
On algebra and arithmetic of binomial and Gaussian coefficients
U. M. Pachev^{} ^{} KabardinoBalkar 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 $\binom{2p1}{p1}$, a new comparability property modulo $p^3 \cdot ( 2p1 )$, which is not equal to the degree of a prime number, where $p$ and $(2p1)$ 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 $(nk)$ 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/10.22405/222683832018193257269
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UDC:
511.17+519.114 Received: 30.07.2018 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 257269
\mathnet{http://mi.mathnet.ru/cheb693}
\crossref{https://doi.org/10.22405/222683832018193257269}
\elib{https://elibrary.ru/item.asp?id=39454402}
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