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

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 hyper...

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Bibliographic Details
Published in:Mathematics (Basel) Vol. 10; no. 20; p. 3903
Main Authors: Dyachenko, Alexander, Karp, Dmitrii
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.10.2022
Subjects:
Asymptotic properties
gauss continued fraction
gauss hypergeometric function
Gaussian processes
Geometrical models
Hypergeometric functions
Integer programming
integral representation
Mathematical analysis
Mathematical research
Parameter estimation
Parameters
Polynomials
Random variables
Ratio and proportion
Ratios
Representations
Israel
ISSN:2227-7390, 2227-7390
Online Access:Get full text
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Summary: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.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math10203903