A NOTE ON THE MODIFIED q-GENOCCHI NUMBERS AND ...

arXiv:1112.5902v2 [math.CO] 27 Jan 2012

A NOTE ON THE MODIFIED q-GENOCCHI NUMBERS AND POLYNOMIALS WITH WEIGHT (α, β) AND THEIR INTERPOLATION FUNCTION AT NEGATIVE INTEGERS ¨ FENG QI, AND HASSAN JOLANY SERKAN ARACI, MEHMET AC ¸ IKGOZ, Abstract. The purpose of this paper concerns to establish modified q-Genocchi numbers and polynomials with weight (α,β). In this paper we investigate special generalized q-Genocchi polynomials and we apply the method of generating function, which are exploited to derive further classes of q-Genocchi polynomials and develop q-Genocchi numbers and polynomials. By using the Laplace-Mellin transformation integral, we define q-Zeta function with weight (α,β) and by presenting a link between q-Zeta function with weight (α,β) and q-Genocchi numbers with weight (α,β) we obtain an interpolation formula for the q-Genocchi numbers and polynomials with weight (α,β). Also we derive distribution formula (Multiplication Theorem) and Witt’s type formula for modified q-Genocchi numbers and polynomials with weight (α,β) which yields a deeper insight into the effectiveness of this type of generalizations for qGenocchi numbers and polynomials. Our new generating function possess a number of interesting properties which we state in this paper.

1. Introduction, Definitions and Notations Recently, q-calculus has served as a bridge between mathematics and physics. Therefore, there is a significant increase of activity in the area of the q-calculus due to applications of the q-calculus in mathematics, statistics and physics. The majority of scientists in the world who use q-calculus today are physicists. q-Calculus is a generalization of many subjects, like hypergeometric series, generating functions, complex analysis, and particle physics. In short, q-calculus is quite a popular subject today. One of Important Branch of q-calculus in number theory is q-type of special generating functions, for instance q-Bernoulli numbers, q-Euler numbers, and q-Genocchi numbers, here we introduce a new class of q-type generating function. We introduce q-Genocchi numbers with weight (α, β). When we define a new class of generating functions like, q-Genocchi numbers with weight (α, β), then we face to with this question that “can we define a new q-Zeta type function in related of this new class of q-type generating function?”. We give a positive answer for our new class of numbers and polynomials. More precisely we show that our q-type generating function is generalization of the Hurwitz Zeta function. Historically many authors have tried to give q-analogues of the Riemann Zeta function ζ (s), and its related functions. By just following the method of Kaneko et al. [M. Kaneko, N. Kurokawa and M. Wakayama, A variation of Euler’s approach to the Riemann Zeta Date: December 12, 2011. 2000 Mathematics Subject Classification. Primary 46A15, Secondary 41A65. Key words and phrases. Genocchi numbers and polynomials, q-Genocchi numbers and polynomials, q-Genocchi numbers and polynomials with weight α. 1

¨ SERKAN ARACI, MEHMET AC ¸ IKGOZ, FENG QI, AND HASSAN JOLANY

2

function, Kyushu J. Math. 57 (2003), 175–192], who mainly used Euler-Maclaurin summation formula to present and investigate a q-analogue of the Riemann zeta function ζ (s), and gave a good and reasonable explanation that their q-analogue may be a best choice. They also commented that q-analogue of ζ (s) can be achieved by modifying their method. Furthermore it is clear that q-Genocchi polynomials of weight (α, β) are in a class of orthogonal polynomials and we know that most such special functions that are orthogonal are satisfied in multiplication theorem, so in this present paper we show this property is true for q-Genocchi polynomials of weight (α, β). In this introductory section, we present the definitions and notations (and some of the Important properties and characteristics) of the various special functions, polynomials and numbers, which are potentially useful in the remainder of the paper. Assume that p be a fixed odd prime number. Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Q denotes the field of rational numbers, Qp denotes the field of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp . Let N be the set of natural numbers and Z+ = N ∪ {0} . Let vp be the normalized exponential valuation of Cp with |p|p = p−vp (p) = p−1 . When one speaks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C or p-adic number q ∈ Cp . If 1 q ∈ C one normally assume that |q| < 1. If q ∈ Cp , we assume that |1 − q|p < p− p−1 so that q x = exp (x log q) for |x|p ≤ 1. We use the following notation as follows: [x]q =

