Integral Representations of Ratios of the Gauss Hypergeometric Functions with Parameters Shifted by Integers - Alexander Dyachenko; Dmitrii Karp (Mathematics, 2022)
Given real parameters <semantics>a,b,c</semantics> and integer shifts <semantics>n1,n2,m</semantics>, we consider the ratio <semantics>R(z)=2F1(a+n1,b+n2;c+m;z)/2F1(a,b;c;z)</semantics> of the Gauss hypergeometric functions. We find a formula for <semantics>ImR(x±i0)</semantics> with <semantics>x>1</semantics> 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.