A New Formula for the Bernoulli Numbers of the Second Kind in Terms of the Stirling Numbers of the First Kind
نویسنده
چکیده
and that the Bernoulli numbers of the second kind bn for n > 0 may be generated by x ln(1+x) = ∑ ∞ n=0 bnx . In combinatorics, the signed Stirling number of the first kind s(n, k) may be defined such that the number of permutations of n elements which contain exactly k permutation cycles is the nonnegative number |s(n, k)| = (−1)s(n, k). The Bernoulli numbers of the second kind bn are also called the Cauchy numbers of the first kind, see [20,27] and closely related references therein. In [14], the following formula for computing the Bernoulli numbers of the second kind in terms of the signed Stirling numbers of the first kind was derived:
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