New Construction Weighted (ℎ,𝑞)-Genocchi Numbers and Polynomials Related to Zeta Type Functions

Abstract

The fundamental aim of this paper is to construct (ℎ,𝑞)-Genocchi numbers and polynomials with weight 𝛼. We shall obtain some interesting relations by using 𝑝-adic 𝑞-integral on ℤ_𝑝 in the sense of fermionic. Also, we shall derive the (ℎ,𝑞)-extensions of zeta type functions with weight 𝛼 from the Mellin transformation of this generating function which interpolates the (ℎ,𝑞)-Genocchi numbers and polynomials with weight 𝛼 at negative integers.

1. Introduction, Definitions, and Notations

Let 𝑝 be a fixed odd prime number. Throughout this paper we use the following notations. ℤ_𝑝 denotes the ring of 𝑝-adic rational integers, ℚ denotes the field of rational numbers, ℚ_𝑝 denotes the field of 𝑝-adic rational numbers, and ℂ_𝑝 denotes the completion of algebraic closure of ℚ_𝑝. Let ℕ be the set of natural numbers and ℕ^∗=ℕ∪{0}. The 𝑝-adic absolute value is defined by |𝑝|_𝑝=1/𝑝. In this paper, we assume |𝑞−1|_𝑝<1 as an indeterminate. In [1–3],   Kim defined the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 as follows:𝐼_−𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁𝑝_−𝑞^𝑁−1_𝑥=0𝑓(𝑥)−𝑞^𝑥. (1.1)

[𝑥]_𝑞 is a 𝑞-extension of 𝑥 which is defined by [𝑥]_𝑞=1−𝑞^𝑥,1−𝑞 (1.2) see [1–15].

Note that lim_𝑞→1[𝑥]_𝑞=𝑥.

Let 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛). By the definition (1.1) we easily get−𝑞𝐼_−𝑞𝑓₁=lim_𝑁→∞1𝑝^𝑁𝑝_−𝑞^𝑁−1_𝑥=0𝑓(𝑥+1)−𝑞^𝑥+1=lim_𝑁→∞1𝑝^𝑁𝑝_−𝑞^𝑁−1_𝑥=0𝑓(𝑥)−𝑞^𝑥−1+𝑞lim_𝑁→∞𝑓𝑝^𝑁𝑞^𝑝𝑁+𝑓(0)1+𝑞^𝑝𝑁=𝐼_−𝑞𝑓−[2]_𝑞𝑓(0). (1.3)

Continuing this process, we obtain easily the relation𝑞^𝑛𝐼_−𝑞𝑓_𝑛+(−1)^𝑛−1𝐼_−𝑞𝑓=[2]_𝑞^𝑛−1_𝑙=0(−1)^{𝑛−𝑙−1}𝑞^𝑙𝑓(𝑙), (1.4)

(ℎ,𝑞)-Genocchi numbers are defined as follows: 𝐺^(ℎ)_0,𝑞=0,𝑞^ℎ−2𝑞𝐺_𝑞^(ℎ)+1^𝑛+𝐺^(ℎ)_𝑛,𝑞=[2]_𝑞,if𝑛=1,0,if𝑛>1, (1.5) with the usual convention about replacing (𝐺_𝑞^(ℎ))^𝑛 by 𝐺^(ℎ)_𝑛,𝑞 (see [6]).

In this paper, we constructed (ℎ,𝑞)-Genocchi numbers and polynomials with weight 𝛼. By using fermionic 𝑝-adic 𝑞-integral equations on ℤ_𝑝, we investigated some interesting identities and relations on the (ℎ,𝑞)-Genocchi numbers and polynomials with weight 𝛼. Furthermore, we derive the 𝑞-extensions of zeta type functions with weight 𝛼 from the Mellin transformation of this generating function which interpolates the (ℎ,𝑞)-Genocchi polynomials with weight 𝛼.

2. On the Weighted (ℎ,𝑞)-Genocchi Numbers and Polynomials

In this section, by using fermionic 𝑝-adic 𝑞-integral equations on ℤ_𝑝, some interesting identities and relation on the (ℎ,𝑞)-Genocchi numbers and polynomials with weight 𝛼 are shown.

