Bernoulli Numbers

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Bernoulli Numbers


a special sequence of rational numbers which figures in various problems of mathematical analysis and the theory of numbers. The values of the first six Bernoulli numbers are

In mathematical analysis, Bernoulli numbers appear as the coefficients of expansion of certain elementary functions in power series—for example,

The Euler-Maclaurin summation formula is one of the most important formulas in which Bernoulli numbers are encountered. The sums of many series and the values of improper integrals are expressed in terms of Bernoulli numbers. Bernoulli numbers first appeared in the posthumous work of Jakob Bernoulli (1713) in connection with the calculation of the sum of identical powers of natural numbers. He proved that

Recurrence formulas that permit the sequential calculation of Bernoulli numbers, as well as explicit formulas (which have a rather complex form), are known for Bernoulli numbers.

There is great interest in the theoretic-numerical properties of the Bernoulli numbers. In 1850 the German mathematician E. Kummer established that Fermat’s equation xp + yp = zp is not solved in integers x, y, and z which are not zero unless a prime number p > 2 divides the numerators of the Bernoulli numbers B1, B2, . . ., B(p - 3)/2. Often (-1)m-1 B2m (m = 1, 2,. . .) is written instead of Bm to designate Bernoulli numbers; furthermore, it is assumed that

B0 = 1, B1 = -½, B3 = B5 = B7 = . . . = 0


Chistiakov, I. I. Bernullievye chisla. Moscow, 1895.
Kudriavtsev, V. A. Summirovanie stepenei chisel natural’nogo riada i chisla Bernulli. Moscow-Leningrad, 1936.
Whittaker, E. T., and G. N. Watson. Kurs sovremennogo analiza, 2nd ed., part 1. Moscow, 1963. (Translated from English.)
Landau, E. Vorlesungenüber Zahlentheorie, vol. 3. New York, 1927.


The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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In [4], Arakawa and Kaneko used analytic continuation of [zeta]([s.sub.1], ..., [s.sub.d]) as a function of one variable [s.sub.d] when [s.sub.1], ..., [s.sub.d-1] are positive integers and discussed the relation among generalized Bernoulli numbers. For a general d, Zhao [12] proved the analytic continuation of [zeta]([s.sub.1], ..., [s.sub.d]) as a function of d variables using the theory of generalized function and Akiyama, Egami and Tanigawa [1] proved the same result by applying the classical Euler-Maclaurin formula to the index of the summation [n.sub.d].
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Let [B.sub.k](t), k[greater than or equal to]0 be the Bernoulli polynomials, and [B.sub.k] = [B.sub.k](0), k[greater than or equal to]0, the Bernoulli numbers. The first few Bernoulli polynomials are