Some generalizations of 2D Bernoulli polynomials
Journal of Inequalities and Applications volume 2013, Article number: 110 (2013)
As a generalization of 2D Bernoulli polynomials, neo-Bernoulli polynomials are introduced from a point of view involving the use of nonexponential generating functions. Their relevant recurrence relations, the differential equations satisfied by them and some other properties are obtained. Especially, we obtain the relationships between them and neo-Hermite polynomials. We also study some other generalizations of 2D Bernoulli polynomials.
MSC:11B68, 33C99, 34A35.
1 Introduction, definitions and motivation
The 2-variable Hermite Kampé de Fériet polynomials (2VHKdFP)  are defined by
with the generating function
The neo-Hermite polynomials  are defined by
with the generating function
where is a continuous and infinitely differentiable function and it can be expanded in series as follows:
And it has an operational definition
As it is well-known, the HKdF polynomials are generated by (1.4) when reduces to an exponential function.
The classical Bernoulli polynomials are defined by 
and consequently, the classical Bernoulli numbers can be obtained by the generating function
It is well known that
The generalized Bernoulli polynomials are defined by 
Clearly, the generalized Bernoulli numbers are given by
In 2005, Luo defined the Apostol-Bernoulli numbers and polynomials as follows:
The generalized Apostol-Bernoulli numbers and polynomials are defined by 
The 2D Bernoulli polynomials are defined  by the generating function
In this paper, we will give some generalizations of the 2D Bernoulli polynomials. And some properties of them will be given.
2 The Bernoulli polynomials from a general point of view
We consider a continuous and infinitely differentiable function and associate it with the following generating function:
As it is well known, the 2D Bernoulli polynomials are generated by (2.1) when reduces to an exponential function.
It is possible to find an explicit form of the polynomials in terms of the neo-Hermite polynomials defined by (1.4).
Theorem 2.1 The following representation formulas hold true:
where denotes the Bernoulli numbers;
Proof Equation (2.2) is obtained starting from the generating function (2.1) by using the Cauchy product of the series expansion (1.4) and (1.7), and then using the identity principle of power series.
Equation (2.3) is obtained in the same way, starting from the equation
By assuming that can be expanded in series as follows:
we introduce the operator defined in such a way that
Thus, we can write
Within the context of such a formalism, we have
Making use of (2.1), (2.7), we can obtain
By using (1.6), (2.5), (2.8) and (2.9), we know that the polynomials satisfy the following partial differential equation:
Thus, the polynomials can be constructed according to the following operational rule:
Then we can derive some relations of the polynomials by using the relations of the Bernoulli polynomials along with the operational rule (2.12).
Remark 2.2 Upon the proof, the series (2.1) is the absolute convergence and uniformly convergence, thus we can differentiate the both sides of (2.1) with respect to the variable x or y.
We recall the following functional equations involving the Bernoulli polynomials :
Now, substituting x with in the above equations and performing the operation on the results, then by using (1.5), (2.5) and (2.11), we get the following identities involving .
Theorem 2.3 We have the following relationships between the neo-Hermite polynomials and the neo-Bernoulli polynomials:
A recurrence relation for the polynomials is given by the following theorem.
Theorem 2.4 For any integral , the following linear homogeneous recurrence relation for the 2D Bernoulli polynomials holds true:
where denotes the Bernoulli numbers.
Proof Differentiating both sides of Eq. (2.1) with respect to t, recalling the generating functions (1.7) of the Bernoulli numbers, and using (2.7) and some elementary algebra and the identity principle of power series, we get recursion (2.13) easily. □
A natural further extension is obtained by considering the following case:
We can also extend the polynomials to defined by
The properties of these polynomials can be obtained by using a similar method.
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Dedicated to Professor Hari M Srivastava.
The present investigation was supported by the National Natural Science Foundation of China under Grant 11226281, Fund of Science and Innovation of Yangzhou University, China under Grant 2012CXJ005, Research Project of Science and Technology of Chongqing Education Commission, China under Grant KJ120625 and Fund of Chongqing Normal University, China under Grant 10XLR017 and 2011XLZ07.
The authors declare that they have no competing interests.
All authors contributed equally in writing this paper, and read and approved the final manuscript.
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Lu, DQ., Luo, QM. Some generalizations of 2D Bernoulli polynomials. J Inequal Appl 2013, 110 (2013). https://doi.org/10.1186/1029-242X-2013-110