arXiv subjects

René Gy

Publications and source records attributed to René Gy.

3 recordsLinked to original sources

arXiv2020

An Aearated Triangular Array of Integers

Congruences modulo prime powers involving generalized Harmonic numbers are known. While looking for similar congruences, we have encountered a curious triangular array of numbers indexed with positive integers $n,k$, involving the Bernoulli and cycle Stirling numbers. These numbers are all integers and they vanish when $n-k$ is odd. This triangle has many similarities with the Stirling triangle. In particular, we show how it can be extended to negative indices and how this extension produces a {\it second kind} of such integers which may be considered as a new generalization of the Genocchi numbers and for which a generating function is easily obtained. But our knowledge of these integers remains limited, especially for those of the {\it first kind}.

math.CO↗

arXiv2020

Congruences for certain lacunary sums of products of binomial coefficients

It is shown that for any prime $p$ and any natural numbers $\ell, m,$ and $s$ such that $0<s<p$, the three following congruences \begin{align*}\sum_{i\ge \ell+1}(-1)^{m-i} {m \choose i}{m+s-1+i(p-1) \choose m+s-1+\ell(p-1)} &\equiv 0 \bmod p\\ \sum_{i\ge 0}(-1)^{m-i} {m \choose i}{\ell+ip \choose m+s-1}&\equiv 0 \bmod p^m\\ \sum_{j,i\ge \ell}(-1)^{j-i}{m \choose j} {j \choose i}{j+s-1+i(p-1) \choose j+s-1+\ell(p-1)}&\equiv 0 \bmod p^{m-\ell} \end{align*} hold true. The corresponding quotients involve Adelberg polynomials which can be computed explicitly, providing closed-form expressions for these sums, valid even if $p$ is not prime, when the congruences do not necessarily hold.

math.NT↗

arXiv2019

Extended Congruences for Harmonic Numbers

We derive $p$-adic expansions for the generalized Harmonic numbers $H^{(j)}_{p-1}$ and $H^{(j)}_{\frac{p-1}{2}}$ involving the Bernoulli numbers $B_j$ and the the base-2 Fermat quotient $q_p$. While most of our results are not new, we obtain them elementarily, without resorting to the theory of $p$-adic L-functions as was the case previously. Moreover, we show that \begin{equation*}\sum_{j=0}^{n-1}\left(\frac{(2^{j+1}-1)}{(j+1)}\frac{(2^{j+2}-1)}{(j+2)}\frac{B_{j+2}}{2^{j}}H^{(j+1)}_{\frac{p-1}{2}}+2(-1)^j\frac{q_p^{j+1}}{j+1}\right)p^j\equiv 0 \pmod {p^n} \end{equation*} holds under the condition that $p >\frac{n+1}{2}$. This is another generalization, modulo any prime power, of the old $p$-congruence $H_{\frac{p-1}{2}}+2q_p \equiv 0 \bmod p$ attributed to Eisenstein, which is stronger than the one which has been published recently.

math.NT↗