On algebra and arithmetic of binomial and Gaussian coefficients

U. M. Pachev

Čebyševskij sbornik, Tome 19 (2018) no. 3, pp. 257-269.

Résumé

In this paper we consider questions relating to algebraic and arithmetic properties of such binomial, polynomial and Gaussian coefficients. For the central binomial coefficients $\binom{2p}{p}$ and $\binom{2p-1}{p-1}$, a new comparability property modulo $p^3 \cdot \left( 2p-1 \right)$, which is not equal to the degree of a prime number, where $p$ and $(2p-1)$ are prime numbers, Wolstenholm's theorem is used, that for $p \geqslant 5$ these coefficients are respectively comparable with the numbers 2 and 1 modulo $p^3$. In the part relating to the Gaussian coefficients $\binom{n}{k}_q$, the algebraic and arithmetic properties of these numbers are investigated. Using the algebraic interpretation of the Gaussian coefficients, it is established that the number of $k$-dimensional subspaces of an $n$-dimensional vector space over a finite field of q elements is equal to the number of $(n-k)$ -dimensional subspaces of it, and the number $q$ on which The Gaussian coefficient must be the power of a prime number that is a characteristic of this finite field. Lower and upper bounds are obtained for the sum $\sum_{k = 0}^{n} \binom{n}{k}_q$ of all Gaussian coefficients sufficiently close to its exact value (a formula for the exact value of such a sum has not yet been established), and also the asymptotic formula for $q \to \infty$. In view of the absence of a convenient generating function for Gaussian coefficients, we use the original definition of the Gaussian coefficient $\binom{n}{k}_q$, and assume that $q>1$. In the study of the arithmetic properties of divisibility and the comparability of Gaussian coefficients, the notion of an antiderivative root with respect to a given module is used. The conditions for the divisibility of the Gaussian coefficients $\binom{p}{k}_q$ and $\binom{p^2}{k}_q$ by a prime number $p$ are obtained, and the sum of all these coefficients modulo a prime number $p$. In the final part, some unsolved problems in number theory are presented, connected with binomial and Gaussian coefficients, which may be of interest for further research.

Détail

Keywords: central binomial coefficients, Wolstenholme's theorem, Gaussian coefficient, the sum of Gaussian coefficients, divisibility by prime number, congruences modulo, primitive roots for this module.

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