## Bernoulli Polynomial

There are two definitions of Bernoulli polynomials in use. The th Bernoulli polynomial is denoted here by , and the archaic Bernoulli polynomial by . These definitions correspond to the Bernoulli Numbers evaluated at 0,

 (1) (2)

They also satisfy
 (3)

and
 (4)

(Lehmer 1988). The first few Bernoulli Polynomials are

Bernoulli (1713) defined the polynomials in terms of sums of the Powers of consecutive integers,
 (5)

Euler (1738) gave the Bernoulli polynomials in terms of the generating function
 (6)

They satisfy the Recurrence Relation
 (7)

(Appell 1882), and obey the identity
 (8)

where is interpreted here as . Hurwitz gave the Fourier Series
 (9)

for , and Raabe (1851) found
 (10)

A sum identity involving the Bernoulli Polynomials is

 (11)

for an Integer and arbitrary Real Numbers and .

References

Abramowitz, M. and Stegun, C. A. (Eds.). Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula.'' §23.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 804-806, 1972.

Appell, P. E. Sur une classe de polynomes.'' Annales d'École Normal Superieur, Ser. 2 9, 119-144, 1882.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 330, 1985.

Bernoulli, J. Ars conjectandi. Basel, Switzerland, p. 97, 1713. Published posthumously.

Euler, L. Methodus generalis summandi progressiones.'' Comment. Acad. Sci. Petropol. 6, 68-97, 1738.

Lehmer, D. H. A New Approach to Bernoulli Polynomials.'' Amer. Math. Monthly. 95, 905-911, 1988.

Lucas, E. Ch. 14 in Théorie des Nombres. Paris, 1891.

Raabe, J. L. Zurückführung einiger Summen und bestimmten Integrale auf die Jakob Bernoullische Function.'' J. reine angew. Math. 42, 348-376, 1851.

Spanier, J. and Oldham, K. B. The Bernoulli Polynomial .'' Ch. 19 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 167-173, 1987.