Novel Identities for 𝑞-Genocchi Numbers and Polynomials

Abstract

The essential aim of this paper is to introduce novel identities for q-Genocchi numbers and polynomials by using the method by T. Kim et al. (article in press). We show that these polynomials are related to p-adic analogue of Bernstein polynomials. Also, we derive relations between q-Genocchi and q-Bernoulli numbers.

1. Preliminaries

Imagine that 𝑝 be a fixed odd prime number. We now start with definition of the following notations. Let ℚ_𝑝 be the field 𝑝-adic rational numbers and let ℂ_𝑝 be the completion of algebraic closure of ℚ_𝑝.

Thus, ℚ_𝑝=𝑥=^∞_𝑛_=−𝑘𝑎_𝑛𝑝^𝑛∶0≤𝑎_𝑛<𝑝.(1.1)

Then ℤ_𝑝 is integral domain, which is defined by ℤ_𝑝=𝑥=^∞_𝑛=0𝑎_𝑛𝑝^𝑛∶0≤𝑎_𝑛<𝑝,(1.2) or ℤ_𝑝=𝑥∈ℚ_𝑝∶|𝑥|_𝑝≤1.(1.3)

We assume that 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<1 as an indeterminate. The 𝑝-adic absolute value |⋅|_𝑝, is normally defined by |𝑥|_𝑝=𝑝^−𝑟,(1.4) where 𝑥=𝑝^𝑟(𝑠/𝑡) with (𝑝,𝑠)=(𝑝,𝑡)=(𝑠,𝑡)=1, and 𝑟∈ℚ.

[𝑥]_𝑞 is a 𝑞-extension of 𝑥, which is defined by [𝑥]_𝑞=1−𝑞^𝑥1−𝑞,(1.5) we note that lim_𝑞→1[𝑥]_𝑞=𝑥 (see [1–17]).

Throughout this paper, we use notation of ℕ^∗∶=ℕ∪{0}, where ℕ denotes set of Natural numbers.

We say that 𝑓 is a uniformly differentiable function at a point 𝑎∈ℤ_𝑝, if the difference quotient, 𝐹_𝑓(𝑥,𝑦)=𝑓(𝑥)−𝑓(𝑦)𝑥−𝑦,(1.6) has a limit 𝑓^(𝑎) as (𝑥,𝑦)→(𝑎,𝑎), and we denote this by 𝑓∈𝑈𝐷(ℤ_𝑝). Then, for 𝑓∈𝑈𝐷(ℤ_𝑝), we can start with the following expression: 1𝑝^𝑁_𝑞0≤_{𝜉<𝑝𝑁}𝑓(𝜉)𝑞^𝜉=_{0≤𝜉<𝑝𝑁}𝑓(𝜉)𝜇_𝑞𝜉+𝑝^𝑁ℤ_𝑝,(1.7) which represents a 𝑝-adic 𝑞-analogue of Riemann sums for 𝑓. The integral of 𝑓 on ℤ_𝑝 will be defined as the limit (𝑁→∞) of these sums, when it exists. The 𝑝-adic 𝑞-integral of function 𝑓∈𝑈𝐷(ℤ_𝑝) is defined by Kim in [7, 12] as 𝐼_𝑞(𝑓)=_ℤ𝑝𝑓(𝜉)𝑑𝜇_𝑞(𝜉)=lim_𝑁→∞1𝑝^𝑁_𝑞𝑝^𝑁−1_𝜉=0𝑓(𝜉)𝑞^𝜉.(1.8)

The bosonic integral is considered as a bosonic limit 𝑞→1, 𝐼₁(𝑓)=lim_𝑞→1𝐼_𝑞(𝑓). Similarly, the fermionic 𝑝-adic integral on ℤ_𝑝 is introduced by Kim as follows: 𝐼_−𝑞(𝑓)=_ℤ𝑝𝑓(𝜉)𝑑𝜇_−𝑞(𝜉)(1.9) (for more details, see [13–16]).

From (1.9), it is well-known equality that 𝑞𝐼_−𝑞𝑓₁+𝐼_−𝑞(𝑓)=[2]_𝑞𝑓(0),(1.10) where 𝑓₁(𝑥)=𝑓(𝑥+1) (for details, see [2, 3, 8, 9, 12, 13, 15–17]).

