Blaschke’s “More down than up” conjecture for generalized hypergeometric series

Prove Blaschke’s conjecture asserting that for the generalized hypergeometric function pFq in which k numerator parameters and m denominator parameters are shifted by ±a, if k < m, then as |a| → ∞ the defining Taylor series in z is an asymptotic expansion for some values of z, with the asymptotic form given by the ratio of Pochhammer products for the shifted parameters.

Background

In discussing the challenges of deriving uniform estimates needed for the proposed "uniformity approach" to asymptotics, the authors point to Blaschke’s conjecture as a central unresolved principle that would underpin such estimates for generalized hypergeometric functions with large parameters.

Confirming this conjecture would provide a theoretical foundation for uniform asymptotic expansions across families of multiple hypergeometric functions and directly facilitate methods used in this paper for Humbert and Appell functions.