A GENERALIZATION OF THE BERNOULLI POLYNOMIALS PIERPAOLO NATALINI

The Bernoulli polynomials have important applications in number theory and classical analysis. They appear in the integral representation of differentiable periodic functions since they are employed for approximating such functions in terms of polynomials. They are also used for representing the remainder term of the composite Euler-MacLaurin quadrature rule (see [15]). The Bernoulli numbers [3, 13] appear in number theory, and in many mathematical expressions, such as


Introduction
The Bernoulli polynomials have important applications in number theory and classical analysis.They appear in the integral representation of differentiable periodic functions since they are employed for approximating such functions in terms of polynomials.They are also used for representing the remainder term of the composite Euler-MacLaurin quadrature rule (see [15]).
The Bernoulli numbers [3,13] appear in number theory, and in many mathematical expressions, such as (i) the Taylor expansion in a neighborhood of the origin of the circular and hyperbolic tangent and cotangent functions; (ii) the sums of powers of natural numbers; (iii) the residual term of the Euler-MacLaurin quadrature rule.
The Bernoulli polynomials B n (x) are usually defined (see, e.g., [7, page xxix]) by means of the generating function and the Bernoulli numbers B n := B n (0) by the corresponding equation The B n are rational numbers.We have, in particular, 3) The following properties are well known: The Bernoulli polynomials are easily computed by recursion since Some generalized forms of the Bernoulli polynomials and numbers already appeared in literature.We recall, for example, the generalized Bernoulli polynomials B α n (x) recalled in the book of Gatteschi [6] defined by the generating function by means of which, Tricomi and Erdélyi [16] gave an asymptotic expansion of the ratio of two gamma functions.
Another generalized forms can be found in [5,11], starting from the generating functions

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where J α is the Bessel function of the first kind of order α and j 1 = j 1 (α) is the first zero of J α , or respectively.
In this paper, we introduce a countable set of polynomials B [m−1] n (x) generalizing the Bernoulli ones, which can be recovered assuming m = 1.To this aim, we consider a class of Appell polynomials [2], defined by using a generating function linked to the so-called Mittag-Leffler function considered in the general form by Agarwal [1] (see also [12]).Furthermore, exploiting the factorization method introduced in [10] and recalled in [8], we derive the differential equation satisfied by these polynomials.It is worth noting that the differential equation for Appelltype polynomials was derived in [14], and more recently recovered in [9] by exploiting the factorization method.It is easily checked that our differential equation matches with the general form of the above mentioned articles [9,14].In particular, when m = 1, the differential equation of the classical Bernoulli polynomials is derived again.
We will show in this paper that the differential equation satisfied by the B [m−1] n (x) polynomials is of order n, so that all the considered families of polynomials can be viewed as solutions of differential operators of infinite order.This is a quite general situation since the Appell-type polynomials, satisfying a differential operator of finite order, can be considered as an exceptional case (see [4]).

A new class of generalized Bernoulli polynomials
The generalized Bernoulli polynomials are defined by means of the generating function, defined in a suitable neighborhood of t = 0

.1)
For m = 1, we obtain, from (2.1), the generating function Since G [m−1] (x, t) = A(t)e xt , the generalized Bernoulli polynomials belong to the class of Appell polynomials.
It is possible to define the generalized Bernoulli numbers assuming From (2.1), we have and therefore

.5)
By comparing the coefficients of (2.5), we obtain Inverting (2.6), it is possible to find explicit expressions for the polynomials The first ones are given by and, consequently, the first generalized Bernoulli numbers are . (2.8) P. Natalini and A. Bernardini 159

Differential equation for generalized Bernoulli polynomials
In this section, we prove the following theorem.
Theorem 3.1.The generalized Bernoulli polynomials In order to prove (3.1), we first derive a recurrence relation for Lemma 3.2.For any integral n ≥ 1, the following linear homogeneous recurrence relation for the generalized Bernoulli polynomials holds true: This relation, starting from n = 1, and taking into account the initial value B [m−1] 0 (x) = m!, allows a recursive formula for the generalized Bernoulli polynomials.
Proof.Differentiation of both sides of (2.1), with respect to t, yields and consequently (3.4) Recalling (2.1), the left-hand side of (3.4) becomes t n n! . (3.5) (x) could be chosen as an arbitrary constant), the following equation is obtained:

P. Natalini and A. Bernardini 161 and moreover
t n n! . (3.8) Then the conclusion immediately follows by the identity principle of power series, equating coefficients in the left-and right-hand side of the last equation (3.8).