On elementary methods in positivity theory

We give a short proof of a result of Askey and Gasper [J. Analyse Math., 31 (1977), pp. 48–68] that $(1 - x - y - z + 4xyz)^{ - \beta } $ has positive power series coefficients for $\beta \geq {{(\sqrt {17} - 3)} / 2}$. We also show how Ismail and Tamhankar’s proof [SIAM J. Math. Anal., 10 (1979), pp. 478–485] that \[ (1 - (1 - \lambda )x - \lambda y - \lambda xz - (1 - \lambda )yz + xyz)^{ - \alpha } \quad (0 \leq \lambda \leq 1) \] has positive power series coefficients for $\alpha = 1$ implies Koornwinder’s result that it does so for $\alpha \geq 1$.