1 − (−q)x 1 − qx , [x]−q = 1−q 1+q

Note that limq→1 [x]q = x; cf. [1-24]. For a fixed positive integer d with (d, f ) = 1, we set X

=

X∗

=

Xd = lim Z/dpN Z, ← − N

∪

a + dpZp

0 1

with the usual convention of replacing (Gq ) by Gn,q . In [6], (h, q)-Genocchi numbers are indicated as: n [2]q , n = 1 (h) (h) (h) h−2 qGq + 1 + Gn,q = G0,q = 0, and q 0, n > 1,


4

n (h) (h) with the usual convention about replacing Gq by Gn,q . Recently, for n ∈ Z+ , Araci et al. are considered weighted q-Genocchi numbers by

n [2]q , n = 1 (α) (α) (α) 1−α e e e q Gq + 1 + Gn,q = (1.5) G0,q = 0, q 0, n 6= 1, n en,q (for more information, eq by G with the usual convention about replacing G

see [1]) For α, n ∈ Z+ and h ∈ N, Araci et al. [2] defined weighted (h, q)-Genocchi numbers as follows: Z e(α,h) = q (h−1)x [x]nqα dµ−q (x) . G n+1,q Zp

Taekyun Kim, by using p-adic q-integral on Zp , introduced a new class of numbers and polynomials. He added a weight on q-Bernoulli numbers and polynomials and defined q-Bernoulli numbers with weight α. He is given some interesting properties concerning q-Bernoulli numbers and polynomials with weight α. After, by using p-adic q-integral on Zp , several mathematicians started to study on this new branch of generating function theory and extended most of the symmetric properties of q-Bernoulli numbers and polynomials to q-Bernoulli numbers and polynomials with weight α (for more informations, see [1],[2],[4],[5],[6],[7],[8],[9],[10],[11],[12]). With the same motivation, we also introduce modified q-Genocchi numbers and polynomials with weight (α, β) . Also, we give some interesting properties this type of polynomials. Furthermore, we derive the q-extensions of zeta type functions with weight (α, β) from the Mellin transformation to this generating function which interpolates the q-Genocchi polynomials with weight (α, β) at negative integers. 2.

Modified q-Genocchi numbers and polynomials with weight (α, β)

In this section, we derive some interesting properties Modified q-Genocchi numbers and polynomials with weight (α, β). Lemma 1. For n ∈ Z+ ,we obtain n−1 X (β) (β) (−1)n−l−1 f (l) , I−q q −βx fn + (−1)n−1 I−q q −βx f = [2]qβ

(2.1)

l=0

(β)

Proof. Let be fn (x) = f (x + n) and I−q (f ) = we easily get −

(β) I−q

q

−βx

f1

Zp

f (x) dµ−qβ (x) by the (1.1),

N

=

= (2.2)

R

=

pX −1 1 x f (x + 1) (−1) lim N →∞ [pN ]−qβ x=0

N pX −1 f pN + f (0) 1 x f (x) (−1) − [2]qβ lim lim N →∞ N →∞ [pN ]−qβ 1 + q βpN x=0 (β) I−q q −βx f − [2]qβ f (0)

ON THE MODIFIED q-GENOCCHI NUMBERS AND POLYNOMIALS WITH WEIGHT (α, β) 5

and (β) I−q

q

−βx

f2

Thus, we have

Z

N

pX −1 1 x q f (x + 2) dµ−qβ (x) = lim = f (x + 2) (−1) N →∞ [pN ]−qβ Zp x=0 −f (0) + f (1) − f pN + f pN + 1 (β) −βx f + [2]qβ lim = I−q q N →∞ 1 + q βpN (β) = I−q q −βx f + [2]qβ (f (1) − f (0)) −βx

1 X (β) (β) 1−l (−1) f (l) I−q q −βx f2 − I−q q −βx f = [2]qβ l=0

By continuing this process, we arrive at the desired result.

