete graph

There is an extensive literature concerning self-avoiding walk on infinite graphs, but the subject is relatively undeveloped on finite graphs. The purpose of this paper is to elucidate the phase transition for self-avoiding walk on the simplest finite graph: the complete graph. We make the elementary observation that the susceptibility of the self-avoiding walk on the complete graph is given exactly in terms of the incomplete gamma function. The known asymptotic behaviour of the incomplete gamma function then yields a complete description of the finite-size scaling of the self-avoiding walk on…