A DETERMINANTAL EXPRESSION AND A RECURRENCE RELATION FOR THE EULER POLYNOMIALS

In the paper, by a very simple approach, the author establishes an expression in terms of a lower Hessenberg determinant for the Euler polynomials. By the determinantal expression, the author finds a recurrence relation for the Euler polynomials. By the way, the author derives the corresponding expression and recurrence relation for the Euler numbers.


Main results
It is known that a matrix H = (h ij ) n×n is called a lower (or an upper, respectively) Hessenberg matrix if h ij = 0 for all pairs (i, j) such that i + 1 < j (or j + 1 < i, respectively).Correspondingly, we can define a lower (or an upper, respectively) Hessenberg determinant.
It is general knowledge that the Bernoulli numbers and polynomials B k and B k (u) can be generated respectively by for |z| < 2π.Because the function x e x −1 − 1 + x 2 is odd in x ∈ R, all of the Bernoulli numbers B 2k+1 for k ∈ N equal 0. The first six Bernoulli numbers B 2k for − ≤ k ≤ 5 are , B 10 = 5 66 .
In [4,Section 21.5] and [5, p. 1], it was listed that In [7, Theorem 1.2], the Bernoulli polynomials B k (u) for k ∈ N were expressed by a lower Hessenberg determinant and, consequently, the Bernoulli numbers B k for k ∈ N were represented by a lower Hessenberg determinant It is common knowledge that the Euler numbers E k and the Euler polynomials E k (x) can be generated respectively by which converge uniformly with respect to t ∈ (−π, π).By these definitions, it is easy to see that Since the generating function At the website [9], the Euler numbers E 2k were represented by a lower Hessenberg determinant In [8, Theorem 1.1], the Euler numbers E 2k for k ∈ N were represented by a lower Hessenberg and sparse determinant There have been many explicit expressions for the Euler numbers B k and the Euler polynomials E k (x).See, for example, the recently-published papers [2,3,8] and plenty of references therein.
In this paper, by a very simple approach, we will establish a new expression in terms of a lower Hessenberg determinant for the Euler polynomials E k (x).By the new determinantal expression, we will find a recurrence relation for the Euler polynomials.By the way, we will derive the corresponding expression and recurrence relation for the Euler numbers E k .
Our main results can be summarized up as two theorems below.
Theorem 1.For k ≥ 0, the Euler polynomials E k (x) can be expressed as and, consequently, the Euler numbers E k can be expressed as As an application of the expression (4), the following recurrence relation for the Euler polynomials and numbers E k (x) and E k can be derived.

Proofs of Theorems 1 and 2
Now we start out to prove our main results.
Considering the relation (3) and taking x = 1 2 in (4) lead to the formula (5) readily.The proof of Theorem 1 is complete.
Remark 1.The expression (5) can also be obtained by applying the formula (8) to the functions u(t) = e t/2 and v(t) = e t + 1 and considering the generating function