On Binomial coefficients of real arguments
Tatiana I. Fedoryaeva

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 for a real number is the value of Euler's gamma function . 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 of real arguments 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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Taxonomy
TopicsHistory and Theory of Mathematics · Mathematics and Applications · Advanced Mathematical Identities
