Families of Two-Point Padé Approximants and Some ${}_4 F_3 (1)$ Identities

Abstract

In this paper, a family of two-point Pade approximants, that is, two polynomials, each of degree n and depending on integer k, l,$0 \leq l \leq k \leq n$, whose ratio approximates one function to order $(z^{n + l + 1} )$ at $z = 0$ and another to order $O(z^{ - n + k} )$ at $z = \infty $ is presented. The functions in question are ratios of Gaussian hypergeometric functions. Explicit closed-form expressions for the polynomials are given. Also, this derivation establishes some hypergeometric identities involving functions of the form ${}_4 F_3 (1)$. Several interesting limiting cases, namely, $[n,n]$ and $[n - 1,n]$ Pade approximants for ratios of confluent hypergeometric functions and Bessel functions are given.