Some Determinantal Expressions and Recurrence Relations of the Bernoulli Polynomials

In the paper, the authors recall some known determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, find alternative determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, and present several new recurrence relations for the Bernoulli numbers and polynomials.


Introduction and Main Results
It is general knowledge that the Bernoulli numbers and polynomials B k and B k (t) can be generated by and for |z| < 2π respectively.It is clear that B k (0) = B k .Because the function x e x −1 − 1 + x 2 is even in x ∈ R, all the Bernoulli numbers B 2k+1 for k ∈ N equal 0. In addition to B 0 = 1 and B 1 = − It is well known that a matrix H = (h ij ) n×n is called a lower (respectively upper) Hessenberg matrix if h ij = 0 for all pairs (i, j) such that i + 1 < j (respectively j + 1 < i).Correspondingly, we can define a Hessenberg determinant.
In [1] (p. 40), it was mentioned that ( ( and for k ∈ N. The determinant in ( 2) is a sub-determinant of the determinant in (3).It was pointed out in [1] (p. 40) that these two determinantal expressions were recorded in [2] and can be traced back to the book [3].The Bernoulli polynomials B k (t) were represented in [1] in terms of the Hessenberg determinants as In [4] (Section 21.5) and [5] (p.1), the determinantal expression for the Bernoulli numbers B k for k ≥ 0 was listed.This expression can also be deduced from taking the limit t → 0 in (4).
Let {a m } 0≤m≤∞ be a sequence of complex numbers and let {D k (a m )} 0≤k≤∞ be a sequence of the Hessenberg determinants such that D 0 (a m ) = 1 and In [6], the determinantal expressions and for k ∈ N were established.In [7] (Theorem 1.1), it was shown that, if are the ordinary generating functions of {a m } 0≤m≤∞ and {b m } 0≤m≤∞ such that A(x)B(x) = 1, then a 0 = 0 and As applications of [7] (Theorem 1.1), among other things, some properties of D k (a m ) were discovered and applied to give an elegant proof of ( 6) and (7).In particular, the Hessenberg determinantal expressions which recovers [8] (Equation ( 4)), and were derived.
In [9] (Theorem 1.2), the Bernoulli polynomials B k (t) for k ∈ N were expressed as a Hessenberg determinant by under the conventions that ( 0 0 ) = 1 and ( p q ) = 0 for q > p ≥ 0. Consequently, the Bernoulli numbers B k for k ∈ N were be represented as The first aim of this paper is to find alternative determinantal expressions in terms of the Hessenberg determinants for the Bernoulli polynomials B k (t) and the Bernoulli numbers B k .The second aim is to derive several recurrence relations for the Bernoulli polynomials B k (t) and the Bernoulli numbers B k .
Our main results can be formulated as a theorem and a corollary below.
Theorem 1.For k ≥ 0, the Bernoulli polynomials B k (t) can be expressed in terms of a Hessenberg determinant as and, consequently, the Bernoulli numbers B k can be expressed as Corollary 1.For k ≥ 1, the Bernoulli polynomials B k (t) satisfy the recurrence relations and Consequently, the Bernoulli numbers B k satisfy the recurrence relations and

Lemmas
In order to obtain our aims and to prove our main results, we need the following lemmas.

Proofs of Theorem 1 and Corollary 1
We are now in a position to prove our main results.
Proof of Corollary 1. Expanding the determinant in ( 10) by the last column consecutively and employing (10) inductively reveal Accordingly, it follows that Further considering the identity the relation ( 12) is thus proved.Taking t = 0 in (12) results in the relation (14).
A straightforward application of the recurrence Equation ( 17) to the determinantal Equations ( 10) and ( 11) leads to (13) and (15), respectively.The proof of Corollary 1 is complete.

Proof of Theorem 1 .
We can write the generating function of the Bernoulli polynomials B k (t) the help of Equation (16) applied to u(t) = e tz and v(t) = e 1 s z−1 d s, we obtain the kth derivative e tz e 1 s z−1 d s