On lambda-linear functionals arising from p-adic integrals on Z(p)

Abstract

The aim of this paper is to determine the.-linear functionals sending any given polynomial p(x) with coefficients in C-p to the p-adic invariant integral of P(x) on Z(p) and also to that of P(x(1) + center dot center dot center dot + x(r)) on Z(p)(r). We show that the former is given by the generating function of degenerate Bernoulli polynomials and the latter by that of degenerate Bernoulli polynomials of order r. For this purpose, we use the lambda-umbral algebra which has been recently introduced by Kim and Kim ( J. Math. Anal. Appl. 493(1):124521 2021).