Integral Representations of Ratios of the Gauss Hypergeometric Functions with Parameters Shifted by Integers

被引：1

作者：

Dyachenko, Alexander ^{[1
]}

Karp, Dmitrii ^{[2
,3
,4
]}

机构：

[1] Keldysh Inst Appl Math, Moscow 125047, Russia

[2] Holon Inst Technol, Dept Math, IL-5810201 Holon, Israel

[3] Far Eastern Fed Univ, Sch Econ & Management, Vladivostok 690922, Russia

[4] Far Eastern Fed Univ, Far Eastern Ctr Res & Educ Math, Vladivostok 690922, Russia

来源：

MATHEMATICS | 2022年 / 10卷 / 20期

关键词：

gauss hypergeometric function; gauss continued fraction; integral representation; JACOBI-POLYNOMIALS; 3-TERM RELATIONS; STIELTJES; ZEROS;

D O I：

10.3390/math10203903

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

Given real parameters a,b,c and integer shifts n1,n2,m, we consider the ratio R(z)=2F1(a+n1,b+n2;c+m;z)/2F1(a,b;c;z) of the Gauss hypergeometric functions. We find a formula for ImR(x +/- i0) with x>1 in terms of real hypergeometric polynomial P, beta density and the absolute value of the Gauss hypergeometric function. This allows us to construct explicit integral representations for R when the asymptotic behaviour at unity is mild and the denominator does not vanish. The results are illustrated with a large number of examples.

引用

页数：26

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