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A symmetry of power sum polynomials and Bernoulli numbers. (English) Zbl 0983.11008
The author proves a nice symmetry relation between the sums $\sigma(k)=\sum_{j=0}^k j^m$ and Bernoulli numbers $B_m$ ($m\ge 0$). This relation may be written in symbolic (or “umbral calculus”) notation as $$ \tfrac 1{a}(aB+b\sigma(a-1))^m = \tfrac 1{b}(bB+a\sigma(b-1))^m, $$ where $a$ and $b$ are any positive integers. If $b=1$, the right hand side equals $B_m$. In this case the formula is known; see, e.g., a problem and its solution in Am. Math. Mon. 96, No. 4, 364-365 (1989).

11B68Bernoulli and Euler numbers and polynomials
11B75Combinatorial number theory
05A40Umbral calculus
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