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 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.