Symmetric identities for Carlitz’s q-Bernoulli numbers and polynomials
© He; licensee Springer. 2013
Received: 24 April 2013
Accepted: 30 July 2013
Published: 14 August 2013
In this paper, a further investigation for the Carlitz’s q-Bernoulli numbers and q-Bernoulli polynomials is performed, and several symmetric identities for these numbers and polynomials are established by applying elementary methods and techniques. It turns out that various known results are derived as special cases.
Clearly, . These numbers and polynomials play important roles in many different areas of mathematics such as number theory, combinatorics, special function and analysis, and numerous interesting properties for them have been explored, see, for example, [1, 2].
with and .
where with and .
The aim of the present paper is to perform a further investigation on the Carlitz’s q-Bernoulli numbers and q-Bernoulli polynomials. By applying elementary methods and techniques, we establish some symmetric identities for these numbers and polynomials, by virtue of which, various known results are derived as special cases.
2 The restatement of results
In this section, a further investigation for the Carlitz’s q-Bernoulli numbers and q-Bernoulli polynomials is performed, and several symmetric identities for them are established. We firstly state the following result for the q-zeta function.
Thus, equating (2.5) and (2.6) gives the desired result. □
respectively, see, for example, .
Now, we are in the position to give some similar symmetric identities for the Carlitz’s q-Bernoulli numbers and q-Bernoulli polynomials.
Thus, the desired result follows by applying (2.14) and (2.16) to Theorem 2.1. □
Based on the observation of Theorem 2.2, we have the following.
Thus, by applying Theorem 2.2 to (2.23), we complete the proof of Theorem 2.3. □
where is called as the sums of powers. It is worth mentioning that the case , in (2.24) can be used to give the proofs of the famous von Staudt-Clausen, Frobenius, Ramanujan, etc.-type theorems, see  for details. For some generalization of (2.24) in other directions, the interested readers may consult [13–16].
We finally present another type symmetric identities for the Carlitz’s q-Bernoulli numbers and q-Bernoulli polynomials.
which together with implies the desired result. We are done. □
In fact, there exists a similar symmetric identity to (2.33), which is as follows.
Thus, putting (2.37) and (2.38) to the right-hand side of (2.36), we are given the desired result, and this completes the proof. □
The author expresses his gratitude to the referees for their helpful comments and suggestions. This work is supported by the Foundation for Fostering Talents in Kunming University of Science and Technology (Grant No. KKSY201307047) and the National Natural Science Foundation of China (Grant No. 11071194).
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