On λ-linear functionals arising from p-adic integrals on Zₚ

The aim of this paper is to determine the λ -linear functionals sending any given polynomial $p(x)$ p ( x ) with coefficients in Cₚ C p to the p -adic invariant integral of $P(x)$ P ( x ) on Zₚ Z p and also to that of P(x₁+⋯ +xᵣ) P ( x 1 + ⋯ + x r ) on Zₚʳ 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).