Further Results on Multiple $q$-Eulerian Integrals for Various $Q$-Hypergeometric Functions

Thomas Ernst

We continue the study of single and multiple $q$-Eulerian integrals in the spirit of Exton, Driver, Johnston, Pandey, Saran and Erdélyi. The method of proof is often the $q$-beta integral method with the correct $q$-power together with the $q$-binomial theorem. By the Totov method we can prove summation theorems as special cases of multiple $q$-Eulerian integrals. The Srivastava $\triangle$ notation for $q$-hypergeometric functions is used to enable the shortest possible form of the long formulas. The various $q$-Eulerian integrals are in fact meromorphic continuations of the various multiple $q$-functions, suitable for numerical computations. In the end of the paper a generalization of the $q$-binomial theorem is used to find $q$-analogues of a multiple integral formulas for $q$-Kampé de Fériet functions.