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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Vydáno v:Mathematics (Basel) Ročník 10; číslo 20; s. 3903
Hlavní autoři: Dyachenko, Alexander, Karp, Dmitrii
Médium: Journal Article
Jazyk:angličtina
Vydáno: Basel MDPI AG 01.10.2022
Témata:
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
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ISSN:2227-7390, 2227-7390
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Shrnutí: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.
Bibliografie:ObjectType-Article-1
SourceType-Scholarly Journals-1
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ISSN:2227-7390
2227-7390
DOI:10.3390/math10203903