A New q-Hypergeometric Symbolic Calculus in the Spirit of Horn, Borngässer, Debiard and Gaveau

The purpose of this article is to introduce a new complete multiple q-hypergeometric symbolic calculus, which leads q-Euler integrals and very similar canonical system q-difference equations for functions. q-analogues recurrence formulas in Horns paper Borngässers thesis lead more exact way find these Frobenius solutions. To the right formulas, parameters q-shifted factorials can be changed negative integers, give no extra q-factors. In proving q-formulas, limit q→1 we obtain versions by Debiard Gaveau solution differential or equations. also correction some statements Gaveau, e.g., Euler other solutions Appell functions, without references page numbers standard work Kampé de Fériet. Sometimes q-binomial theorem used simplify q-integral formulas. By Horn method, another Φ1 function partial equation, was not mentioned Le Vavasseur 1893.