Recursion formulae for basic hypergeometric functions
It is shown that the basic hypergeometric functions \[ \begin{multlined} F_k(\omega):=\frac{\prod_{i=1}^r(a_i;q)_k}{(q;q)_k\prod_{j=1}^s(b_j;q)_k} \omega^k\left[(-1)^kq^{\binom{k}{2}}\right]^{s+1-r}\\ {}_{r+1}\phi_{s+1} \left(\left. a_1q^k,\ldots,a_rq^k,\alpha q^{k+1} b_1q^k,\ldots,b_sq^k,\alpha\beta q^{2k+2}\right|q;\omega q^{k(s+1-r)}\right)\end{multlined} \] satisfy a recurrence relation of the form \[ \sum_{i=0}^{\vartheta}\left[A_i(k)+\frac{1}{\omega}B_i(k)\right] F_{k+i}(\omega)=0,\quad\vartheta=\max(r+1,s+2), \] where \(A_i(k)\) and \(B_i(k)\) are rational functions of \(q^k\), \(B_0(k)\equiv 0\) and \(B_{\vartheta}(k)\equiv 0\). In the case that \(r=s+1\) and \(\omega=q\) this result can be refined to~: the basic hypergeometric functions \[ F_k(q):=\frac{\prod_{i=1}^{s+1}(a_i;q)_k}{(q;q)_k\prod_{j=1}^s(b_j;q)_k} q^k {}_{s+2}\phi_{s+1}( a_1q^k,\ldots,a_{s+1}q^k,\alpha q^{k+1} b_1q^k,\ldots,b_sq^k,\alpha\beta q^{2k+2}|q;q) \] satisfy a recurrence relation of order \(s+1\), \[ \sum_{i=0}^{s+1}C_i(k)F_{k+i}(q)=0 \] with rational coefficients in \(q^k\). This can be seen as a partial analogue of a result by J.~Wimp for the generalized hypergeometric functions \[ \begin{multlined} U_k(\omega):=\frac{\prod_{i=1}^r(a_i)_k}{(k+\gamma)_k\prod_{j=1}^s(b_j)_k} \omega^k\\ {}_{r+1}F_{s+1}(a_1+k,\dots,a_r+k,\alpha+k b_1+k,\dots,b_s+k,2k+\gamma+1|\omega)\end{multlined} \] derived in [\textit{J.~Wimp}, Math. Comput. 22, 363-373 (1968; Zbl 0186.10401].
scientific article; zbMATH DE number 1533138
scientific article