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A modified Bernoulli number. (English) Zbl 0964.11015
In the note under review the author studies the rational numbers $$ B_n^*=\sum_{r=0}^n {{n+r}\choose{2r}}\frac{B_r}{n+r}, $$ 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 $\displaystyle\sum_{(p+1)|n\atop p\ \text{ prime}}\frac 1p$ ; c) $B_n^*\approx (-1)^{n/2}\pi Y_n(4\pi)$, $n\to\infty$, $n$ even, where $Y_n(x)$ denotes the $n$th 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.

11B68Bernoulli and Euler numbers and polynomials