Some Properties of a Sequence Similar to Generalized Euler Numbers

We introduce the sequence given by generating function and establish some explicit formulas for the sequence . Several identities involving the sequence , Stirling numbers, Euler polynomials, and the central factorial numbers are also presented.

In fact, ( ) 2 ( ∈ Z) are the Euler numbers of order , Z being the set of integers. The numbers (1) 2 = 2 are the ordinary Euler numbers.
Zhi-Hong Sun introduces the sequence { } similar to Euler numbers as follows (see [6,7]): The sequence { } is related to the classical Bernoulli polynomials ( ) (see [8][9][10][11]) and the classical Euler polynomials ( ). Zhi-Hong Sun gets the generating function of 2 ISRN Discrete Mathematics { } and deduces many identities involving { }. As example, (see [6]), Similarly, we can define the generalized sequence { ( ) }. For a real or complex parameter , the generalized sequence { ( ) } is defined by the following generating function: Obviously, By using (10), we can obtain We now return to the Stirling numbers ( , ) of the first kind, which are usually defined by (see [2,5,8,11,12]) or by the following generating function: It follows from (13) or (14) that and that The central factorial numbers ( , ) are given by the following expansion formula (see [3,5,13]): or by means of the generating function It follows from (17) or (18) that with (0, 0) = 1, We also find from (18) that The main purpose of this paper is to prove some formulas for the generalized sequence { ( ) } and ( ). Some identities involving the sequence { ( ) }, Stirling numbers ( , ), and the central factorial numbers ( , ) are deduced.

Main Results
From Corollary 3, we may immediately deduce the following results.

Proofs of Theorems
Proof of Theorem 1. By (10), (13), and (18), we have which readily yields This completes the proof of Theorem 1.
Proof of Theorem 5. By (10), we have By Theorem 1 and comparing the coefficient of 2 /(2 )! on both sides of (35), we get Proof of Theorem 6. By applying Theorem 1, we have On the other hand, it follows from (10) that By using (38) and (39), we find that We now note that Hence, Comparing the coefficient of 2 /(2 )! on both sides of (43), we immediately get (28). This completes the proof of Theorem 6.
Proof of Theorem 8. By using (7), we have Thus ISRN Discrete Mathematics 5 That is, Comparing the coefficient of / ! on both sides of (48), we get the following: By (49) we immediately obtain (30). This completes the proof of Theorem 8.
Proof of Theorem 9. By integrating (7) with respect to from 0 to 1, we have