- Open Access
Sums of products of the degenerate Euler numbers
© Wu and Pan; licensee Springer. 2014
Received: 2 October 2013
Accepted: 8 January 2014
Published: 27 January 2014
The paper focuses on the degenerate Euler numbers, the degenerate Euler polynomials and the degenerate Bernoulli polynomials. By adopting the method of recurrences, two explicit expressions have been established for sums of products of the degenerate Euler polynomials and the degenerate Bernoulli polynomials. As a special case of the degenerate Euler polynomials, an expression can be obtained for .
MSC:11B68, 11B65, 11B73.
for any .
and are the Stirling numbers of the first kind (see, e.g., ). Furthermore, Dilcher also extended (1.2) to the Bernoulli polynomials.
In , Wang obtained an explicit expression for sums of products of l Bernoulli polynomials and Euler polynomials.
In particular, here we set .
In fact, we shall prove a polynomial extension of (1.6) in the next section. In the third section, we also establish a generalization of (1.4) for the degenerate Bernoulli polynomial (see, e.g., ).
2 Degenerate Euler numbers and polynomials
Note that . So (1.6) is evidently a consequence of the following theorem.
This concludes our proof. □
Thus (2.1) immediately follows from (2.3).
3 Degenerate Bernoulli numbers and polynomials
In this section, we shall give a generalization of (3.1) for .
In particular, we set .
All proofs thus are done. □
The authors are grateful to two anonymous referees for their helpful suggestions. The second author is supported by National Natural Science Foundation of China (Grant No.11271185). The second author is the corresponding author.
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