Matrix Algebraic Infinite Product Representation for Generalized Hypergeometric Functions of Type p+1Fp

Metin Demi̇ralp, Sevda Üsküplü
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引用次数: 3

Abstract

We present a novel representation for generalized hypergeometric functions of type p+1Fp which is in fact defined by an infinite series in nonnegative integer powers of its argument. We first construct a first order vector differential equation such that the unknown vector's coefficient is the sum of a constant matrix and a matrix premultiplied by the reciprocal of the independent variable whereas its first order derivative has unit matrix coefficient. An infinite process of factor extractions and power annihilations is employed yielding finally a vector differential equation that can be easily and analytically solved. Truncation of this scheme can be used to get approximations to hypergeometric functions of type p+1Fp. These functions have regular singularities at 0 and 1 values of the independent variable together with another regular singularity at infinity. Hence the factors are chosen to reflect the expected behavior of the functions at the singular point in a descending contribution order. Factorization is realized also for regular points. A simple, yet meaningful, implementation seems to give quite promising results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

广义超几何函数p+1Fp的矩阵代数无穷积表示
本文给出了p+1Fp型广义超几何函数的一种新表示,该函数实际上是由其参数的非负整数幂无穷级数定义的。我们首先构造一个一阶向量微分方程,使得未知向量的系数是一个常数矩阵和一个预乘自变量倒数的矩阵的和,而它的一阶导数具有单位矩阵系数。通过无穷次的因子提取和幂元湮没,最终得到一个易于解析求解的矢量微分方程。该格式的截断可用于得到p+1Fp型超几何函数的近似。这些函数在自变量的0和1处有规则奇点,在无穷远处有另一个规则奇点。因此,选取的因子反映了函数在奇点处的期望行为,其贡献顺序由高到低。对正则点也实现了因式分解。一个简单而有意义的实现似乎会产生相当有希望的结果。(©2005 WILEY-VCH Verlag GmbH &KGaA公司,Weinheim)
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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