On Quadruple q-Hypergeometric Functions and Diverse Generalizations to n Variables in the Spirit of Exton

The purpose of this article is to study convergence regions and q-integral representations of certain non-symmetric q-Lauricella functions and quadruple functions in the spirit of Exton. In the process, we slightly improve Exton’s original formulas, notation, and convergence regions. There are three so-called q-real numbers, which are briefly introduced. These numbers occur both in the q-integrals and in the convergence regions.When q-integral expressions for the symmetric (n) A and (n) D are used in the proofs, in the first case, third q-real numbers occur in the q-integrals. When (n) D is used in the proofs, the formulas are simpler, because the latter function has greater convergence region. Similarly, multiple q-Horn functions are briefly discussed.