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.
Hinweise
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in writing this paper, and read and approved the final manuscript.
1 Introduction, definitions and motivation
The 2-variable Hermite Kampé de Fériet polynomials (2VHKdFP) [1] are defined by
where is a continuous and infinitely differentiable function and it can be expanded in series as follows:
And it has an operational definition
(1.5)
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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 [3]
(1.6)
and consequently, the classical Bernoulli numbers can be obtained by the generating function
(1.7)
It is well known that
(1.8)
The generalized Bernoulli polynomials are defined by [4]
(1.9)
Clearly, the generalized Bernoulli numbers are given by
and
In 2005, Luo defined the Apostol-Bernoulli numbers and polynomials as follows:
(1.10)
(1.11)
The generalized Apostol-Bernoulli numbers and polynomials are defined by [5]
(1.12)
(1.13)
The 2D Bernoulli polynomials are defined [6] by the generating function
(1.14)
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:
(2.1)
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 2.1The following representation formulas hold true:
(2.2)
wheredenotes the Bernoulli numbers;
(2.3)
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:
(2.4)
we introduce the operator defined in such a way that
(2.5)
Thus, we can write
(2.6)
Within the context of such a formalism, we have
(2.7)
(2.8)
Making use of (2.1), (2.7), we can obtain
(2.9)
(2.10)
By using (1.6), (2.5), (2.8) and (2.9), we know that the polynomials satisfy the following partial differential equation:
(2.11)
with
Thus, the polynomials can be constructed according to the following operational rule:
(2.12)
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 [4]:
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 2.3We 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 2.4For any integral , the following linear homogeneous recurrence relation for the 2D Bernoulli polynomialsholds true:
(2.13)
wheredenotes 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:
(2.14)
where
(2.15)
We can also extend the polynomials to defined by
(2.16)
where
(2.17)
The properties of these polynomials can be obtained by using a similar method.
Acknowledgements
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.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in writing this paper, and read and approved the final manuscript.