(P, Q)-HURWITZ ZETA FUNCTION 1. Introduction Tmany Mathematicians

J. Appl. Math. & Informatics Vol. 37(2019), No. 1 - 2, pp. 97 - 103 https://doi.org/10.14317/jami.2019.097

SOME IDENTITIES FOR (p, q)-HURWITZ ZETA FUNCTION†

CHEON SEOUNG RYOO

Abstract. In this paper, we give some interesting symmetric identities of the (p, q)-Hurwitz zeta function. We also give some new interesting properties, explicit formulas, a connection with (p, q)-Bernoulli numbers and polynomials.

AMS Mathematics Subject Classiﬁcation : 11B68, 11S40, 11S80. Key words and phrases : Bernoulli numbers and polynomials, q-Bernoulli numbers and polynomials, (p, q)-analogue of Bernoulli numbers and polynomials, (p, q)-Hurwitz zeta function.

1. Introduction TMany mathematicians have studied special polynomials such as Bernoulli numbers and polynomials, tangent numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials (see [1-11]). The Bernoulli numbers and polynomials have been extensively worked in many diﬀerent contexts in such branches of mathematics as, for instance, el- ementary number theory, complex analytic number theory, q-adic analytic number theory, diﬀerential topology, and quantum physics. Therefore, the study of extended Bernoulli numbers and polynomials is very meaningful and interesting. Using q-numbers, the study of q-Bernoulli numbers and polynomials was done by some mathematicians(see [4, 5, 6, 7, 11]). Recently, we introduced (p, q)- Bernoulli numbers and polynomials, which extend this number and polynomials, using (p, q)-numbers(see [1, 8, 9, 10]). In this paper, we obtain symmetric properties of the (p, q)-Hurwitz zeta function associated with (p, q)-Bernoulli polynomials. As applications of these properties, we study some interesting identities for the (p, q)-Bernoulli polynomials

Received April 20, 2018. Revised July 22, 2018. Accepted January 4, 2019. †This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2017R1A2B4006092). c 2019 Korean SIGCAM and KSCAM. 97 98 Cheon Seoung Ryoo and numbers. In this paper, we always use the following notations. • N = {1, 2, 3,...} denotes the set of natural numbers. − • Z0 = {0, −1, −2, −3,...} denotes the set of nonpositive integers. • C denotes the set of complex numbers.

We remember that the classical Bernoulli numbers Bn and Bernoulli polynomials Bn(x) are defined by the following generating functions ∞ t X tn = B , (|t| < 2π). (1) et − 1 n n! n=0 and ∞ t X tn ext = B (x) , (|t| < 2π). (2) et − 1 n n! n=0 respectively. The q-numbers defined by 1 − qn [n] = = 1 + q + q2 + ··· + qn−3 + qn−2 + qn−1. q 1 − q The (p, q)-number is defined as pn − qn [n] = = pn−1 + pn−2q + pn−3q2 + ··· + p2qn−3 + pqn−2 + qn−1. p,q p − q It is clear that (p, q)-number contains symmetric property, and this number is q-number when p = 1. In particular, we can see limq→1[n]p,q = n with n−1 p = 1. Since [n] = p [n] q , we note that p-numbers and (p, q)-numbers are p,q p q different. That is, we can not have (p, q)-number by assigning q to p in the definition q-number. Using (p, q)-numbers, Bernoulli numbers and polynomials, Euler numbers and polynomials, tangent numbers and polynomials, and various mathematical theories are being studied(see [1, 8, 9, 10]). In [8], we define the Carlitz’s type (p, q)-Bernoulli polynomials and numbers, which generalized the previously known numbers and polynomials, including the Bernoulli numbers and polynomials and the Carlitz’s type q-Bernoulli numbers and polynomials. We begin by recalling here the Carlitz’s type (p, q)-Bernoulli numbers and polynomials(see [8]). Definition 1.1. For 0 < q < p ≤ 1, the Carlitz’s type (p, q)-Bernoulli numbers Bn,p,q and polynomials Bn,p,q(x) are defined by means of the generating functions ∞ ∞ X tn X B = −t qme[m]p,q t, (3) n,p,q n! n=0 m=0 and ∞ ∞ X tn X B (x) = −t qme[m+x]p,q t, (4) n,p,q n! n=0 m=0 respectively. Some identities for (p, q)-Hurwitz zeta function 99

Many kinds of of generalizations of the Riemann zeta function, Dirichlet eta function, and Hurwitz zeta function have been presented in the literature(see [1-11]). Based on this idea, we generalize the q-Hurwitz zeta function. It follows that we deﬁne the following (p, q)-Hurwitz zeta function(see [8]).