The 𝑞-Genocchi polynomials with wegiht 0 are introduced as 𝐺_𝑛+1,𝑞(𝑥)𝑛+1=_ℤ𝑝(𝑥+𝜉)^𝑛𝑑𝜇_−𝑞(𝜉).(1.11)

From (1.11), we have 𝐺_𝑛,𝑞(𝑥)=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝑥_𝑙𝐺_{𝑛−𝑙,𝑞},(1.12) where 𝐺_𝑛,𝑞(0)∶=𝐺_𝑛,𝑞 are called 𝑞-Genocchi numbers with weight 0. Then, 𝑞-Genocchi numbers are defined as 𝐺_0,𝑞=0,𝑞𝐺_𝑞+1^𝑛+𝐺_𝑛,𝑞=0[2,]^𝑞,iiff𝑛𝑛≠=11,,(1.13) with the usual convention about replacing (𝐺_𝑞)^𝑛 by 𝐺_𝑛,𝑞 is used (for details, see [3]).

Let 𝑈𝐷(ℤ_𝑝) be the space of continuous functions on ℤ_𝑝. For 𝑓∈𝑈𝐷(ℤ_𝑝), 𝑝-adic analogue of Bernstein operator for 𝑓 is defined by 𝐵_𝑛(𝑓,𝑥)=^𝑛_𝑘=0𝑓𝑘𝑛𝐵^𝑘,𝑛(𝑥)=^𝑛_𝑘=0𝑓𝑘𝑛⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝑥^𝑘(1−𝑥)^𝑛−𝑘,(1.14) where 𝑛,𝑘∈ℕ^∗. Here, 𝐵_𝑘,𝑛(𝑥) is called 𝑝-adic Bernstein polynomials, which are defined by 𝐵_𝑘,𝑛(𝑥)=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝑥^𝑘(1−𝑥)^𝑛−𝑘,𝑥∈[0,1](1.15) (for details, see [1, 4, 5, 7]).

The 𝑞-Bernoulli polynomials and numbers with weight 0 are defined by Kim et al., respectively, 𝐵_𝑛,𝑞(𝑥)=lim_𝑛→∞1[𝑝^𝑛]_𝑞𝑝^𝑛−1_𝑦=0𝐵(𝑥+𝑦)^𝑛𝑞^𝑦=_ℤ𝑝(𝑥+𝜉)^𝑛𝑑𝜇_𝑞(𝜉),_𝑛,𝑞=_ℤ𝑝𝜉^𝑛𝑑𝜇_𝑞(𝜉)(1.16) (for more information, see [10]).

The author, by using derivative operator, will investigate some interesting identities on the 𝑞-Genocchi numbers and polynomials arising from their generating function. Also, the author derives some relations between 𝑞-Genocchi numbers and 𝑞-Bernoulli numbers by using Kim’s 𝑞-Volkenborn integral and fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝.

2. Novel Properties of 𝑞-Genocchi Numbers and Polynomials with Weight 0

Let 𝑓(𝑥)=𝑒^{𝑡(𝑥+𝑦)}. Then, by using (1.10), we easily procure the following: _ℤ𝑝𝑒^{𝑡(𝑥+𝜉)}𝑑𝜇_−𝑞(𝜉)=[2]^𝑞𝑞𝑒^𝑡+1𝑒^𝑥𝑡.(2.1)

From the last equality, by (1.11), we get Araci, Acikgoz, and Qi’s 𝑞-Genocchi polynomials with weight 0 in [3] as follows: [2]_𝑞𝑡𝑞𝑒^𝑡+1𝑒^𝑥𝑡=^∞_𝑛=0𝐺_𝑛,𝑞(𝑥)𝑡^𝑛𝑛!,||log𝑞+𝑡||<𝜋.(2.2)

Here, we assume that 𝑥 is a fixed parameter. Let 𝐹_𝑞(𝑥,𝑡)=[2]^𝑞𝑞𝑒^𝑡+1𝑒^𝑥𝑡=^∞_𝑛=0𝐺_𝑛,𝑞(𝑥)𝑡^𝑛−1𝑛!.(2.3) Thus, by expression of (2.3), we can readily see the following: 𝑞𝑒^𝑡𝐹_𝑞(𝑥,𝑡)+𝐹_𝑞(𝑥,𝑡)=[2]_𝑞𝑒^𝑥𝑡.(2.4)

