Some Generalizations and Basic (or q-) Extensions of the Bernoulli, Euler and Genocchi Polynomials

In the vast literature in Analytic Number Theory, one can find systematic and extensive investigations not only of the classical Bernoulli, Euler and Genocchi polynomials and their corresponding numbers, but also of their many generalizations and basic (or q-) extensions. Our main object in this presentation is to introduce and investigate some of the principal generalizations and unifications of each of these polynomials by means of suitable generating functions. We present several interesting properties of these general polynomial systems including some explicit series representations in terms of the Hurwitz (or generalized) zeta function and the familiar Gauss hypergeometric function. By introducing an analogue of the Stirling numbers of the second kind, that is, the so-called λ-Stirling numbers of the second kind, we derive several properties and formulas and consider some of their interesting applications to the family of the Apostol type polynomials. We also give a brief expository and historial account of the various basic (or q-) extensions of the classical Bernoulli polynomials and numbers, the classical Euler polynomials and numbers, the classical Genocchi polynomials and numbers, and also of their such generalizations as (for example) the above-mentioned families of the Apostol type polynomials and numbers. Relevant connections of the definitions and results presented here with those in earlier as well as forthcoming investigations will be indicated.