\(G\)-continued fractions for basic hypergeometric functions (Q1974231)

Tail values \(f^{(n)}\) of a convergent continued fraction \(K(a_n/b_n)\) satisfy \[ f^{(n-1)}=a_n/(b_n+f^{(n)})\qquad\text{for}\quad n=1,2,3,\dots. \] Hence, if \(f^{(n)}\) can be written \(X_n/X_{n+1}\) for all \(n\) for some sequence \(\{X_n\}\), then \(\{X_n\}\) is a solution of the three term recurrence relation \[ X_n+b_nX_{n-1}-a_nX_{n-2}=0\qquad\text{for}\quad n=1,2,3,\dots. \] G-continued fractions have the same kind of relationship to \(N\) term recurrence relations for an \(N>3\). In 1870 Thomae showed that certain basic hypergeometric functions are solutions to the (m+1) term recurrence relation \[ (1-\alpha_0 z q^n)X_n+(\beta_1-\alpha_1 z q^n)X_{n+1}+\dots +(\beta_m-\alpha_m z q^n)X_{n+m}=0.\tag{1} \] By means of an analogue to Pincherles theorem, found by Zahar in 1968, the authors show that the G-continued fraction corresponding to (1) converges to the ratio \(X_0/X_1\) of the proper Thomae solution \(\{X_n\}\).