Open Access
February 2008 Generating functions and generalized Euler numbers
Guodong Liu
Proc. Japan Acad. Ser. A Math. Sci. 84(2): 29-34 (February 2008). DOI: 10.3792/pjaa.84.29


In this paper we shall give an explicit formula for the coefficient of the expansion of a given generating function raised to an arbitrary power, when that function has an appropriate form. One of the many examples is the generalized Euler numbers and we shall clarify the situation surrounding the congruence $E_{\frac{p-1}{2}}\not\equiv 0 (\text{mod}\ p), p\equiv 1 (\text{mod}\ 4)$, a prime, and restore the priority. At the same time we shall state the true meaning of such a congruence.


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Guodong Liu. "Generating functions and generalized Euler numbers." Proc. Japan Acad. Ser. A Math. Sci. 84 (2) 29 - 34, February 2008.


Published: February 2008
First available in Project Euclid: 4 February 2008

zbMATH: 1204.11050
MathSciNet: MR2386962
Digital Object Identifier: 10.3792/pjaa.84.29

Primary: 11B68
Secondary: 05A10 , 11B83

Keywords: congruences , Explicit formula , generating functions , the generalized Euler numbers

Rights: Copyright © 2008 The Japan Academy

Vol.84 • No. 2 • February 2008
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