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

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Dae San Kim ◽  
Taekyun Kim ◽  
Jongkyum Kwon ◽  
Si-Hyeon Lee ◽  
Seongho Park
Keyword(s):  
Generating Function ◽  
Bernoulli Polynomials ◽  
Linear Functionals ◽  
Invariant Integral ◽  
Umbral Algebra

AbstractThe 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).

scholarly journals A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 648
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran ◽  
Deena Al-Kadi
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Laguerre Polynomials ◽  
Hermite Polynomials ◽  
Bernoulli Polynomials ◽  
Higher Order ◽  
New Class

The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange–Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller–Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite–Miller–Lee polynomials.

scholarly journals IDENTITIES OF SYMMETRY FOR GENERALIZED TWISTED BERNOULLI POLYNOMIALS TWISTED BY RAMIFIED ROOTS OF UNITY

2014 ◽  
Vol 60 (1) ◽  
pp. 19-36
Author(s):  
Dae San Kim
Keyword(s):  
Generating Function ◽  
Bernoulli Polynomials ◽  
Roots Of Unity ◽  
Integral Expression ◽  
Power Sums ◽  
Exponential Generating Function

Abstract We derive eight identities of symmetry in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by ramified roots of unity. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the p-adic integral expression of the generating function for the generalized twisted Bernoulli polynomials and the quotient of p-adic integrals that can be expressed as the exponential generating function for the generalized twisted power sums.

scholarly journals Some Identities of Symmetry for the Generalized Bernoulli Numbers and Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Taekyun Kim ◽  
Seog-Hoon Rim ◽  
Byungje Lee
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Invariant Integral ◽  
Generalized Bernoulli Polynomials ◽  
Symmetric Properties

By the properties ofp-adic invariant integral onℤp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties ofp-adic invariant integral onℤp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

scholarly journals Obtaining Easily Sums of Powers on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus

2017 ◽  
Vol 9 (5) ◽  
pp. 73
Author(s):  
Do Tan Si
Keyword(s):  
Differential Operator ◽  
Generating Function ◽  
Arithmetic Progression ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Arithmetic Progressions ◽  
Operator Calculus ◽  
Sums Of Powers ◽  
The Way

We show that a sum of powers on an arithmetic progression is the transform of a monomial by a differential operator and that its generating function is simply related to that of the Bernoulli polynomials from which consequently it may be calculated. Besides, we show that it is obtainable also from the sums of powers of integers, i.e. from the Bernoulli numbers which in turn may be calculated by a simple algorithm.By the way, for didactic purpose, operator calculus is utilized for proving in a concise manner the main properties of the Bernoulli polynomials. 

scholarly journals Applications of q-Umbral Calculus to Modi…ed Apostol Type q-Bernoulli Polynomials

2017 ◽  
Author(s):  
Mehmet Acikgoz ◽  
Resul Ates ◽  
Ugur Duran ◽  
Serkan Araci
Keyword(s):  
Linear Combination ◽  
Generating Function ◽  
Bernoulli Polynomials ◽  
Umbral Calculus

This article aims to identify the generating function of modi…ed Apostol type q-Bernoulli polynomials. With the aid of this generating function, some properties of modi…ed Apostol type q-Bernoulli polynomials are given. It is shown that aforementioned polynomials are q-Appell. Hence, we make use of these polynomials to have applications on q-Umbral calculus. From those applications, we derive some theorems in order to get Apostol type modi…ed q-Bernoulli polynomials as a linear combination of some known polynomials which we stated in the paper.

scholarly journals Multifarious Results for q-Hermite Based Frobenius Type Eulerian Polynomials

2019 ◽  
Author(s):  
Waseem Khan ◽  
Idrees Ahmad Khan ◽  
Mehmet Acikgoz ◽  
Ugur Duran
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Series Representation ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Eulerian Polynomials ◽  
New Class ◽  
Genocchi Polynomials

In this paper, a new class of q-Hermite based Frobenius type Eulerian polynomials is introduced by means of generating function and series representation. Several fundamental formulas and recurrence relations for these polynomials are derived via different generating methods. Furthermore, diverse correlations including the q-Apostol-Bernoulli polynomials, the q-Apostol-Euler poynoomials, the q-Apostol-Genocchi polynomials and the q-Stirling numbers of the second kind are also established by means of the their generating functions.

scholarly journals Some Hermite base polynomials on q-umbral algebra

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 961-967 ◽  
Author(s):  
Rahime Dere
Keyword(s):  
Bernoulli Polynomials ◽  
Umbral Calculus ◽  
New Type ◽  
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.

scholarly journals Relations for Bernoulli–Barnes numbers and Barnes zeta functions

2014 ◽  
Vol 10 (05) ◽  
pp. 1321-1335 ◽  
Author(s):  
Abdelmejid Bayad ◽  
Matthias Beck
Keyword(s):  
Generating Function ◽  
Zeta Functions ◽  
Bernoulli Polynomials

The Barnes ζ-function is [Formula: see text] defined for [Formula: see text], Re (x) > 0, and Re (z) > n and continued meromorphically to ℂ. Specialized at negative integers -k, the Barnes ζ-function gives [Formula: see text] where Bk(x; a) is a Bernoulli–Barnes polynomial, which can be also defined through a generating function that has a slightly more general form than that for Bernoulli polynomials. Specializing Bk(0; a) gives the Bernoulli–Barnes numbers. We exhibit relations among Barnes ζ-functions, Bernoulli–Barnes numbers and polynomials, which generalize various identities of Agoh, Apostol, Dilcher, and Euler.

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