Definition 2.1. Let 𝛼,𝑛∈ℕ^∗ and ℎ∈ℕ. Then the (ℎ,𝑞)-Genocchi numbers with weight 𝛼 defined by as follows: 𝐺^(𝛼,ℎ)_𝑛+1,𝑞=[2]𝑛+1^𝑞∞_𝑚=0(−1)^𝑚𝑞^𝑚ℎ[𝑚]^𝑛_𝑞𝛼. (2.1)

If we take ℎ=1 to (2.1), then we have, 𝐺^(𝛼,1)_𝑛+1,𝑞=𝐺^(𝛼)_𝑛+1,𝑞 (see [5]).

From (2.1), we obtain 𝐺^(𝛼,ℎ)_𝑛+1,𝑞=[2]𝑛+11−𝑞^{𝑞𝛼𝑛∞}_𝑚=0(−1)^𝑚𝑞^𝑚ℎ1−𝑞^𝑚𝛼𝑛=[2]_𝑞1−𝑞^𝛼𝑛∞_𝑚=0(−1)^𝑚𝑞^𝑛𝑚ℎ_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝑚𝛼𝑙=[2]_𝑞1−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙∞_𝑚=0(−1)^𝑚𝑞^{𝑚𝛼𝑙+𝑚ℎ}=[2]_𝑞1−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙11+𝑞^𝛼𝑙+ℎ (2.2)

Therefore, we obtain the following theorem.

Theorem 2.2. For 𝛼,𝑛∈ℕ^∗ and ℎ∈ℕ. Then 𝐺^(𝛼,ℎ)_𝑛+1,𝑞=[2]𝑛+11−𝑞^{𝑞𝛼𝑛𝑛}_𝑙=0𝑛𝑙(−1)^𝑙11+𝑞^𝛼𝑙+ℎ. (2.3)

In (1.1), one takes 𝑓(𝑥)=𝑞^{(ℎ−1)𝑥}[𝑥]^𝑛_𝑞𝛼,_ℤ𝑝𝑞^{(ℎ−1)𝑥}[𝑥]^𝑛_𝑞𝛼𝑑𝜇_−𝑞(𝑥)=11−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙_ℤ𝑝𝑞^{𝑥(𝛼𝑙+ℎ−1)}𝑑𝜇_−𝑞(𝑥)=11−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙lim_𝑁→∞1𝑝^𝑁𝑝_−𝑞^𝑁−1_𝑥=0−𝑞^{𝛼𝑙+ℎ𝑥}=11−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙1+𝑞1+𝑞^𝛼𝑙+ℎlim_𝑁→∞1+𝑞^{𝛼𝑙+ℎ𝑝𝑁}1+𝑞^𝑝𝑁=[2]_𝑞1−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙11+𝑞^𝛼𝑙+ℎ=𝐺^(𝛼,ℎ)_𝑛+1,𝑞.𝑛+1 (2.4)

From [12], we obtain (ℎ,𝑞)-Genocchi numbers with weight 𝛼 witt's type formula as follows.

Theorem 2.3. For 𝛼,𝑛∈ℕ^∗ and ℎ∈ℕ. Then 𝐺^(𝛼,ℎ)_𝑛+1,𝑞=𝑛+1_ℤ𝑝𝑞^{(ℎ−1)𝑥}[𝑥]^𝑛_𝑞𝛼𝑑𝜇_−𝑞(𝑥). (2.5)

From (2.1), one easily gets_ℤ𝑝𝑞^{(ℎ−1)𝑥}𝑒^{𝑡[𝑥]𝑞𝛼}𝑑𝜇_−𝑞(𝑥)=𝑡[2]_𝑞^∞_𝑚=0(−1)^𝑚𝑞^𝑚ℎ𝑒^{𝑡[𝑚]𝑞𝛼}. (2.6)

By (2.6), one has ^∞_𝑛=0𝐺^(𝛼,ℎ)_𝑛,𝑞𝑡^𝑛[2]𝑛!=𝑡_𝑞^∞_𝑚=0(−1)^𝑚𝑞^𝑚ℎ𝑒^{𝑡[𝑚]𝑞𝛼}. (2.7)

Therefore, we obtain the following corollary.

