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

The aim of this paper is to determine the λ-linear functionals sending any given polynomial $p(x)$ p ( x ) with coefficients in $\mathbb{C}_{p}$ C p to the p-adic invariant integral of $P(x)$ P ( x ) on $\mathbb{Z}_{p}$ Z p and also to that of $P(x_{1}+\cdots +x_{r})$ P ( x 1 + ⋯ + x r ) on $\mathbb{Z}_{p}^{r}$ 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).

Some Hermite base polynomials on q-umbral algebra

The aim of this paper is to investigate the q-Hermite type polynomials by using umbral calculus methods. Using this method, we derive new type polynomials which are related to the q-Bernoulli polynomials and the q-Hermite type polynomials. Furthermore, we also derive some new identities of those polynomials which are derived from q-umbral calculus.

Applications of Umbral Calculus Associated withp-Adic Invariant Integrals onZp

Recently, Dere and Simsek (2012) have studied the applications of umbral algebra to some special functions. In this paper, we investigate some properties of umbral calculus associated withp-adic invariant integrals onZp. From our properties, we can also derive some interesting identities of Bernoulli polynomials.