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Date: 2-9-2019
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Date: 2-5-2019
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Date: 28-4-2019
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The Euler polynomial is given by the Appell sequence with
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(1) |
giving the generating function
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(2) |
The first few Euler polynomials are
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(3) |
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(4) |
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(5) |
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(6) |
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(7) |
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(8) |
Roman (1984, p. 100) defines a generalization for which
. Euler polynomials are related to the Bernoulli numbers by
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(9) |
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(10) |
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(11) |
where is a binomial coefficient. Setting
and normalizing by
gives the Euler number
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(12) |
The first few values of 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 series
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(13) |
The Bernoulli numbers for
can be expressed in terms of
by
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(14) |
The Newton expansion of the Euler polynomials is given by
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(15) |
where is a binomial coefficient,
is a falling factorial, and
is a Stirling number of the second kind (Roman 1984, p. 101).
The Euler polynomials satisfy the identities
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(16) |
and
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(17) |
for a nonnegative integer.
REFERENCES:
Abramowitz, M. and Stegun, I. 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, 6th ed. San Diego, CA: Academic Press, 2000.
Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Generalized Zeta Function , Bernoulli Polynomials
, Euler Polynomials
, and Polylogarithms
." §1.2 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 23-24, 1990.
Roman, S. "The Euler Polynomials." §4.2.3 in The Umbral Calculus. New York: Academic Press, pp. 100-106, 1984.
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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