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Convolution identities and lacunary recurrences for Bernoulli numbers

✍ Scribed by Takashi Agoh; Karl Dilcher

Publisher: Elsevier Science
Year: 2007
Tongue: English
Weight: 169 KB
Volume: 124
Category: Article
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2006.08.009

No coin nor oath required. For personal study only.

✦ Synopsis

We extend Euler's well-known quadratic recurrence relation for Bernoulli numbers, which can be written in symbolic notation as (B 0 + B 0 ) n = -nB n-1 -(n -1)B n , to obtain explicit expressions for (B k + B m ) n with arbitrary fixed integers k, m 0. The proof uses convolution identities for Stirling numbers of the second kind and for sums of powers of integers, both involving Bernoulli numbers. As consequences we obtain new types of quadratic recurrence relations, one of which gives B 6k depending only on B 2k , B 2k+2 , . . . , B 4k .

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