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# Some generalizations of 2D Bernoulli polynomials

- Da-Qian Lu
^{1}and - Qiu-Ming Luo
^{2}Email author

**2013**:110

https://doi.org/10.1186/1029-242X-2013-110

© Lu and Luo; licensee Springer. 2013

**Received:**10 December 2012**Accepted:**24 February 2013**Published:**19 March 2013

## Abstract

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.

## Keywords

- 2D Bernoulli polynomials
- neo-Bernoulli polynomials
- generating function
- recurrence relation
- umbral calculus

## 1 Introduction, definitions and motivation

As it is well-known, the HKdF polynomials are generated by (1.4) when $f(x)$ reduces to an exponential 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

As it is well known, the 2D Bernoulli polynomials ${B}_{n}^{(2)}(x,y)$ are generated by (2.1) when $f(x)$ reduces to an exponential function.

It is possible to find an explicit form of the polynomials ${b}_{n}^{(2)}(x,y)$ in terms of the neo-Hermite polynomials ${\varphi}_{n}(x,y)$ defined by (1.4).

**Theorem 2.1**

*The following representation formulas hold true*:

*where*${B}_{k}$

*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.

□

Then we can derive some relations of the polynomials ${b}_{n}^{(2)}(x,y)$ by using the relations of the Bernoulli polynomials ${B}_{n}(x)$ 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*.

Now, substituting *x* with $\stackrel{\u02c6}{f}x$ in the above equations and performing the operation $exp[({\stackrel{\u02c6}{f}}^{-1}y\frac{{\partial}^{2}}{\partial {x}^{2}})]$ on the results, then by using (1.5), (2.5) and (2.11), we get the following identities involving ${b}_{n}^{(2)}(x,y)$.

**Theorem 2.3**

*We have the following relationships between the neo*-

*Hermite polynomials and the neo*-

*Bernoulli polynomials*:

A recurrence relation for the polynomials ${b}_{n}^{(2)}(x,y)$ is given by the following theorem.

**Theorem 2.4**

*For any integral*$n\ge 1$,

*the following linear homogeneous recurrence relation for the*2D

*Bernoulli polynomials*${b}_{n}^{(2)}(x,y)$

*holds true*:

*where* ${B}_{k}$ *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. □

The properties of these polynomials can be obtained by using a similar method.

## Declarations

### 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.

## Authors’ Affiliations

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## Copyright

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