“Euler Polynomial”的概念、定义、习题解题方法及翻译-数学辞典

单词

Euler Polynomial

释义

Euler Polynomial

A Polynomial given by the sum

(1)

Euler polynomials are related to the Bernoulli Numbers by

			(2)
			(3)

			(4)

where

is a Binomial Coefficient. Setting

and normalizing by

gives the Euler Number

(5)

Call

, then the first few terms are

, 0, 1/4,

, 0, 17/8, 0, 31/2, 0, .... The terms arethe same but with the Signs reversed if

. These values can be computed using the double sum

(6)

The Bernoulli Numbers

for

can be expressed in terms of the

(7)

See also Bernoulli Polynomial, Euler Number, Genocchi Number

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.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1979.

Spanier, J. and Oldham, K. B. ``The Euler Polynomials .'' Ch. 20 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 175-181, 1987.

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