Journal of the
Korean Mathematical Society JKMS

Several addition formulas for a general class of $q$-Appell sequences are proved. The $q$-addition formulas, which are derived, involved not only the generalized $q$-Bernoulli, the generalized $q$-Euler and the generalized $q$-Genocchi polynomials, but also the $q$-Stirling numbers of the second kind and several general families of hypergeometric polynomials. Some $q$-umbral calculus generalizations of the addition formulas are also investigated.

Keywords: $q$-addition theorem, $q$-Appell polynomials, $q$-Bernoulli, $q$-Euler and $q$-Genocchi polynomials, generating functions, $q$-orthogonal polynomials

MSC numbers: 33D15, 33D45, 11B68, 11B73, 11B83