Bivariate Basic Hypergeometric Series

By combining series rearrangement, basic hypergeometric series transformations, suchas Sears transformation formulae, this dissertation investigates double basic hypergeometricseries identities. The contents are as follows:1. By applying Sears transformation formulae and series rearrangement technique, we es-tablish several general transformation theorems related to q-Clausen seriesΦ_1:1;μ^0:3;λ andΦ_0:2;μ^1:2;λ. Moreover, we show many transformation, reduction and summation formulaeonΦ₀₂₂¹²²,Φ₀₂₃¹²³,Φ₁₁₁⁰³³ andΦ₁₁₂⁰³⁴ as special cases. These identities not only provide a fullcoverage of the q-analogues of the double classical hypergeometric series results, but alsoinclude several new formulae.2. As continuation of the work in the first chapter, we apply the Sears transformationsagain to deal with general transformations on other two tvDe q-Clausen seriesΦ_2:0;μ^2:1:λ andΦ_1:1;μ^1:2:λ. Furthermore, several double series identities forΦ₂₀₁²¹²,Φ₂₀₂²¹³,Φ₂₀₃²¹⁴,Φ₁₁₁¹²²,Φ₁₁₂¹²³ andΦ₁₁₃¹²⁴ are derived through appropriate parameter specialization. The variety of the resultsobtained show that the Sears transformations are indeed powerful for studying q-Clausenseries.3. We present several classes of double series identities by using summation theorems andtransformation formulae. And we also show how these general results would apply toyield some reduction and summation formulae for q-Kampéde Fériet functions. Someof these q-series identities may be considered as q-analogues of the results that appearedin [20, Eq. 13] and [21, Eqs. 2.1, 2.10, 3.8].