New approach to twisted q-Bernoulli polynomials
© Kim et al.; licensee Springer. 2013
Received: 31 July 2013
Accepted: 5 September 2013
Published: 7 November 2013
By using the theory of basic hypergeometric series, we present some formulas for q-consecutive integers, and we find certain new identities for twisted q-Bernoulli polynomials and q-consecutive integers (Simsek in Adv. Stud. Contemp. Math. 16(2):251-278, 2008).
MSC: 11B68, 05A30.
where , , and x is a natural number. In this paper, we first study relations among q-consecutive integers, q-Bernoulli numbers, and q-Euler numbers.
with and .
In Section 2, we recall some necessary identities for basic hypergeometric series . Further, we obtain a generalization of Proposition 2.1, and accordingly, we obtain q-consecutive integers for . These new results are similar to the ones presented in some other studies [7–9] and .
Here, we note that these are related to .
with . Finally, we shall relate through Theorem 4.7 and Remark 4.14, q-Bernoulli polynomials with the third-order mock theta functions introduced by Ramanujan.
Throughout this paper, we adopt the following notations:
ω: the r th root of unity.
2 Identities of basic hypergeometric series and
Throughout this paper, q denotes a fixed complex number of absolute value less than 1, so that we may write , where τ is a complex number with a positive imaginary part. We use to denote . The partial product converges for all values of a, as may be easily seen from the absolute convergence of . Hence, if b is not one of the values , the coefficients are bounded, and the series (2.1) converges for all t inside the unit circle, and represents an analytic function therein. Hence, the function on the right-hand of (2.2) is regular in the domain , except for a simple pole at . Therefore, we obtain the continuation of F to a larger circle. Then, it is easy to apply (2.2) again to the continuation of F to the circle , and thus, we conclude that for , , the only possible singularities of F occur at the points (), which are simple poles in general. As a function of b, F is regular, except possibly at the simple poles (), provided that b and t do not have one of the singular values mentioned above. First, we derive Theorem 2.2 by generalizing the following proposition.
To prove this, we need some identities from .
Thus, we deduce the identity as desired. □
Next, we present alternative proofs of the following results of Kim  as an application of Theorem 2.2.
These are the q-analogues of .
- (2)If we put in (1), we have the first equality
- (3)It follows from (2) that(2.3)
Thus, by combining (2.3), (2.4), and (2.5), we reach the conclusion. □
3 q-Consecutive and q-analogue of Eulerian numbers and
We have studied the infinite sum with linear coefficients for q-numbers in the previous section. In this section, we consider the sum with quadratic coefficients, i.e., the following equation.
Since , we get the first equality.
by the definition of . Therefore, the last equality follows. □
The other cases will be studied in greater detail in the next section. This was previously proved by Schlosser by using Bailey’s terminating very-well-poised balanced transformation.
Then, the sum of formulas after setting and shows that our corollary is true. □
A similar result is in [, Lemma 2.1]. Thus, we get the results for and 2 as follows.
Comparing this with (3.1), we can prove (1).
as desired. □
for any positive integer k from [, Proposition 1]. The proof of Theorem 3.4 is obtained without the help of the summations above.
4 Difference equation and q-consecutive integer
As mentioned in Theorem 3.3, in this section, we study for more general cases l and its similar sum with q-binomial coefficients. In addition, we show the relations between these and twisted q-Bernoulli numbers. To this end, we need the following lemma.
Proof See Section 20, . □
Using this lemma, we generalize the identities considered in the previous two sections.
If l is 1 (respectively, 2), this would be the result of Corollary 2.4(1) (respectively, Lemma 3.1).
Therefore, if we let , we are done, which amounts to recovering Corollary 2.4, Lemma 3.1, and Theorem 3.2. □
Then, we get the following theorem.
- (2)Since , we have
Replacing t by in Theorem 4.4(1), we can deduce one of Simsek’s relations [, Proposition 3.1].
By (S2) and Theorem 4.5, we get a corollary.
Substituting , , and for l, n, and t in (4.3), respectively, we establish the last identities. □
As its immediate corollary, we have the following.
Moreover, we can deduce the following corollary, which is analogous to Theorem 4.4.
Thus, by substituting q for t, we conclude (3). □
Proof We see from (6.22) in  that . Thus, the proposition follows from Theorem 4.4(2). □
Henceforth, we concentrate on introduced in Section 1.
Here, we consider as 1.
Multiplying both sides by and replacing t by , we complete the proof. □
Further, it gives rise to a third-order mock theta function , where the explicit formula of was conjectured by both Andrews  and Dragonette , and later proved by Bringmann and Ono . Let (resp., ) be the infinite sum (resp., ). Then, we can find the coefficients of the infinite sum , because the formula for the partition function is already known and .
By utilizing our notations, we interpret them as follows.
with and .
The first author was supported by the National Institute for Mathematical Science (NIMS) grant funded by the Korean government (B21303), the second named author partially was supported by the NRF of Korea grant funded by the MISP (2013042157) and the corresponding author was supported by NRF 2012-0006901.
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