Bernoulli Numbers or Bernoullian Num Bers

BERNOULLI NUMBERS or BERNOULLIAN NUM BERS, a series of fractions beginning with -, , , , and designated respectively by the symbols • • The name was given to them by Abraham De Moivre (q.v.) and Leonard Euler (q.v.) in honour of their discoverer, Jacques Ber noulli (1654-1705), who introduced them into analysis in his posthumous work, Ars Conjectandi (Basle, He used them in summing the nth powers of the first x integers, his formula being:— For upwards of two centuries these numbers have been the object of study of some of the best mathematical minds. It has been shown that they have numerous important applications, particularly in the theory of numbers, the calculus of differences, and the theory of definite integrals. They are also intimately connected with the theory of tangential coefficients, and with such important families of numbers as Euler's (see EULER NUMBERS) and Stirling's (see STIRLING NUMBERS) of the first and second methods of calculation. Bernoulli himself gave only five of these numbers, but since his time, Euler has given 15; Ohm, 3 I ; Adams, 62; and Serebrennikow, 90.

The first of the recurring formulae for the numbers was given

by De Moivre in his Miscellanea Analytica, Complementum, (1730), as follows:— (2m+I)Bm-(2m+I)3Bm-1+(2m+I)5Bm-2+ . .( • (- I)m-1 (2n1+')2»:-1 B1+ (- I 2) = o. This is based on a rule given by Bernoulli. Jacobi (1834) gave the following:— (2112+ 2)2 - (2151+ 2)4 (2M+ 2)6 Bm-2 + • . (— I)m-1 m=O.

Many other recurring formulae have been developed, but

they are all subject to one great disadvantage—that the calcula tion of any special Bernoulli number, say the nth, requires a knowledge of all the previous n-- numbers. Since 1887, how ever, various reversion formulae have been developed—notably by Seidel, Stern, and Saalschiitz—in which some of the preced ing numbers were missing. In particular, Haussner developed a set of formulae in which none appeared except those with the indices nq+v.

Formulae for these numbers have been given in determinant

notation. For example, Glaisher gave the following:— One of the most interesting discoveries made by Euler in this connection is that the Bernoulli numbers occur also in the sum of the even powers of the reciprocals of the natural numbers; i.e., I I I 22m-1..2m S2m = -}- 2 -}- 32 2 m + • • • = 2112 which gives the for mula = 22m-1 ,2m . They also occur as coefficients in the ex pansion of both x • ex+I and x • 2 ex — I Of the many properties of these numbers that have been dis covered by various investigators, one of the most interesting is Von Staudt's or Clausen's theorem, as it is called from its simultaneous discoverers. It states that, if a, (3, -y, ... are all the prime numbers greater by I than the divisors of 2m, then +. (- 0'4+1 L 2 I + I + I + • • • + x I is an integer. a Q y From this it is apparent that the denominator of any of the num bers can be found. For example, to find the denominator of B11, observe that the divisors of 22 are I, 2, I I, 22. Increasing each by 1, we have 2, 3; 12, 23 of which only 2, 3, and 23 are prime. Hence the denominator is 2, 3, 23, or 138, and the fractional part of is + -}- or H8. Later investigators have found recurring formulae for the integral parts of the numbers, the most important ones being due to Hermite, Stern, and Lipschitz. The third of these writers gave the most general one.

Besides the relation of these numbers to certain other families

already mentioned, Frobenius considered one that is so funda mental that he used it for a new definition of the numbers them selves. It states that when the Stirling number of the second kind, of order n, is expressed as a polynomial in x (see STIRLING NUMBERS), the value of the polynomial for x = - I is equal to and the value of its derivative for x= o is I (- I) nB,,,.

Many generalizations of these numbers have been made,

a discussion of the subject by Prof. E. T. Bell appearing in the American Journal of Mathematics (1925). These generalizations vary according to the branches to which the numbers are appli cable, many owing their appearance to the function used to de fine the numbers themselves. For example, if the numbers are defined as the value of a certain polynomial for x= - I, then the values of T for x= -2,-3, ... ; i.e., T(- 2), T( - 3), ... , will lead to what are known as ultra-Bernoulli numbers or generalized Bernoulli numbers, a term that will probably persist.

BIBLIOGRAPHY.

L. Saalschutz, Vorlesung fiber die Bernoullischen Bibliography.—L. Saalschutz, Vorlesung fiber die Bernoullischen Zahlen (1893) ; N. Nielson, Traite Elementaire des nombres de Bernoulli (1923) . See also K. G. von Staudt in vol. xxi. (1840) and C. J. Malmsten in vol. xxxv. (1847) of A. L. Crelle, Zeitschrift fur die refine Mathematik; Th. Clausen in Astronomische Nachrichten, vol. xvii. (1840) ; G. S. Ely, "Bibliography of Bernoulli's Numbers" in American Journal of Mathematics (Baltimore, 1882) ; J. W. L. Glaisher, numerous articles in the Messenger of Mathematics, the Quarterly Journal of Pure and Applied Mathematics.