On the coefficients of power sums of arithmetic progressions

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Some notes on alternating power sums of arithmetic progressions

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Some notes on alternating power sums of arithmetic progressions

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Singmaster-type results for Stirling numbers and some related diophantine equations

Motivated by the work of David Singmaster, we study the number of times an integer can appear among the Stirling numbers of both kinds. We provide an upper bound for the occurrences of all the positive integers, and present certain questions for further study. Some numerical results and conjectures concerning the related diohantine equations are collected

On equal values of products and power sums of consecutive elements in an arithmetic progression

In this paper we study the Diophantine equation \begin{align*} b^k + \left(a+b\right)^k + &\left(2a+b\right)^k + \ldots + \left(a\left(x-1\right) + b\right)^k = \\ &y\left(y+c\right) \left(y+2c\right) \ldots \left(y+ \left(\ell-1\right)c\right), \end{align*} where $a,b,c,k,\ell$ are given integers under natural conditions. We prove some effective results for special values for $c,k$ and $\ell$ and obtain a general ineffective result based on Bilu-Tichy method

On the coefficients of power sums of arithmetic progressions

We investigate the coefficients of the polynomial $$S_{m,r}^n(\ell)=r^n+(m+r)^n+(2m+r)^n+\cdots+((\ell-1)m+r)^n.$$ We prove that these can be given in terms of Stirling numbers of the first kind and $r$-Whitney numbers of the second kind. Moreover, we prove a necessary and sufficient condition for the integrity of these coefficients