Ratios of the Gauss hypergeometric functions with parameters shifted by integers as Stieltjes and Nevanlinna N-kappa functions

Using a contiguous relation Gauss derived a continued fraction for the ratio of hypergeometric functions with two parameters shifted by unity. Under certain restrictions the elements of this continued fraction are all positive, so that it represents a function from the Stieltjes class. All such functions possess an integral representation as the Stieltjes transform of a positive measure. Surprisingly enough, an explicit form of this representing measure for the Gauss continued fraction was only computed in 1984 by Vitold Belevitch. In this talk we will discuss more general ratios of hypergeometric functions, namely, with parameters of the numerator and denominator hypergeometric functions shifted by arbitrary integers. We present an explicit integral representation for such ratios under the restriction of analyticity in the complex plane cut along the ray from 1 to infinity and assuming certain asymptotic behavior near the singular points 1 and infinity. Furthermore, we show that for arbitrary real values parameters these ratios belong to some Nevanlinna N-kappa class. We give an estimate of kappa (the number of negative squares of the appropriate kernel) in some cases. We illustrate our results with numerous examples with specific values of the parameter shifts.