Reflection Symmetries of q-Bernoulli Polynomials

By B. Kupershmidt. 2005

89 citations recorded by the source corpus.

Abstract A large part of the theory of classical Bernoulli polynomials B n (x)’s follows from their reflection symmetry around x = 1/2: B n (1 − x) = (−1) n B n (x). This symmetry not only survives quantization but has two equivalent forms, classical and quantum, depending upon whether one reflects around 1/2 the classical x or quantum [x] q .

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Spectroscopy

View Paper: https://www.semanticscholar.org/paper/db8e1d7f3af9dea8df91646e28706b01438dd76b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Spectroscopy

Taekyun Kim

Citations in the corpus

Symmetry p-adic invariant integral on ℤ p for Bernoulli and Euler polynomials (2007, background): “In the recent the p-adic invariant integral on Zp are studied by several researchers in the area of number theory and mathematical physics (see [1-25]).”
Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on ℤp (2009, unknown): “[1–22]).”
An identity of the symmetry for the Frobenius-Euler polynomials associated with the Fermionic $p$-adic invariant $q$-integrals on ${\bf Z}_p$ (2008, background): “When one talks of q-extension, q is variously considered as an indeterminate, a complex q ∈ C, or a p-adic number q ∈ Cp, see [9-22].”
q-Generalized Euler numbers and polynomials (2006, background): “In a recent paper (see [ 12 ]), B. A. Kupershmidt constructed reflection symmetries of q-Bernoulli polynomials which differ from Carlitz’ q-Bernoulli numbers.”
Euler Numbers and Polynomials Associated with Zeta Functions (2008, unknown): “From this zeta function, he derived the following formula, cf.[1-33]: ζ(−n) = − Bn + 1 n + 1 , n ∈ N = {1, 2, 3, · · · }.”
A note on q-Bernstein polynomials (2010, background): “In the last decade some new generalizations of well known positive linear operators based on q-integers were introduced and studied by several authors(see [1-21]).”
On the weighted q-Bernoulli numbers and polynomials (2010, methodology): “In this paper we use the following notation: [x]q = 1− q 1− q , (see [1-16]).”
Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials (2011, unknown): “Zp qedμ−1(x) = 2 qet + 1 , ( see [18, 19, 4, 21]).”