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A modified Bernoulli number. (English) Zbl 0964.11015

In the note under review the author studies the rational numbers

B n * = r=0 n n+r 2rB r n+r,

where B n are the classical famous Bernoulli numbers. He shows that these modified Bernoulli numbers have some very interesting properties similar to the classical ones, namely

a) The value of B n * for odd n is periodic mod 12;

b) The fractional part of 2nB n * -B n for n even is given by (p+1)|n pprime 1 p ;

c) B n * (-1) n/2 πY n (4π), n, n even, where Y n (x) denotes the nth Bessel function of the second kind.

The proofs are really fun, as the author points out, and are examples of his virtuos handling of generating functions.

MSC:
11B68Bernoulli and Euler numbers and polynomials