| 释义 |
Bernoulli PolynomialThere 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,
They also satisfy
 | (3) |
 | (4) |
Bernoulli (1713) defined the polynomials in terms of sums of the Powers of consecutive integers,
 | (5) |
 | (6) |
 | (7) |
 | (8) |
 is interpreted here as  . Hurwitz gave the Fourier Series
 | (9) |
 , and Raabe (1851) found
 | (10) |
A sum identity involving the Bernoulli Polynomials is
 | (11) |
 and arbitrary Real Numbers  and  .See also Bernoulli Number, Euler-Maclaurin Integration Formulas, Euler Polynomial
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.
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