On multivariate \(p\)-adic \(q\)-integrals (Q2766261)

The authors define the \(p\)-adic \(q\)-integral (\(q\in \mathbb C_p\), \(|q-1|_p<p^{-1/(p-1)}\)) using \(q\)-analogs of the Riemann sums; a similar construction in the usual \(p\)-adic analysis is called the Volkenborn integral [see \textit{W. Schikhof}, Ultrametric calculus, Cambridge University Press (1984; Zbl 0553.26006)]. It is proved that the \(q\)-integral of a uniformly differentiable function exists and defines a locally analytic function of \(q\). NEWLINENEWLINENEWLINEThe \(p\)-adic \(q\)-integral is then used to define the \(p\)-adic \(q\)-\(L\)-functions of several variables and the \(q\)-extension of the \(p\)-adic multiple log-gamma function. Expressions for the values of the \(p\)-adic \(q\)-\(L\)-functions at positive integers are found.