Deﬁnition 1.2. We deﬁne the (p, q)-Hurwitz zeta function for s ∈ C with − Re(s) > 0 and x∈ / Z0 by ∞ X qn ζ (s, x) = . (5) p,q [n + x]s n=0 p,q

Note that ζp,q(s, x) is a meromorphic function on C. Obverse that, if p = 1 and q → 1, then ζp,q(s, x) = ζ(s, x) which is the Hurwitz zeta function(see [2, 4, 7, 8]). The relationship between ζp,q(s, x) and Bk,p,q(x) is given by the following theorem.

Theorem 1.3. For k ∈ N, we have B (x) ζ (1 − k, x) = − k,p,q . (6) p,q k Deﬁnition 1.4. For s ∈ C with Re(s) > 1, we deﬁne (p, q)-Riemann zeta function ζp,q(s) by ∞ X qn ζ (s) = . (7) p,q [n]s n=1 p,q

Note that ζp,q(s) is a meromorphic function on C. Note that, if p = 1, q → 1, then ζp,q(s) = ζ(s) which is the Riemann zeta function(see [2, 4, 7, 8]). Relation between ζp,q(s) and Bk,p,q is given by the following theorem.

Theorem 1.5. For k ∈ N, we have B ζ (1 − k) = − k,p,q . p,q k

2. Symmetric properties about (p, q)-Hurwitz zeta function In this section, we are going to obtain the main results of (p, q)-Hurwitz zeta function. We also establish some interesting symmetric identities for (p, q)- Bernoulli polynomials by using (p, q)-Hurwitz zeta function. w1i For any x, y ∈ , observe that [xy]p,q = [x] y y [y]p,q. By substitute w1x+ C p ,q w2 for x in Deﬁnition 1.2, replace p by pw2 and replace q by qw2 , respectively, we derive ∞ w2n w1i X q ζpw2 ,qw2 s, w1x + = w1i s w2 [w1x + + n] w2 w2 n=0 w2 p ,q ∞ X qw2n = [w ]s . 2 p,q [w w x + w i + w n]s n=0 1 2 1 2 p,q 100 Cheon Seoung Ryoo

Since for any non-negative integer m and positive integer w1, there exist unique non-negative integer r such that m = w1r + j with 0 ≤ j ≤ w1 − 1. Hence, this can be written as ∞ w2(w1r+j) w1i s X q w w ζp 2 ,q 2 s, w1x + = [w2]p,q s w2 [w2(w1r + j) + w1w2x + w1i]p,q w1r+j=0 0≤j≤w1−1 w −1 ∞ X1 X qw2(w1r+j) = [w ]s . 2 p,q [w w (r + x) + w i + w j]s j=0 r=0 1 2 1 2 p,q It follows from the above equation that

w2−1 w i s X w1i 1 [w ] q ζ w2 w2 s, w x + 1 p,q p ,q 1 w i=0 2 (8) w −1 w −1 ∞ X2 X1 X qw1w2r+w1i+w2j = [w ]s [w ]s . 1 p,q 2 p,q [w w (r + x) + w i + w j]s i=0 j=0 r=0 1 2 1 2 p,q From the similar method, we can have that ∞ w1n w2j X q ζpw1 ,qw1 s, w2x + = w2j s w1 [w2x + + n] w1 w1 n=0 w1 p ,q ∞ X qw1n = [w ]s . 1 p,q [w w x + w j + w n]s n=0 1 2 2 1 p,q After some calculations in the above, we have

w1−1 w j s X w2j (h) 2 [w ] q ζ w w s, w x + 2 p,q p 1 ,q 1 2 w j=0 1 (9) w −1 w −1 ∞ X2 X1 X qw1w2r+w1i+w2j = [w ]s [w ]s . 1 p,q 2 p,q [w w (r + x) + w i + w j]s i=0 j=0 r=0 1 2 1 2 p,q Thus, we have the following theorem from (8) and (9).

