Simple continued fraction expansions of some values of certain hypergeometric functions. (Q1411744)

This paper studies regularities in the continued fraction expansions of certain hypergeometric series. More precisely, let \(F(a,b,c;z)\) denote the hypergeometric function defined by the power series \[ F(a,b,c;z)=1 +\frac{ab}{c}+\frac{a(a+1)b(b+1)}{c(c+1)} \frac{z ^2}{ 2!} + \frac{a(a+1)(a+2)b(b+1)(b+2)}{c(c+1)(c+2)} \frac{z ^3}{ 3!} +\cdots, \] where \(a,b,c\) are complex constants and \(c \) does not belong to the set of non-positive integers. The continued fraction expansion of certain values of this hypergeometric function (as well as certain values of the confluent hypergeometric functions) are shown in this paper to present some regularities (in the sense of Hurwitz continued fraction expansions) up to some order in the partial quotients.