- Open Access
Some results for the q-Bernoulli, q-Euler numbers and polynomials
© Kim and Kim; licensee Springer. 2011
Received: 2 September 2011
Accepted: 23 December 2011
Published: 23 December 2011
The q-analogues of many well known formulas are derived by using several results of q-Bernoulli, q-Euler numbers and polynomials. The q-analogues of ζ-type functions are given by using generating functions of q-Bernoulli, q-Euler numbers and polynomials. Finally, their values at non-positive integers are also been computed.
2010 Mathematics Subject Classification: 11B68; 11S40; 11S80.
Carlitz [1, 2] introduced q-analogues of the Bernoulli numbers and polynomials. From that time on these and other related subjects have been studied by various authors (see, e.g., [3–10]). Many recent studies on q-analogue of the Bernoulli, Euler numbers, and polynomials can be found in Choi et al. , Kamano , Kim [5, 6, 12], Luo , Satoh , Simsek [13, 14] and Tsumura .
For a fixed prime p, ℤ p , ℚ p , and ℂ p denote the ring of p-adic integers, the field of p-adic numbers, and the completion of the algebraic closure of ℚ p , respectively. Let | · | p be the p-adic norm on ℚ with |p| p = p -1. For convenience, | · | p will also be used to denote the extended valuation on ℂ p .
where B k are Bernoulli numbers. The Euler numbers and polynomials (so-named by Scherk in 1825) appear in Euler's famous book, Institutiones Calculi Differentialis (1755, pp. 487-491 and p. 522).
In this article, we derive q-analogues of many well known formulas by using several results of q-Bernoulli, q-Euler numbers, and polynomials. By using generating functions of q-Bernoulli, q-Euler numbers, and polynomials, we also present the q-analogues of ζ-type functions. Finally, we compute their values at non-positive integers.
This article is organized as follows.
In Section 2, we recall definitions and some properties for the q-Bernoulli, Euler numbers, and polynomials related to the bosonic and the fermionic p-adic integral on ℤ p .
In Section 3, we obtain the generating functions of the q-Bernoulli, q-Euler numbers, and polynomials. We shall provide some basic formulas for the q-Bernoulli and q-Euler polynomials which will be used to prove the main results of this article.
In Section 4, we construct the q-analogue of the Riemann's ζ-functions, the Hurwitz ζ-functions, and the Dirichlet's L-functions. We prove that the value of their functions at non-positive integers can be represented by the q-Bernoulli, q-Euler numbers, and polynomials.
2. q-Bernoulli, q-Euler numbers and polynomials related to the Bosonic and the Fermionic p-adic integral on ℤ p
In this section, we provide some basic formulas for p-adic q-Bernoulli, p-adic q-Euler numbers and polynomials which will be used to prove the main results of this article.
where i ∈ ℤ. We easily see that if |q - 1| p < 1, then q x = 1 for x ≠ 0 if and only if q is a root of unity of order p N and x ∈ p N ℤ p (see ).
where k ≥ 0 and log is the p-adic logarithm. Note that in (2.10), the term with i = 0 is understood to be 1/log q (the limiting value of the summand in the limit i → 0).
where E k are Euler numbers (see (1.5) above).
Similarly, the first identity follows.□
where q ∈ ℂ p tends to 1 in |q - 1| p < 1. Here B i (x) and E i (x) denote the classical Bernoulli and Euler polynomials, see [17, 15] and see also the references cited in each of these earlier works.
Proof. Use Lemma 2.2, the proof can be obtained by the similar way to [, Lemma 2.3]. □
We can now obtain the multiplication formulas by using p-adic integrals.
which is true for any positive integer k and any positive integer n > 1 (see [, (2)]).
From (2.18) and (2.21), we can obtain Proposition 2.5 below.
3. Construction generating functions of q-Bernoulli, q-Euler numbers, and polynomials
where the symbol B k (q) and E k (q) are interpreted to mean that (B(q)) k and (E(q)) k must be replaced by B k (q) and E k (q) when we expanded the one on the right, respectively.
This completes the proof. □
Remark 3.2. The remarkable point is that the series on the right-hand side of Lemma 3.1 is uniformly convergent in the wider sense.
Hence, we have
The second identity leads at once to Lemma 3.1. Hence, the lemma follows. □
Proof. By similar method of Lemma 3.3, we prove this lemma by (3.1), (3.3), and Lemma 3.1. □
Similarly we prove the second part by (3.3) and Lemma 3.4. This proof is complete.
Proof. Use Lemma 2.4 and Corollary 3.5, the proof can be obtained by the similar way to [, Lemma 2.4]. □
for k ≥ 0.
In Corollary 3.5, let x = 0. We arrive at the following proposition.
This proof is complete. □
Remark 3.8. (1). We note here that quite similar expressions to the first identity of Proposition 3.7 are given by Kamano [, Proposition 2.4], Rim et al. [, Theorem 2.7] and Tsumura [, (1)].
4. q-analogues of Riemann's ζ-functions, the Hurwitz ζ-functions and the Didichlet's L-functions
for k ≥ 0.
Note that ζ q (s) is a meromorphic function on ≤ with only one simple pole at s = 1 and ζ q ,E(s) is a analytic function on ≤.
(In [, p. 1070], our ζ E (s) is denote ϕ(s).)
The values of ζ q (s) and ζ q ,E(s) at non-positive integers are obtained by the following proposition.
Proof. It is clear by (4.1) and (4.2). □
We can investigate the generating functions F q (t, x) and G q (t, x) by using a method similar to the method used to treat the q-analogues of Riemann's ζ-functions in Definition 4.1.
Note that ζ q (s, x) is a meromorphic function on ≤ with only one simple pole at s = 1 and ζ q ,E(s, x) is a analytic function on ≤.
The values of ζ q (s, x) and ζ q ,E(s, x) at non-positive integers are obtained by the following proposition.
for k ≥ 1. We obtain the desired result by (3.2). Similarly the second form follows by Lemma 3.4 and (3.3). □
where we use . So we have the first form. Similarly the second form follows by Lemma 3.4. □
From (3.2), (3.3), Propositions 4.4 and 4.5, we obtain the following:
By using the definitions of ζ q (s, x) and ζq,E (s, x), we can define the q-analogues of Dirichlet's L-function.
Similarly, we can compute the values of L q (s, x, χ) at non-positive integers.
Similarly the second identity follows. This completes the proof. □
This study was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (2011-0001184).
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