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Symmetric identities for Carlitz’s q-Bernoulli numbers and polynomials
Advances in Difference Equations volume 2013, Article number: 246 (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.
The classical Bernoulli numbers and Bernoulli polynomials are usually defined by the following generating functions
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].
In the present paper, we will be concerned with the Carlitz’s q-Bernoulli numbers and q-Bernoulli polynomials. Throughout this paper, it is supposed that with and ℂ being a complex number field. For , the q-number is defined by (see )
We now recall the q-Bernoulli numbers and q-Bernoulli polynomials , which were introduced by Carlitz , as follows
with the usual convention about replacing by . Since the above Carlitz’s q-Bernoulli numbers and q-Bernoulli polynomials appeared, different properties of the q-Bernoulli numbers and q-Bernoulli polynomials have been well studied by many authors, see  for a good introduction. In fact, the Carlitz’s q-Bernoulli numbers and q-Bernoulli polynomials can be defined by the following generating functions (see [6, 7])
From (1.6) and (1.7) one can easily obtain
If the left-hand side of (1.7) is denoted by , then by the Mellin transform,
with and .
Based on the observation on (1.9), Ryoo et al.  extended the classical Hurwitz zeta function
to the following q-zeta function
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.
Theorem 2.1 Let with . Then for any positive integers a, b,
Proof By substituting for x in (1.11), we have
Note that for any and positive integer n, . Hence, by replacing q by in (2.2), we derive
Since for any non-negative integer n and a positive integer b, there exist unique non-negative integers r and i such that with . So, the above identity (2.3) can be rewritten as
It follows from (2.4) that
In the same way,
Thus, equating (2.5) and (2.6) gives the desired result. □
We next discuss some special cases of Theorem 2.1. Setting in Theorem 2.1, we have the following distribution formula for the q-zeta function
In particular, the case in (2.7) gives the duplication formula for the q-zeta function
Letting in (2.7) and (2.8) leads to the familiar distribution formula for the classical Hurwitz zeta function
and the duplication formula for the classical Hurwitz zeta function
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.
Theorem 2.2 Let a, b be positive integers. Then for any non-negative integer n,
Proof By applying the exponential series to the generating function of the q-Bernoulli polynomials, we have
Comparison of the coefficients of in (2.12) yields
So, from (1.11), (2.13) and the analytic continuation of , one can easily obtain that for any non-negative integer n,
we get the symmetric distribution for the q-Bernoulli polynomials
Thus, the desired result follows by applying (2.14) and (2.16) to Theorem 2.1. □
It follows that we show some special cases of Theorem 2.2. Setting in Theorem 2.2, we derive the multiplication theorem for the q-Bernoulli polynomials due to Carlitz 
which is a q-analogue of Raabe’s multiplication theorem for the classical Bernoulli polynomials (see, e.g., ). Letting in Theorem 2.2, one can immediately obtain the generalized multiplication theorem for the classical Bernoulli polynomials (see, e.g., [9–11])
Based on the observation of Theorem 2.2, we have the following.
Theorem 2.3 Let a, b be positive integers. Then for any non-negative integer n,
Proof Since for any , then (1.7) can be rewritten as
Again, applying the exponential series and (1.7) to the left-hand side of (2.20), with help of the Cauchy product, we get
Hence, comparison of the coefficients of in (2.21) gives the addition theorem for the q-Bernoulli polynomials
Since for any and a positive integer a, by (2.22), we obtain
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.
Theorem 2.4 Let m, n be any non-negative integers. Then
Proof Observe that
It follows from (2.26) that
We next consider the left-hand side of (2.27). In light of (1.7), we have
Applying to the summation on the right-hand side of (2.28), and then changing the order of summation, we get
Hence, putting (2.29) into (2.28), and then using the exponential series, with the help of the Cauchy product, we obtain
Similarly, we have
Thus, by equating (2.30) and (2.31) and comparing the coefficients of , we get
which together with implies the desired result. We are done. □
It becomes obvious that (2.22) is a special case of Theorem 2.4 by setting and replacing m by n. As another special case, in view of (2.16), we discover that for any non-negative integers m, n,
In particular, the case of in (2.33) is an q-analogue of the formula due to Gessel 
In fact, there exists a similar symmetric identity to (2.33), which is as follows.
Theorem 2.5 Let m, n be any non-negative integers. Then
Proof Since for any non-negative integers k and m, then
It follows from (2.33) that
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. □
Noticing that the case in Theorem 2.5 above can be regarded as a q-analogue of the following formula
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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).
The author declares that they have no competing interests.
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He, Y. Symmetric identities for Carlitz’s q-Bernoulli numbers and polynomials. Adv Differ Equ 2013, 246 (2013). https://doi.org/10.1186/1687-1847-2013-246
- Bernoulli numbers and polynomials
- q-Bernoulli numbers and polynomials
- combinatorial identities