Corollary 2.4. If 𝐺^(𝛼,ℎ)_0,𝑞=0. Let 𝐷_𝑞^(𝛼,ℎ)(𝑡)=∑^∞_𝑛=0𝐺^(𝛼,ℎ)_𝑛,𝑞(𝑡^𝑛/𝑛!). Then 𝐷_𝑞^(𝛼,ℎ)(𝑡)=𝑡[2]_𝑞^∞_𝑚=0(−1)^𝑚𝑞^𝑚ℎ𝑒^{𝑡[𝑚]𝑞𝛼}. (2.8)

Now, one considers the (ℎ,𝑞)-Genocchi polynomials with weight 𝛼 as follows: 𝐺^(𝛼,ℎ)_𝑛+1,𝑞(𝑥)=𝑛+1_ℤ𝑝𝑞^{(ℎ−1)𝑦}𝑥+𝑦^𝑛_𝑞𝛼𝑑𝜇_−𝑞𝑦,𝑛∈ℕ,𝛼∈ℕ^∗. (2.9)

From (2.9), one sees that 𝐺^(𝛼,ℎ)_𝑛+1,𝑞(𝑥)=[2]𝑛+11−𝑞^{𝑞𝛼𝑛𝑛}_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥11+𝑞^𝛼𝑙+ℎ=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝑞^𝑚ℎ[𝑚+𝑥]^𝑛_𝑞𝛼. (2.10) Let 𝐷_𝑞^(𝛼,ℎ)(𝑡,𝑥)=∑^∞_𝑛=0𝐺^(𝛼,ℎ)_𝑛,𝑞(𝑥)(𝑡^𝑛/𝑛!). Then, one has 𝐷_𝑞^(𝛼,ℎ)(𝑡,𝑥)=𝑡[2]_𝑞^∞_𝑚=0(−1)^𝑚𝑞^𝑚ℎ𝑒^{𝑡[𝑚+𝑥]𝑞𝛼}=^∞_𝑛=0𝐺^(𝛼,ℎ)_𝑛,𝑞(𝑥)𝑡^𝑛.𝑛! (2.11)

By (1.4), one sees that 𝑞^ℎ𝑛𝐺^(𝛼,ℎ)_𝑚+1,𝑞(𝑛)+()𝑚+1−1^𝑛−1𝐺^(𝛼,ℎ)_𝑚+1,𝑞=[2]𝑚+1^𝑞𝑛−1_𝑙=0(−1)^{𝑛−𝑙−1}𝑞^ℎ𝑙[𝑙]^𝑚_𝑞𝛼. (2.12)

Therefore, we obtain the following theorem.

Theorem 2.5. For 𝑚,ℎ∈ℕ, and 𝛼,𝑛∈ℕ^∗, one has 𝑞^ℎ𝑛𝐺^(𝛼,ℎ)_𝑚+1,𝑞(𝑛)+()𝑚+1−1^𝑛−1𝐺^(𝛼,ℎ)_𝑚+1,𝑞=[2]𝑚+1^𝑞𝑛−1_𝑙=0(−1)^{𝑛−𝑙−1}𝑞^ℎ𝑙[𝑙]^𝑚_𝑞𝛼. (2.13)

In (1.3), it is known that 𝑞𝐼_−𝑞𝑓₁+𝐼_−𝑞𝑓=[2]_𝑞𝑓(0). (2.14)

If we take 𝑓(𝑥)=𝑞^{(ℎ−1)𝑥}𝑒^{𝑡[𝑥]𝑞𝛼}, then one has[2]_𝑞=𝑞_ℤ𝑝𝑞^{(ℎ−1)(𝑥+1)}𝑒^{𝑡[𝑥+1]𝑞𝛼}𝑑𝜇_−𝑞(𝑥)+_ℤ𝑝𝑞^{(ℎ−1)𝑥}𝑒^{𝑡[𝑥]𝑞𝛼}𝑑𝜇_−𝑞(𝑥)=^∞_𝑚=0𝑞^ℎ𝐺^(𝛼,ℎ)_𝑚+1,𝑞(1)+𝐺𝑚+1^{(𝛼,ℎ)𝑚+1,𝑞}𝑡𝑚+1^𝑚.𝑚! (2.15)

Therefore, by (2.15), we obtain the following theorem.

