On λ-linear functionals arising from p-adic integrals on Z p $\mathbb{Z}_{p}$

The aim of this paper is to determine the λ-linear functionals sending any given polynomial $p(x)$ with coefficients in $\mathbb{C}_{p}$ to the p-adic invariant integral of $P(x)$ on $\mathbb{Z}_{p}$ and also to that of $P(x_{1}+\cdots +x_{r})$ on $\mathbb{Z}_{p}^{r}$ . 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 λ-umbral algebra which has been recently introduced by Kim and Kim (J. Math. Anal. Appl. 493(1):124521 2021).