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Zbl 0983.11008
Tuenter, Hans J.H.
A symmetry of power sum polynomials and Bernoulli numbers. (English)
[J] Am. Math. Mon. 108, No.3, 258-261 (2001). ISSN 0002-9890

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).
[Tauno Metsänkylä (Turku)]

MSC 2000:: *11B68 Bernoulli numbers, etc.
11B75 Combinatorial number theory
05A40 Umbral calculus

Keywords: Bernoulli numbers; Bernoulli polynomials; recurrences; sums of powers of integers

Cited in: Zbl 1217.11022 Zbl 1133.11015 Zbl 1160.11313

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