Last from equality, taking derivative operator 𝐷 as 𝐷=𝑑/𝑑𝑡 on the both sides of (2.4), then, we easily see that 𝑞𝑒^𝑡(𝐷+𝐼)^𝑘𝐹_𝑞(𝑥,𝑡)+𝐷^𝑘𝐹_𝑞(𝑥,𝑡)=[2]_𝑞𝑥^𝑘𝑒^𝑥𝑡,(2.5) where 𝑘∈ℕ^∗ and 𝐼 is identity operator. By multiplying 𝑒^−𝑡 on both sides of (2.5), we get 𝑞(𝐷+𝐼)^𝑘𝐹_𝑞(𝑥,𝑡)+𝑒^−𝑡𝐷^𝑘𝐹_𝑞(𝑥,𝑡)=[2]_𝑞𝑥^𝑘𝑒^{(𝑥−1)𝑡}.(2.6)

Let us take derivative operator 𝐷^𝑚(𝑚∈ℕ) on the both sides of (2.6). Then, we get 𝑞𝑒^𝑡𝐷^𝑚(𝐷+𝐼)^𝑘𝐹_𝑞(𝑥,𝑡)+𝐷^𝑘(𝐷−𝐼)^𝑚𝐹_𝑞(𝑥,𝑡)=[2]_𝑞𝑥^𝑘(𝑥−1)^𝑚𝑒^𝑥𝑡.(2.7)

Let 𝐺[0] (not 𝐺(0)) be the constant term in a Laurent series of 𝐺(𝑡) in (2.3). Then, we get ^𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝑞𝑒_𝑡𝐷^{𝑘+𝑚−𝑗}𝐹_𝑞(𝑥,𝑡)[0]+^𝑚_𝑗=0⎛⎜⎜⎝𝑚𝑗⎞⎟⎟⎠(−1)_𝑗𝐷^{𝑘+𝑚−𝑗}𝐹_𝑞(𝑥,𝑡)[0]=[2]_𝑞𝑥^𝑘(𝑥−1)^𝑚.(2.8)

By (2.3), we easily see 𝐷^𝑁𝐹_𝑞(𝑥,𝑡)[0]=𝐺^𝑁+1,𝑞(𝑥)𝑁+1,𝑒^𝑡𝐷^𝑁𝐹_𝑞(𝑥,𝑡)[0]=𝐺^𝑁+1,𝑞(𝑥)𝑁+1.(2.9)

We see that the members of (2.11) are proportional to the Bernstein polynomials with the following theorem.

Theorem 2.1. For 𝑘,𝑚∈ℕ, one has [2]_𝑞(−1)^𝑚𝐵_{𝑘,𝑚+𝑘}(𝑥)^{𝑘𝑚+𝑘}=^{max{𝑘,𝑚}}_𝑗=0𝑞^𝑘𝑗+(−1)^𝑗^𝑚𝑗𝑘+𝑚−𝑗+1𝐺^{𝑘+𝑚−𝑗+1,𝑞}(𝑥).(2.10)

Proof. By expressions of (2.8) and (2.9), we see that ^max^{𝑘,𝑚}_𝑗=0𝑞^𝑘𝑗+(−1)^𝑗^𝑚𝑗𝑘+𝑚−𝑗+1𝐺^{𝑘+𝑚−𝑗+1,𝑞}(𝑥)=[2]_𝑞𝑥^𝑘(𝑥−1)^𝑚.(2.11)
By applying basic operations to above equality, we can easily reach to the desired result.

As a special case, we derive the following.

Corollary 2.2. For 𝑘∈ℕ, one has [2]_𝑞(−1)^𝑘𝐵_𝑘,2𝑘(𝑥)^𝑘2𝑘=[2]^𝑞[𝑘/2]_𝑗=0^𝑘2𝑗2𝑘−2𝑗+1𝐺^{2𝑘−2𝑗+1,𝑞}(𝑥)+⎛⎜⎜⎝2𝑘𝑘⎞⎟⎟⎠(𝑞−1)^{[(𝑘−1)/2]}_𝑗=0^𝑘2𝑗+12𝑘−2𝑗𝐺^{2𝑘−2𝑗,𝑞}(𝑥).(2.12)

Proof. When 𝑘=𝑚 into (2.10), we derive the following identity: (−1)^𝑘𝐵_𝑘,2𝑘(𝑥)=^𝑘2𝑘1+𝑞^𝑘_𝑗=0⎡⎢⎢⎣𝑞⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠+(−1)_𝑗⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠⎤⎥⎥⎦𝐺_{2𝑘−𝑗+1,𝑞}(𝑥)=⎛⎜⎜⎝2𝑘𝑘⎞⎟⎟⎠_[𝑘/2]2𝑘−𝑗+1_𝑗=0^𝑘2𝑗2𝑘−2𝑗+1𝐺^{2𝑘−2𝑗+1,𝑞}+⎛⎜⎜⎝2𝑘𝑘⎞⎟⎟⎠𝑞−1(𝑥)𝑞+1^{[(𝑘−1)/2]}_𝑗=0^𝑘2𝑗+12𝑘−2𝑗𝐺^{2𝑘−2𝑗,𝑞}(𝑥).(2.13) Here, [𝑥] is greatest integer ≤𝑥. Then, we complete the proof of Theorem.

