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Date: 25-3-2019
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Date: 22-7-2019
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The Euler polynomial is given by the Appell sequence with
(1) |
giving the generating function
(2) |
The first few Euler polynomials are
(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
(9) |
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(10) |
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(11) |
where is a binomial coefficient. Setting and normalizing by gives the Euler number
(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
(13) |
The Bernoulli numbers for can be expressed in terms of by
(14) |
The Newton expansion of the Euler polynomials is given by
(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
(16) |
and
(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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كيف تساهم الأطعمة فائقة المعالجة في تفاقم مرض يصيب الأمعاء؟
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مشروع ضخم لإنتاج الهيدروجين الأخضر يواجه تأخيرًا جديدًا
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المجمع العلمي يختتم دورته القرآنية في فن الصوت والنغم بالطريقة المصرية
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