Theorem 2.1. Let s ∈ C with Re(s) > 0, w1, and w2 be positive integers. Then one has w2−1 w i s X w1i 1 [w ] q ζ w2 w2 s, w x + 1 p,q p ,q 1 w i=0 2 w1−1 w j s X w2j 2 = [w ] q ζ w1 w1 s, w x + . 2 p,q p ,q 2 w j=0 1 Next, we observe some special cases of Theorem 2.1. In Theorem 2.1, we get the following distribution formula for the (p, q)-Hurwitz zeta function. Some identities for (p, q)-Hurwitz zeta function 101

Corollary 2.2. Let w2 = 1 in Theorem 2.1. Then we get

w1−1 −s X j x + j ζ (s, x) = [w ] q ζ w1 w1 s, . (10) p,q 1 p,q p ,q w j=0 1

In particular, the case w1 = 2 in (10) gives the duplication formula for the (p, q)- zeta function.

Corollary 2.3. Let w1 = 2 in Corollary 2.2. Then we have

x x + 1 s ζ 2 2 s, + qζ 2 2 s, = [2] ζ (s, x). (11) p ,q 2 p ,q 2 p,q p,q Letting p = 1 and q → 1 in (10) and (11) leads to the distribution formula for the classical Hurwitz zeta function m−1 1 X x + j ζ(s, x) = ζ s, , ms m j=0 and the duplication formula for the classical Hurwitz zeta function x x + 1 ζ s, + ζ s, = 2sζ(s, x), 2 2 respectively(see, e.q., [2, 4, 5, 6, 7]). For n ∈ N, we have B (x) ζ (1 − n, x) = − n,p,q , (see Theorem 1.3). p,q n

By substituting Bn,p,q(x) for ζp,q(s, x) in Theorem 2.1, we can derive that

w2−1 w i 1−n X w1i 1 n[w ] q ζ w2 w2 1 − n, w x + 1 p,q p ,q 1 w i=0 2 w2−1 w i 1−n X w1i 1 = −[w ] q B w2 w2 w x + , 1 p,q n,p ,q 1 w i=0 2 and w1−1 w j 1−n X w2j 2 n[w ] q ζ w1 w1 1 − n, w x + 2 p,q p ,q 2 w j=0 1 w1−1 w j 1−n X w2j 2 = −[w ] q B w1 w1 w x + . 2 p,q n,p ,q 2 w j=0 1 Thus, we obtain the following theorem. 102 Cheon Seoung Ryoo

Theorem 2.4. Let w1, w2 be any positive integers. Then for non-negative integers n, one has w2−1 w i n−1 X w1i 1 [w ] q B w2 w2 w x + 2 p,q n,p ,q 1 w i=0 2 w1−1 w j n−1 X w2j 2 = [w ] q B w1 w1 w x + . 1 p,q n,p ,q 2 w j=0 1

Considering w1 = 1 in the Theorem 2.4, we derive the multiplication theorem for the (p, q)-Bernoulli polynomials(see [8])

w2−1 n−1 X j x + j B (x) = [w ] q B w2 w2 . (12) n,p,q 2 p,q n,p ,q w j=0 2 Setting p = 1 and q → 1 in (12), we derive the multiplication theorem for the Bernoulli polynomials w −1 X2 x + j B (x) = wn−1 B . n 2 n w j=0 2 Setting x = 0 in the Theorem 9, we obtain as below symmetric identity for the (p, q)-Bernoulli numbers.

Theorem 2.5. Let w1 and w2 be any positive integers. Then for non-negative integers n, one has w2−1 w i w1−1 w j n−1 X w1i 1 n−1 X w2j 2 [w ] q B w2 w2 = [w ] q B w1 w1 . 2 p,q n,p ,q w 1 p,q n,p ,q w i=0 2 j=0 1 Setting x = 0 in the Theorem 2.5, we obtain as below equation.

w2−1 n−1 X j j B = [w ] q B w2 w2 . n,p,q 2 p,q n,p ,q w j=1 2

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Cheon Seoung Ryoo received Ph.D. degree from Kyushu University. His research inter- ests focus on the numerical veriﬁcation method, scientiﬁc computing and p-adic functional analysis. Department of Mathematics, Hannam University, Daejeon, 306-791, Korea e-mail:[email protected]