## 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 are the 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 by
 (7)

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.