A few remarks on Euler and Bernoulli polynomials and their connections with binomial coefficients and modified Pascal matrices

TL;DR

This paper explores identities involving Euler and Bernoulli polynomials and reveals their connections to binomial coefficients and modified Pascal matrices, providing new interpretations of these classical numbers.

Contribution

It introduces new identities and interpretations linking Euler and Bernoulli numbers with inverses of certain Pascal matrices, expanding understanding of their algebraic structure.

Findings

Identified identities involving Euler and Bernoulli polynomials

Connected Euler and Bernoulli numbers to inverse binomial coefficient matrices

Provided interpretations of these numbers via modified Pascal matrices

Abstract

We prove certain identities involving Euler and Bernoulli polynomials that can be treated as recurrences. We use these and also other known identities to indicate connection of Euler and Bernoulli numbers with entries of inverses of certain lower triangular built of binomial coefficients. Another words we interpret Euler and Bernoulli numbers in terms of modified Pascal matrices.