From (2.2), we note that 𝑑𝑑𝑥𝐺^𝑛,𝑞(𝑥)=𝑛^𝑛−1_𝑙=0⎛⎜⎜⎝𝑛−𝑙1⎞⎟⎟⎠𝐺_𝑙,𝑞𝑥^{𝑛−1−𝑙}=𝑛𝐺_{𝑛−1,𝑞}(𝑥).(2.14)

By (2.14) and (1.11), we easily see that ¹⁰𝐺_𝑛,𝑞(𝑥)𝑑𝑥=𝐺^𝑛+1,𝑞(1)−𝐺_𝑛+1,𝑞𝑛+1=−[2]^𝑞−1𝐺_𝑛+1,𝑞𝑛+1=−[2]^𝑞−1_ℤ𝑝𝜉^𝑛𝑑𝜇_−𝑞(𝜉).(2.15)

Now, let us consider definition of integral from 0 to 1 in (2.11), then we have −[2]_𝑞−1^{max{𝑘,𝑚}}_𝑗=0𝑞^𝑘𝑗+(−1)^𝑗^𝑚𝑗𝑘+𝑚−𝑗+1𝑘𝐺+𝑚−𝑗+^{𝑘+𝑚−𝑗+2,𝑞}2=[2]^𝑞=[2](−1)^𝑚𝐵(𝑘+1,𝑚+1)_𝑞(−1)^𝑚Γ(𝑘+1)Γ(𝑚+1)Γ(𝑘+𝑚+2),(2.16) where 𝐵(𝑘+1,𝑚+1) is beta function which is defined by 𝐵(𝑘+1,𝑚+1)=¹⁰𝑥^𝑘(1−𝑥)^𝑚𝑑𝑥=1(𝑘+𝑚+1)^{𝑚𝑘+𝑚},𝑘>0,𝑚>0.(2.17)

As a result, we obtain the following theorem.

Theorem 2.3. For 𝑚,𝑘∈ℕ, one has ^max^{𝑘,𝑚}_𝑗=1𝑞^𝑘𝑗+(−1)^𝑗^𝑚𝑗𝑘+𝑚−𝑗+1𝑘𝐺+𝑚−𝑗+^{𝑘+𝑚−𝑗+2,𝑞}2=𝑞(𝑘+𝑚(−+1)1)^{𝑚+1𝑘𝑘+𝑚}−𝑘+[2𝑚]^𝑞+1𝑘𝐺+𝑚+^{𝑘+𝑚+2,𝑞}2.(2.18)

Proof. By taking integral from 0 to 1 in (2.11), we easily reach to desired result.

Substituting 𝑚=𝑘+1 into Theorem 2.1, we readily get ^𝑘+1_𝑗=1𝑞^𝑘𝑗+(−1)^𝑗^𝑗𝑘+12𝑘−𝑗+3𝐺^{2𝑘−𝑗+3,𝑞}=𝑞(−1)^𝑘(2𝑘+2)^𝑘2𝑘+1−2𝑘[2+]^𝑞2𝐺2𝑘+3^2𝑘+3,𝑞.(2.19)

By differentiating both sides of (2.11) with respect to 𝑡, we have the following: ^max^{𝑘,𝑚}_𝑗=0⎧⎪⎨⎪⎩𝑞⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠+(−1)_𝑗⎛⎜⎜⎝𝑚𝑗⎞⎟⎟⎠⎫⎪⎬⎪⎭𝐺_{𝑘+𝑚−𝑗,𝑞}(𝑥)=[2]_𝑞𝑥^𝑘−1(𝑥−1)^𝑚−1((𝑘+𝑚)𝑥−𝑘).(2.20)

We now give interesting theorem for 𝑞-Genocchi numbers with weight 0 as follows.

