Some generalizations of 2D Bernoulli polynomials
© Lu and Luo; licensee Springer. 2013
Received: 10 December 2012
Accepted: 24 February 2013
Published: 19 March 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.
Keywords2D 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 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 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).
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 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.
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 .
A recurrence relation for the polynomials is given by the following theorem.
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. □
The properties of these polynomials can be obtained by using a similar method.
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.
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