Definition 1. Let α, n, β ∈ Z+ . We define modified q-Genocchi numbers with weight (α, β) as follows: (α,β) ∞ X gn+1,q m n = [2]qβ (−1) [m]qα n+1 m=0

(2.3)

Theorem 1. For α, n, β ∈ Z+ , we get (2.4)

(α,β) n [2]qβ X gn+1,q 1 n = (−1)l n n+1 1 + q αl l (1 − q α ) l=0

Proof. By (2.3), we develop as follows: (α,β)

= = = =

gn+1,q n+1 [2]qβ (1 −

q α )n

[2]qβ n

(1 − q α ) [2]qβ

(1 − q α )n [2]qβ (1 −

n qα )

∞ X

m=0 ∞ X

m

n

(−1) (1 − q mα ) (−1)m

m=0

n X n

l

l=0

n X n

l

l=0

n X n l=0 l

(−1)

l

∞ X

(−1)l (q mα )l m

(−1) q mαl

m=0 l

(−1)

1 . 1 + q αl

Thus, we complete the proof of Theorem.

By the following Theorem, we get Witt’s type formula of this type polynomials. Theorem 2. For β, α, n ∈ Z+ , we get (α,β)

(2.5)

gn+1,q = n+1

Z

Zp

n

q −βx [x]qα dµ−qβ (x) .


6

n

Proof. By using p-adic q-integral on Zp , namely, replace f (x) by q −βx [x]qα and µ−q x + pN Zp by µ−qβ x + pN Zp into (1.1), we get Z Z n X n 1 l n q αlx−βx dµ−qβ (x) (−1) q −βx [x]qα dµ−qβ (x) = α )n l (1 − q Zp Zp l=0

=

=

(2.6)

=

N pX −1 n X x 1 1 n l −q αl (−1) lim N →∞ [pN ]−qβ (1 − q α )n l x=0 l=0 pN n X 1 + q αl 1 n l [2]qβ lim (−1) n 1 + q αl N →∞ 1 + q βpN l (1 − q α ) l=0 n [2]qβ X 1 n (−1)l n α l 1 + q αl (1 − q )

l=0

Use of (2.4) and (2.6), we arrive at the desired result.

The Witt’s type formula of modified q-Genocchi numbers with weight (α, β) asserted by Theorem 2, do aid in translating the various properties and results involving q-Genocchi numbers with weight (α, β) which we state some of them in (1,1)

this section. We put α → 1 and β → 1 into (2.5), we readily see (α,β)

Corollary 1. Let Cq

(t) =

P∞

n=0

(α,β) tn n! .

gn,q

Cq(α,β) (t) = [2]qβ t

∞ X

gn+1,q n+1

= εn,q .

Then we have m

(−1) et[m]qα .

m=0

Proof. From (2.3) we easily get, Z ∞ X q −βx et[x]qα dµ−qβ (x) = [2]qβ t (−1)m et[m]qα (2.7) Zp

m=0

By expression (2.7), we have ∞ X

(α,β) gn,q

n=0

∞ X tn m = [2]qβ t (−1) et[m]qα n! m=0

Thus, we complete the proof of Theorem.

Now, we consider the modified q-Genocchi polynomials polynomials with weight α as follows: Z (α,β) gn+1,q (x) q −βt [x + t]nqα dµ−qβ (t) , n ∈ N and α ∈ Z+ = (2.8) n+1 Zp From expression (2.8), we see readily (α,β)

gn+1,q (x) n+1 (2.9)

=

[2]qβ n

(1 − q α )

= [2]qβ

∞ X

m=0

n X n

(−1) q αlx

m

n

l

l=0

l

(−1) [m + x]qα

1 1 + q αl

ON THE MODIFIED q-GENOCCHI NUMBERS AND POLYNOMIALS WITH WEIGHT (α, β) 7 (α,β)

Let Cq

(t, x) =

P∞

(α,β) tn n=0 gn,q (x) n! .

Cq(α,β) (t, x)

Then we have

= [2]qβ t

∞ X

(−1)m et[m+x]qα

m=0

(2.10)

=

∞ X

(α,β) gn,q (x)

n=0

tn . n!

