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Sums of products of the degenerate Euler numbers
Advances in Difference Equations volume 2014, Article number: 40 (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.
The Bernoulli numbers are defined by
The study of the Bernoulli numbers has a very long history. In fact, Euler had found that
for any .
In 1996, Dilcher  generalized (1.1) to the sums of products of N Bernoulli numbers:
where are the multinomial coefficients defined by
and are the Stirling numbers of the first kind (see, e.g., ). Furthermore, Dilcher also extended (1.2) to the Bernoulli polynomials.
Carlitz showed that is a polynomial in λ. And the explicit formula for was obtained by Howard . Note that tends to as . So we have . Furthermore, we know that tends to as , where is the Bernoulli number of the second kind given by
The Euler numbers are another important kind of numbers, which are defined by
The Euler numbers have many similar properties as the Bernoulli numbers. For example, in the same paper, Dilcher also proved that
where and the Euler polynomials are given by
In , Wang obtained an explicit expression for sums of products of l Bernoulli polynomials and Euler polynomials.
In , Carlitz also defined the degenerate Euler numbers by
Motivated by (1.4), we shall establish a generalization of (1.5) for in this note. We define a class of generalized Stirling-like polynomials of the first kind as follows:
In particular, here we set .
Theorem 1.1 For , we have
where the degenerate Euler polynomials are given by
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.
Theorem 2.1 Let . Then for , we have
The degenerate Euler polynomials of order m are defined by
Lemma 2.1 For , we have
Proof Observe that
Comparing the coefficients of in both sides of the above equation, we have
Below we use induction on m to show (2.3). It is easy to see that (2.3) holds for . Now let and assume that (2.3) holds for the smaller values of m. Then, by the induction hypothesis, we have
It is easy to verify that
for , and
So by (2.4), we get
This concludes our proof. □
Let us turn to the proof of Theorem 2.1. Clearly,
Hence we have
Thus (2.1) immediately follows from (2.3).
3 Degenerate Bernoulli numbers and polynomials
The Bernoulli polynomials are defined by
Clearly, . In , Dilcher proved that
The degenerate Bernoulli polynomials are defined by
In this section, we shall give a generalization of (3.1) for .
In particular, we set .
Theorem 3.1 Let . Then for , we have
Proof Similarly as in the proof of Theorem 2.1, it suffices to show that
for , where the degenerate Bernoulli polynomials of order m are defined by
We shall use induction on m. Clearly (3.4) holds for . Let and assume that (3.4) holds for m. Note that
Comparing the coefficients of in both sides of the above equation, we get
Applying (3.4) for and , we have
It is not difficult to check that
for , and
It follows from (3.6) that
All proofs thus are done. □
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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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Wu, M., Pan, H. Sums of products of the degenerate Euler numbers. Adv Differ Equ 2014, 40 (2014). https://doi.org/10.1186/1687-1847-2014-40
- degenerate Euler number
- degenerate Euler polynomial
- degenerate Bernoulli polynomial