A Criterion for Finite Hypergeometric Reductions of Lauricella's $F_C$ with Arguments Related by Cube Roots of Unity

Abstract

We consider the germ at the origin of Lauricella's function $F_C^{(3)}(a,b;c_0,c_1,c_2;\kappa ,\omega \kappa ,{\omega }^2\kappa )$, where $\omega =\exp (2\pi \sqrt{-1}/3)$. We assume that $a,b,c_0,c_1,c_2$ avoid nonpositive integers and that $c_0,c_1,c_2$ also avoid half-integers. We consider finite linear combinations of ordinary generalized hypergeometric series with arguments $\alpha {\kappa }^3$, where $\alpha$ is a nonzero complex constant that may vary between summands, each multiplied by a nonnegative integer power of $\kappa$. Every target series has lower parameters outside the nonpositive integers and converges near the origin or terminates. We prove that such a representation exists if and only if, for each $j=0,1,2$, either $c_j-c_0$ or $c_j+c_0$ is an integer, and construct finite sums of ordinary ${}_9{F}_8$ functions when this condition holds. We prove necessity from irreducible differential equations of order $24$ for the generating functions of the three coefficient subsequences and from their formal solutions at infinity.

arXiv comment19 pages