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 Chebyshevskii Sb., 2018, Volume 19, Issue 3, Pages 257–269 (Mi cheb693)

On algebra and arithmetic of binomial and Gaussian coefficients

U. M. Pachev

Kabardino-Balkar State University

Abstract: 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 $p^3 \cdot ( 2p-1 )$, 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.

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.

DOI: https://doi.org/10.22405/2226-8383-2018-19-3-257-269

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UDC: 511.17+519.114
Accepted:15.10.2018

Citation: U. M. Pachev, “On algebra and arithmetic of binomial and Gaussian coefficients”, Chebyshevskii Sb., 19:3 (2018), 257–269

Citation in format AMSBIB
\Bibitem{Pac18} \by U.~M.~Pachev \paper On algebra and arithmetic of binomial and Gaussian coefficients \jour Chebyshevskii Sb. \yr 2018 \vol 19 \issue 3 \pages 257--269 \mathnet{http://mi.mathnet.ru/cheb693} \crossref{https://doi.org/10.22405/2226-8383-2018-19-3-257-269} \elib{https://elibrary.ru/item.asp?id=39454402}