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Some results for the q-Bernoulli, q-Euler numbers and polynomials
Advances in Difference Equations volume 2011, Article number: 68 (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 .
The Bernoulli polynomials, denoted by B n (x), are defined as
where B k are the Bernoulli numbers given by the coefficients in the power series
From the above definition, we see B k 's are all rational numbers. Since is an even function (i.e., invariant under x ↦ - x), we see that B k = 0 for any odd integer k not smaller than 3. It is well known that the Bernoulli numbers can also be expressed as follows
The Euler polynomial E k (x) is defined by (see [, p. 25]):
which holds for all nonnegative integers k and all real x, and which was obtained by Raabe  in 1851. Setting x = 1/2 and normalizing by 2 k gives the Euler numbers
where E 0 = 1, E 2 = -1, E 4 = 5, E 6 = -61,.... Therefore, E k ≠ E k (0), in fact ([, p. 374 (2.1)])
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.
Let UD(ℤ p , ℂ p ) denote the space of all uniformly (or strictly) differentiable ℂ p -valued functions on ℤ p . The p-adic q-integral of a function f ∈ UD(ℤ p ) on ℤ p is defined by
where [x] q = (1 - q x )/(1 - q), and the limit taken in the p-adic sense. Note that
From (2.1), we have the fermionic p-adic integral on ℤ p as follows:
In particular, setting in (2.3) and in (2.4), respectively, we get the following formulas for the p-adic q-Bernoulli and p-adic q-Euler numbers, respectively, if q ∈ ℂ p with 0 < |q - 1| p < 1 as follows
For |q - 1| p < 1 and z ∈ ℤ p , we have
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 ).
By (2.3) and (2.7), we obtain
since the series log converges at |x| p < 1. Similarly, by (2.4), we obtain (see [, p. 4, (2.10)])
From (2.5), (2.6), (2.8) and (2.9), we obtain the following explicit formulas of B k (q) and E k (q):
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).
We now move on to the p-adic q-Bernoulli and p-adic q-Euler polynomials. The p-adic q-Bernoulli and p-adic q-Euler polynomials in q x are defined by means of the bosonic and the fermionic p-adic integral on ℤ p :
where q ∈ ℂ p with 0 < |q - 1| p < 1 and x ∈ ℤ p , respectively. We will rewrite the above equations in a slightly different way. By (2.5), (2.6), and (2.12), after some elementary calculations, we get
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, since and
where E k are Euler numbers (see (1.5) above).
Lemma 2.2 (Addition theorem).
Proof. Applying the relationship to (2.14) for x α x + y, we have
Similarly, the first identity follows.□
Remark 2.3. From (2.12), we obtain the not completely trivial identities
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.
Lemma 2.4. Let n be any positive integer. Then
Proof. Use Lemma 2.2, the proof can be obtained by the similar way to [, Lemma 2.3]. □
We note here that similar expressions to those of Lemma 2.4 are given by Luo [, Lemma 2.3]. Obviously, Lemma 2.4 are the q-analogues of
We can now obtain the multiplication formulas by using p-adic integrals.
From (2.3), we see that
is equivalent to
If we put x = 0 in (2.18) and use (2.13), we find easily that
Obviously, Equation (2.19) is the q-analogue of
which is true for any positive integer k and any positive integer n > 1 (see [, (2)]).
From (2.4), we see that
By (2.12) and (2.20), we find easily that
From (2.18) and (2.21), we can obtain Proposition 2.5 below.
Proposition 2.5 (Multiplication formulas). Let n be any positive integer. Then
3. Construction generating functions of q-Bernoulli, q-Euler numbers, and polynomials
In the complex case, we shall explicitly determine the generating function F q (t) of q-Bernoulli numbers and the generating function G q (t) of q-Euler numbers:
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.
Proof. Combining (2.10) and (3.1), F q (t) may be written as
Here, the term with i = 0 is understood to be 1/log q (the limiting value of the summand in the limit i → 0). Specifically, by making use of the following well-known binomial identity
Thus, we find that
Next, by (2.11) and (3.1), we obtain the result
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.
From (2.13)and (2.14), we define the q-Bernoulli and q-Euler polynomials by
Hence, we have
Proof. From (3.1) and (3.2), we note that
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. □
Corollary 3.5 (Difference equations).