Theorem 2.6. For 𝛼∈ℕ^∗ and 𝑚,ℎ∈ℕ, one has 𝐺^(𝛼,ℎ)_0,𝑞=0,𝑞^ℎ𝐺^(𝛼,ℎ)_𝑚+1,𝑞(1)+𝐺𝑚+1^{(𝛼,ℎ)𝑚+1,𝑞}=[2]𝑚+1^𝑞,if𝑚=0,0,if𝑚≠0. (2.16)

From (2.9), one can easily derive the following:_ℤ𝑝𝑞^{(ℎ−1)𝑦}𝑥+𝑦^𝑛_𝑞𝛼𝑑𝜇_−𝑞𝑦=[𝑑]^𝑛_𝑞𝛼[𝑑]_−𝑞^𝑑−1_𝑎=0(−1)^𝑎𝑞^ℎ𝑎_ℤ𝑝𝑞^{𝑑𝑦(ℎ−1)}𝑥+𝑎𝑑+𝑦^𝑛_𝑞𝑑𝛼𝑑𝜇_(−𝑞)^𝑑𝑦=[𝑑]^𝑛_𝑞𝛼[𝑑]_−𝑞^𝑑−1_𝑎=0(−1)^𝑎𝑞^ℎ𝑎𝐺^(𝛼,ℎ)_{𝑛+1,𝑞𝑑}((𝑥+𝑎)/𝑑).𝑛+1 (2.17)

Therefore, by (2.17), we obtain the following theorem.

Theorem 2.7. For 𝑑≡1(mod2), 𝑛∈ℕ^∗ and 𝛼,ℎ∈ℕ𝐺^(𝛼,ℎ)_𝑛+1,𝑞(𝑥)=[𝑑]^𝑛_𝑞𝛼[𝑑]_−𝑞^𝑑−1_𝑎=0(−1)^𝑎𝑞^ℎ𝑎𝐺^(𝛼,ℎ)_{𝑛+1,𝑞𝑑}𝑥+𝑎𝑑. (2.18)

3. Interpolation Function of the Polynomials 𝐺^(𝛼,ℎ)_𝑛,𝑞(𝑥)

In this section, we give interpolation function of the generating functions of (ℎ,𝑞)-Genocchi polynomials with weight 𝛼. For 𝑠∈ℂ and ℎ∈ℕ, by applying the Mellin transformation to (2.11), we obtain ℑ_𝑞^(𝛼,ℎ)()=1𝑠,𝑥Γ(𝑠)^∞₀𝑡^𝑠−2−𝐷_𝑞^(𝛼,ℎ)(−𝑡,𝑥)𝑑𝑡=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝑞^𝑚ℎ1Γ(𝑠)^∞₀𝑡^𝑠−1𝑒^{−𝑡[𝑚+𝑥]𝑞𝛼}𝑑𝑡, (3.1) so we have ℑ_𝑞^(𝛼,ℎ)(𝑠,𝑥)=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝑞^𝑚ℎ[𝑚+𝑥]^𝑠_𝑞𝛼. (3.2)

We define 𝑞-extension zeta type function as follows.

Theorem 3.1. For 𝑠∈ℂ, ℎ∈ℕ, and 𝛼∈ℕ^∗. One has ℑ_𝑞^(𝛼,ℎ)(𝑠,𝑥)=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝑞^𝑚ℎ[𝑚+𝑥]^𝑠_𝑞𝛼. (3.3)ℑ_𝑞^(𝛼,ℎ)(𝑠,𝑥) can be continued analytically to an entire function.

By subsituting 𝑠=−𝑛 into (3.3) one easily gets ℑ_𝑞^(𝛼,ℎ)()=𝐺−𝑛,𝑥^{(𝛼,ℎ)𝑛+1,𝑞}(𝑥).𝑛+1 (3.4) We obtain the following theorem.

Theorem 3.2. For ℎ∈ℕ and 𝑞,𝑠∈ℂ, |𝑞|<1. Then one defines ℑ_𝑞^(𝛼,ℎ)()=𝐺−𝑛,𝑥^{(𝛼,ℎ)𝑛+1,𝑞}(𝑥).𝑛+1 (3.5)