On Binomial coefficients of real arguments

T. I. Fedoryaeva

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 1, pp. 514-523

Résumé

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. We prove by elementary means 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.

Détail

Keywords: factorial, gamma function
Mots-clés : binomial coefficient, real binomial coefficient.