A new generalization of binomial coefficients

TL;DR

This paper introduces a new generalization of binomial coefficients parameterized by t, exploring their properties and inductive definitions, expanding the mathematical understanding of binomial coefficient generalizations.

Contribution

It proposes a novel generalization of binomial coefficients with a parameter t and studies their properties and inductive definitions.

Findings

Defined generalized binomial coefficients with parameter t

Derived properties and inductive relations for these coefficients

Extended understanding of binomial coefficient generalizations

Abstract

Let $t$ be a fixed parameter and $x$ some indeterminate. We give some properties of the generalized binomial coefficients $\genfrac{<}{>}{0pt}{}{x}{k}$ inductively defined by $k/x \genfrac{<}{>}{0pt}{}{x}{k}= t\genfrac{<}{>}{0pt}{}{x-1}{k-1} +(1-t)\genfrac{<}{>}{0pt}{}{x-2}{k-2}$.