Proof. By applying (3.2) and Lemma 3.3, we obtain (3.4)
By comparing the coefficients of both sides of (3.4), we have B 0(x, q) = 1 and
Similarly we prove the second part by (3.3) and Lemma 3.4. This proof is complete.
From Lemma 2.2 and Corollary 3.5, we obtain for any integer k ≥ 0,
which are the q-analogues of the following familiar expansions (see, e.g., [, p. 9]):
Corollary 3.6 (Difference equations). Let k ≥ 0 and n ≥ 1. Then
Proof. Use Lemma 2.4 and Corollary 3.5, the proof can be obtained by the similar way to [, Lemma 2.4]. □
Letting n = 1, Corollary 3.6 reduces to Corollary 3.5. Clearly, the above difference formulas in Corollary 3.6 become the following difference formulas when q → 1:
respectively (see [, (2.22), (2.23)]). If we now let n = 1 in (3.6) and (3.7), we get the ordinary difference formulas
for k ≥ 0.
In Corollary 3.5, let x = 0. We arrive at the following proposition.
Proof. The first identity follows from (2.13). To see the second identity, setting x = 0 and x = 1 in (2.14) we have
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)].
(2). Letting q → 1 in Proposition 3.7, the first identity is the corresponding classical formulas in [, (1.2)]:
and the second identity is the corresponding classical formulas in [, (1.1)]:
4. q-analogues of Riemann's ζ-functions, the Hurwitz ζ-functions and the Didichlet's L-functions
Now, by evaluating the k th derivative of both sides of Lemma 3.1 at t = 0, we obtain the following
for k ≥ 0.
Definition 4.1 (q-analogues of the Riemann's ζ-functions). For s ∈ ≤, define
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 ≤.
Also, we have
(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.
Proposition 4.2. For k ≥ 1, we have
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.
Definition 4.3 (q-analogues of the Hurwitz ζ-functions). For s ∈ ≤ and 0 < x ≤ 1, define
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.
Proposition 4.4. For k ≥ 1, we have
Proof. From Lemma 3.3 and Definition 4.3, we have
for k ≥ 1. We obtain the desired result by (3.2). Similarly the second form follows by Lemma 3.4 and (3.3). □
Proposition 4.5. Let d be any positive integer. Then
Proof. Substituting m = nd + i with n = 0, 1,... and i = 0,..., d - 1 into Lemma 3.3, we have
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:
Corollary 4.6. Let d and k be any positive integer. Then
Let χ be a primitive Dirichlet character of conductor f ∈ ℕ. We define the generating function F q,χ (x, t) and G q,χ (x, t) of the generalized q-Bernoulli and q-Euler polynomials as follows:
where B k,χ (x, q) and E k,χ (x, q) are the generalized q-Bernoulli and q-Euler polynomials, respectively. Clearly (4.4) and (4.5) are equal to
respectively. As q → 1 in (4.6) and (4.7), we have F q,χ (t, x) → F χ (t, x) and G q,χ (t, x) → G χ (t, x), where F χ (t, x) and G χ (t, x) are the usual generating function of generalized Bernoulli and Euler numbers, respectively, which are defined as follows :
From (3.2), (3.3), (4.4) and (4.5), we can easily see that
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.
Definition 4.7 (q-analogues of the Dirichlet's L-functions). For s ∈ ℂ and 0 < x ≤ 1,
Similarly, we can compute the values of L q (s, x, χ) at non-positive integers.
Theorem 4.8. For k ≥ 1, we have
Proof. Using Lemma 3.3 and (4.4), we obtain
where we use and . Therefore, we obtain
Hence for k ≥ 1
Similarly the second identity follows. This completes the proof. □
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This study was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (2011-0001184).
The authors declare that they have no competing interests.
The authors have equal contributions to each part of this paper. All the authors read and approved the final manuscript.
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Kim, D., Kim, MS. Some results for the q-Bernoulli, q-Euler numbers and polynomials. Adv Differ Equ 2011, 68 (2011). https://doi.org/10.1186/1687-1847-2011-68
- Bosonic p-adic integrals
- Fermionic p-adic integrals
- q-Bernoulli polynomials
- q-Euler polynomials
- generating functions
- q-analogues of ζ-type functions
- q-analogues of the Dirichlet's L-functions