By Lemma 1, we get the following Theorem: Theorem 3. For m ∈ N,and α, β, n ∈ Z+ , we get (α,β) (α,β) n−1 X gm+1,q n−1 gm+1,q (n) n−l−1 m + (−1) = [2]qβ (−1) [l]qα m+1 m+1 l=0

Proof. By applying Lemma 1 the methodology and techniques used above in getting some identities for the generating functions of the modified q-Genocchi numbers and polynomials with weight (α, β) , we arrive at the desired result. Theorem 4. The following identity holds: (α,β)

g0,q

(α,β) (α,β) = 0, and gn,q (1) + gn,q =

[2]qβ , if n = 1, 0, if n > 1.

Proof. In (2.2) it is known that (β) (β) I−q q −βx f1 + I−q q −βx f = [2]qβ f (0)

If we take f (x) = et[x]qα , then we have Z Z [2]qβ = q −βx et[x+1]qα dµ−qβ (x) + (2.11)

=

Zp ∞ X

q −βx et[x]qα dµ−q−β (x)

Zp

(α,β) (α,β) gn,q (1) + gn,q

n=0

tn−1 n!

Therefore, we get the Proof of Theorem.

Theorem 5. For d ≡ 1 (mod 2), α, β ∈ Z+ and n ∈ N, we get, (α,β) gn,q

(dx) =

n−1 d−1 X

[d]qα

[d]−qβ

a=0

a

(α,β)

(−1) gn,qd

x+

a . d

Proof. From (2.8), we can easily derive the following (2.12) n Z Z n X [d]qα d−1 a n −βt x + a −βt (−1) +t dµ(−qd )β (t) q q [x + t]qα dµ−qβ (t) = [d]−qβ a=0 d Zp Zp qdα (α,β) n x+a X g [d]qα d−1 d a n+1,qd = (2.12) . (−1) [d]−qβ a=0 n+1 So, by applying expression (2.12), we get at the desired result and proof is complete.

8

¨ SERKAN ARACI, MEHMET AC ¸ IKGOZ, FENG QI, AND HASSAN JOLANY (α,β)

3. Interpolation function of the polynomials gn,q

(x)

In this section, we derive the interpolation function of the generating functions of modified q-Genocchi polynomials with weight α and we give the value of qextension zeta function with weight (α, β) at negative integers explicitly. For s ∈ C, by applying the Mellin transformation to (2.10), we obtain Z ∞ n o 1 ts−2 −Cq(α,β) (−t, x) dt ξ (α,β) (s, x | q) = Γ (s) 0 Z ∞ ∞ X 1 m = [2]qβ (−1) ts−1 e−t[m+x]qα dt Γ (s) 0 m=0 where Γ (s) is Euler gamma function. We have ξ (α,β) (s, x | q) = [2]qβ

∞ X

m

(−1) s [m + x]qα m=0

So, we define q-extension zeta function with weight (α, β) as follows: Definition 2. For s ∈ C and α, β ∈ N, we have ∞ m X (−1) (3.1) ξ (α,β) (s, x | q) = [2]qβ s [m + x]qα m=0

ξ (α,β) (s, x | q) can be continued analytically to an entire function. Observe that, if q → 1, then ξ (α,β) (s, x | 1) = ζ (s, x) which is the Hurwitz(α,β) Euler zeta functions. Relation between ξ (α,β) (s, x | q) and gn,q (x) are given by the following theorem: Theorem 6. For α, β ∈ N and n ∈ N, we get (α,β)

ξ

(α,β)

gn+1,q (x) (−n, x | q) = . n+1

Proof. By substituting s = −n into (3.1), we arrive at the desired result.

