Recurrence Relations for Hypergeometric Functions of Unit Argument

Abstract

We show that the generalized hypergeometric function \[ {P_n}:{ = _{p + 3}}{F_{p + 2}}\left ( {\left . {\begin {array}{*{20}{c}} { - n,n + \lambda ,{a_p},1} \\ {{b_{p + 2}}} \\ \end {array} } \right |1} \right )\quad (n \geqslant 0)\] satisfies a nonhomogeneous recurrence relation of order $p + \sigma$, where $\sigma = 0$ when $_{p + 3}{F_{p + 2}}(1)$ is balanced, and $\sigma = 1$ otherwise. Also, for \[ {U_n}: = \frac {{{{({c_{q + 1}})}_n}}}{{{{({d_q})}_n}{{(n + \lambda )}_n}}}{ _{q + 2}}{F_{q + 1}}\left ( {\left . {\begin {array}{*{20}{c}} {n + {c_{q + 2}}} \\ {n + {d_q},2n + \lambda + 1} \\ \end {array} } \right |1} \right )\quad (n \geqslant 0)\] a homogeneous recurrence relation of order $q + 1$ is given.