A note on Kummer's congruences
It is proved that theorem 5 of author's paper [Duke Math. J. 20, 423--431 (1953; Zbl 0051.27604)] can be simplified and that a complicated hypothesis can be replaced by another, which also occurs in analogous theorems. His result now is: Let \(p\) be a prime, \(\{a_m\}\) and \(\{b_m\}\) be sequences of rational numbers, that are integral mod \(p\) and let \(c_m =\sum_{s=0}^m {m\choose s} a_mb_{m-s}\). Suppose \(a_p\equiv b_p\pmod p\) and assume that \(\{a_m\}\) and \(\{b_m\}\) satisfy Kummer's congruences (this means \[ \sum_{s=0}^r (-1)^{r-s}{r\choose s} a_p^{r- s} a_{m+s(p-1)}\equiv 0 \pmod{p^r}\quad (m\geq r\geq 1) \] and the same with \(\{b_m\}\) instead of \(\{a_m\})\). Then \(\sum_{s=0}^r (-1)^{r-s}{r\choose s} k^{r-s} c_{m+s(p-1)}\equiv 0 \pmod{p^r}\) holds for all \(k \equiv a_p \equiv b_p \pmod p\) and all \(m\geq r\geq 1\). Other former results are now again inspected. In addition it is shown that if \(\{a_m\}\) satisfies Kummer's congruences and if \(\sum_{s=0}^m {m\choose s}a_s c_{m-s}=0\) \((a_0 = c_0 = 1)\) for \(m\geq 1\), then \(\{c_n\}\) satisfies the above mentioned congruences with \(k = a_p\).
scientific article; zbMATH DE number 3125799