Power series and asymptotic series associated with the \(q\)-analog of the two-variable \(p\)-adic \(L\)-function. (English) Zbl 1190.11049

Summary: We construct a two-variable \(p\)-adic \(q-L\)-function which interpolates the generalized \(q\)-Bernoulli polynomials associated with a primitive Dirichlet character \(c\). Indeed, this function is the \(q\)-extension of the two-variable \(p\)-adic \(L\)-function, due to Fox, corresponding to the case \(q=1\). Finally, we give a \(p\)-adic integral representation for this two-variable \(p\)-adic \(q-L\)-function and derive a \(q\)-extension of the generalized Diamond-Ferro-Greenberg formula for the two-variable \(p\)-adic \(L\)-function in terms of the \(p\)-adic gamma and log gamma functions.


11M41 Other Dirichlet series and zeta functions
05A30 \(q\)-calculus and related topics
11B68 Bernoulli and Euler numbers and polynomials
11M35 Hurwitz and Lerch zeta functions
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)