Duals of Bernoulli Numbers and Polynomials and Euler Number and Polynomials

TL;DR

This paper introduces a unified framework using pseudo-involution of Riordan arrays to establish dual relationships among Bernoulli and Euler numbers and polynomials, leading to new identities and constructions.

Contribution

It presents a novel approach to construct dual relationships via Riordan arrays, unifying and extending previous results on Bernoulli and Euler sequences.

Findings

Four dual relationships for Bernoulli and Euler numbers are derived.

Dual sequences of Bernoulli and Euler polynomials are constructed.

Applications include new identities for Bernoulli and Euler polynomials.

Abstract

A sequence inverse relationship can be defined by a pair of infinite inverse matrices. If the pair of matrices are the same, they define a dual relationship. Here presented is a unified approach to construct dual relationships via pseudo-involution of Riordan arrays. Then we give four dual relationships for Bernoulli numbers and Euler numbers, from which the corresponding dual sequences of Bernoulli polynomials and Euler polynomials are constructed. Some applications in the construction of identities of Bernoulli numbers and polynomials and Euler numbers and polynomials are discussed based on the dual relationships.