On Binomial coefficients of real arguments
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 $Γ(1+x)$. 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 $\binom{r}α$ of real arguments $r,\,α\in {\mathbb R}$ 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.
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