References [1] Araci, S., Erdal, D., and Seo, J-J., A study on the fermionic p-adic q-integral representation on Zp associated with weighted q-Bernstein and q-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages. [2] Araci, S., Seo, J-J., and Erdal, D., New construction weighted (h, q)-Genocchi numbers and polynomials related to zeta type functions, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 487490, 7 pages, doi:10.1155/2011/487490. [3] Araci, S., Erdal, D.,and Kang, D-J., Some new properties on the q-Genocchi numbers and polynomials associated with q-Bernstein polynomials, Honam Mathematical J. 33 (2011), no. 2, pp. 261-270. [4] Araci, S., Aslan, N., and Seo, J-J., A Note on the weighted Twisted Dirichlet’s type q-Euler numbers and polynomials, Honam Mathematical J. 33 (2011), no. 3, pp. 311-320. [5] Araci, S., A¸cıkg¨ oz, M., and Seo, J-J., A study on the weighted q-Genocchi numbers and polynomials with their Interpolation Function, Accepted in Honam Mathematical Journal. [6] Araci, S., and A¸cıkg¨ oz, M., and Park, K-H., Some identities concerning the (h, q)-Genocchi numbers and polynomials via the p-adic q-integral on Zp and q-Bernstein polynomials,(submitted). [7] Kim, T., On the weighted q-Bernoulli numbers and polynomials, Advanced Studies in Contemporary Mathematics 21(2011), no.2, p. 207-215, http://arxiv.org/abs/1011.5305.

ON THE MODIFIED q-GENOCCHI NUMBERS AND POLYNOMIALS WITH WEIGHT (α, β) 9

[8] Kim, T., Lee, S. H., Dolgy, D. V., Ryoo, C. S., A note on the generalized q-Bernoulli measures with weight α, Abstract and Applied Analysis, Article ID 867217, 9 pages, doi:10.1155/2011/867217. [9] Kim, T., Choi, J., Kim, Y. H., Ryoo, C. S., A note on the weighted p-adic q-Euler measure on Zp , Advn. Stud. Contemp. Math. 21 (2011), 35-40. [10] Hwang, K-W., Dolgy, D-V., Lee, S. H., and Kim, T., On the Higher-Order q-Euler numbers and polynomials with weight α, Discrete Dynamics in Nature and Society, Volume (2011), Article ID 354329, 12 pages. [11] Kim, T., and Choi, J., On the q-Bernoulli numbers and polynomials with weight α, Abstract and Applied Analysis, Volume 2011 (2011), Article ID 392025, 14 pages. [12] Kim, T., Dolgy, D-V., Lee, B., and Rim S-H., Identities on the weighted q-Euler numbers of higher order, Discrete Dynamics in Nature and Society, Volume 2011 (2011), Article ID 918364, 6 pages. [13] Kim, T., On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007) 1458-1465. [14] Kim, T., On the multiple q-Genocchi and Euler numbers, Russian J. Math. Phys. 15 (4) (2008) 481-486. arXiv:0801.0978v1 [math.NT] [15] Kim, T., The modified q-Euler numbers and polynomials, Advn. Stud. Contemp. Math. 16 (2008), 161-170. [16] Ozden, H., Simsek, Y., Rim, S-H., Cangul, I-N., A note on p-adic q-euler measure, Adv. Stud. Contemp. Math. 14 (2007), 233-239. [17] Kim, T., q-Volkenborn integration, Russ. J. Math. phys. 9(2002) , 288-299. [18] Kim, T., q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russ. J. Math. Phys. 15(2008) , 51-57. [19] Kim, T., An invariant p-adic q-integrals on Zp , Applied Mathematics Letters, vol. 21, pp. 105-108,2008. [20] Kim, T., A Note on the q-Genocchi Numbers and Polynomials, Journal of Inequalities and Applications, Article ID 71452, 8 pages, doi:10.1155/2007/71452. [21] Kim, T. Choi, J. Kim, Y. H. and Jang, L. C., On p-Adic Analogue of q-Bernstein Polynomials and Related Integrals, Discrete Dynamics in Nature and Society, Article ID 179430, 9 pages, doi:10.1155/2010/179430. [22] Kim, T., q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math. Phys., 14 (2007), no. 1, 15–27. [23] Kim, T., New approach to q-Euler polynomials of higher order, Russ. J. Math. Phys., 17 (2010), no. 2, 218–225. [24] Kim, T., Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp , Russ. J. Math. Phys., 16 (2009), no.4,484–491. University of Gaziantep, Faculty of Science and Arts, Department of Mathematics, 27310 Gaziantep, TURKEY E-mail address: [email protected] University of Gaziantep, Faculty of Science and Arts, Department of Mathematics, 27310 Gaziantep, TURKEY E-mail address: [email protected] Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin 300160, China E-mail address: [email protected] School of Mathematics, Statistics and Computer Science, University of Tehran, Iran E-mail address: [email protected]