Ratios of the Gauss hypergeometric functions with parameters shifted by integers: part I

Abstract

Given real parameters a, b, c and integer shifts n 1 , n 2 , m, we consider the ratio R(z) = 2 F 1 (a + n 1 , b + n 2 ; c + m; z)/ 2 F 1 (a, b; c; z) of the Gauss hypergeometric functions. We find a formula for Im R(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. Moreover, for arbitrary a, b, c and ω ≤ 1 the product P (z + ω)R(z + ω) is proved to belong to the generalized Nevanlinna class N λ κ . We give an in-depth analysis of the case n 1 = 0, n 2 = m = 1 known as the Gauss ratio. Furthermore, we establish a few general facts relating generalized Nevanlinna classes to Jacobi and Stieltjes continued fractions, as well as to factorization formulae for these classes. The results are illustrated with a large number of examples.