Recurrence relations for hypergeometric functions of unit argument

Abstract

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