Bernoulli Number of the Second Kind

A number defined by b_n=b_n(0), where b_n(x) is a Bernoulli polynomial of the second kind (Roman 1984, p. 294), also called Cauchy numbers of the first kind. The first few for n=0, 1, 2, ... are 1, 1/2, -1/6, 1/4, -19/30, 9/4, ... (OEIS A006232 and A006233). They are given by


where (x)_n is a falling factorial, and have exponential generating function


See also

Bernoulli Number, Bernoulli Polynomial of the Second Kind

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Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 294, 1974.Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 259, 1988.Roman, S. The Umbral Calculus. New York: Academic Press, p. 114, 1984.Sloane, N. J. A. Sequences A006232/M5067 and A006233/M1558 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Bernoulli Number of the Second Kind

Cite this as:

Weisstein, Eric W. "Bernoulli Number of the Second Kind." From MathWorld--A Wolfram Web Resource.

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