On Binomial coefficients of real arguments

TL;DR

This paper explores properties and asymptotic behavior of binomial coefficients with real arguments, extending classical combinatorial concepts through elementary proofs.

Contribution

It provides new elementary proofs of properties like unimodality, symmetry, and Pascal's rule for binomial coefficients of real numbers, and analyzes their asymptotic behavior.

Findings

Proved analogs of classical binomial coefficient properties for real arguments.

Established asymptotic formulas for specific forms of generalized binomial coefficients.

Demonstrated elementary methods to analyze binomial coefficients beyond integer values.

Abstract

As is well-known, a generalization of the classical concept of the factorial $n!$ for a real number $x\in {\mathbb R}$ is the value of Euler's gamma function $\Gamma(1+x)$. In this connection, the notion of a binomial coefficient naturally arose for admissible values of the real arguments. By elementary means, it is proved a number of properties of binomial coefficients $\binom{r}{\alpha}$ of real arguments $r,\,\alpha\in {\mathbb R}$ such as analogs of unimodality, symmetry, Pascal's triangle, etc. for classical binomial coefficients. The asymptotic behavior of such generalized binomial coefficients of a special form is established.