Bibasic Summation, Transformation and Expansion Formulas, q-Analogues of Clausen’s Formula, and Nonnegative Basic Hypergeometric Series

Abstract

Recent work on bibasic summation, transformation and expansion formulas and on inequalities and quadratic, cubic and quartic summation and transformation formulas for basic hy-pergeometric series is discussed. In particular, the Gasper and Rahman q-analogues of Clausen’s formula and of the Ramanujan and Bailey extension of Clausen’s formula are considered and a q-analogue of Clausen’s formula is employed to prove the nonnegativity of certain basic hypergeometric series and to derive q-analogues of the inequalities used by de Branges in his proof of the Bieberbach conjecture.