On \u03bb-linear functionals arising from p-adic integrals on mathbb{Z}_{p}

Dae San Kim,Seongho Park,Jongkyum Kwon,Si-Hyeon Lee,Taekyun Kim

doi:10.1186/s13662-021-03634-z

Dae San Kim, Seongho Park + Show 3 more

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https://doi.org/10.1186/s13662-021-03634-z

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Abstract

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).

Highlights

It turns out that the one for Zp is given by the generating function of the degenerate Bernoulli numbers and the other for Zrp by that of the degenerate Bernoulli polynomials of order r
We are concerned with linear functionals on Cp[x] arising from p-adic invariant integrals on Zp and on Zrp
We determined the linear functional given by P(x) → Zp P(x) dμ0(x) and that given by P(x) → Zp · · · Zp P(x1 + · · · + xr) dμ0(x1) · · · dμ0(xr). This means that we have to determine those series corresponding to the two linear functionals arising from p-adic invariant integrals

Summary

Introduction

The impetus for studying λ-umbral calculus was the recent regained interest in degenerate special numbers and polynomials. 3, we determine the λ-linear functional associated with degenerate Bernoulli numbers. 4, we consider the λ-linear functional associated with higher-order degenerate Bernoulli numbers. Note that the degenerate Bernoulli polynomials βn,λ(x) here are different from those introduced by Carlitz [2]. Ak ∈ Cp k=0 be the algebra of all formal power series in t with coefficients in Cp. Let P = Cp[x] be the ring of all polynomials in x with coefficients in Cp, and let P∗ denote the vector space of all linear functionals on P (see [6]). (f ) This follows from (e) by noting that (x)n,λ ∼ (1, t)λ

The λ-linear functional associated with degenerate Bernoulli numbers

Conclusion