Some identities involving polynomial coefficients

TL;DR

This paper explores identities involving polynomial coefficients, extending classical binomial coefficient formulas to powers of the polynomial 1+t+...+t^m, including positive and negative powers.

Contribution

It establishes new identities and summation formulas for polynomial coefficients, generalizing classical binomial coefficient properties to a broader class of polynomial expansions.

Findings

Derived several identities for polynomial coefficients.

Established summation formulas analogous to binomial coefficient identities.

Extended classical binomial identities to polynomial coefficients with positive and negative powers.

Abstract

By polynomial (or extended binomial) coefficients, we mean the coefficients in the expansion of integral powers, positive and negative, of the polynomial $1+t +\cdots +t^{m}$; $m\geq 1$ being a fixed integer. We will establish several identities and summation formul\ae\ parallel to those of the usual binomial coefficients.