Theorem 2.4. For 𝑘∈ℕ, one has [2]_𝑞[𝑘/2]_𝑗=0^𝑘2𝑗2𝑘−2𝑗+12𝐺𝑘−2𝑗+^{2𝑘−2𝑗+2,𝑞}2+(𝑞−1)^{[(𝑘−1)/2]}_𝑗=0^𝑘2𝑗+12𝑘−2𝑗2𝐺𝑘−2𝑗+^{2𝑘−2𝑗+1,𝑞}1=(2𝑘𝑞(+−11))^{𝑘+1𝑘2𝑘}.(2.21)

Proof. It is proved by using definition of integral on the both sides in the following equality, that is, ^𝑘_𝑗=0𝑞^𝑘𝑗+(−1)^𝑗^𝑘𝑗2𝑘−𝑗+1¹⁰𝐺_{2𝑘−𝑗+1,𝑞}(𝑥)𝑑𝑥=[2]_𝑞¹⁰𝑥^𝑘(𝑥−1)^𝑘𝑑𝑥.(2.22) Last from equality, we discover the following: [2]_𝑞[𝑘/2]_𝑗=0⎛⎜⎜⎝2𝑘𝑗⎞⎟⎟⎠∫¹⁰𝐺_{2𝑘−2𝑗+1,𝑞}(𝑥)𝑑𝑥2𝑘−2𝑗+1+(𝑞−1)^{[(𝑘−1)/2]}_𝑗=0⎛⎜⎜⎝2𝑗𝑘+1⎞⎟⎟⎠∫¹⁰𝐺_{2𝑘−2𝑗,𝑞}(𝑥)𝑑𝑥=[2]2𝑘−2𝑗_𝑞(−1)^𝑘¹⁰𝑥^𝑘(1−𝑥)^𝑘𝑑𝑥.(2.23) Then, taking integral from 0 to 1 both sides of last equality, we get −[2]_𝑞−1[2]_𝑞[𝑘/2]_𝑗=0^𝑘2𝑗2𝑘−2𝑗+12𝐺𝑘−2𝑗+^{2𝑘−2𝑗+2,𝑞}2+[2]^𝑞−1(1−𝑞)^{[(𝑘−1)/2]}_𝑗=0^𝑘2𝑗+1=[2]2𝑘−2𝑗2𝐺𝑘−2𝑗+^{2𝑘−2𝑗+1,𝑞}1_𝑞(−1)^𝑘𝐵(𝑘+1,𝑘+1)=[2]^𝑞(−1)^𝑘(2𝑘+1)^𝑘2𝑘.(2.24)
Thus, we complete the proof of the theorem.

Theorem 2.5. For 𝑘∈ℕ, one has [2]_{𝑞[(𝑘+1)/2]}_𝑗=0^𝑘2𝑗2𝑘−2𝑗+1𝐺^{2𝑘−2𝑗+1,𝑞}(𝑥)+^[𝑘/2]_𝑗=1^{𝑘2𝑗−1}2𝑘−2𝑗+1𝐺^{2𝑘−2𝑗+1,𝑞}(𝑥)−^[𝑘/2]_𝑗=0^𝑘2𝑗2𝑘−2𝑗𝐺^{2𝑘−2𝑗,𝑞}+(𝑞−1)[2]_𝑞[𝑘/2](𝑥)_𝑗=0⎛⎜⎜⎝2𝑗𝑘+1⎞⎟⎟⎠𝐺_{2𝑘−2𝑗,𝑞}(𝑥)[2]_𝑞(2𝑘−2𝑗)+𝐺^{2𝑘−2𝑗+1,𝑞}(𝑥)[2]^2𝑞(2𝑘−2𝑗+1)=𝑥^𝑘(𝑥−1)^𝑘[2]_𝑞𝑥−𝑞.(2.25)

Proof. In view of (2.2) and (2.23), we discover the following applications: ^𝑘+1_𝑗=0⎡⎢⎢⎣𝑞⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠+(−1)_𝑗⎛⎜⎜⎝𝑘+𝑗1⎞⎟⎟⎠⎤⎥⎥⎦𝐺_{2𝑘−𝑗+1,𝑞}(𝑥)2𝑘−𝑗+1=[2]^𝑞𝐺_2𝑘+1,𝑞(𝑥)+2𝑘+1_[(𝑘+1)/2]_𝑗=1⎡⎢⎢⎣𝑞⎛⎜⎜⎝2𝑘𝑗⎞⎟⎟⎠+⎛⎜⎜⎝2𝑘𝑗⎞⎟⎟⎠+⎛⎜⎜⎝2𝑗𝑘−1⎞⎟⎟⎠⎤⎥⎥⎦𝐺_{2𝑘−2𝑗+1,𝑞}+2𝑘−2𝑗+1(𝑥)_[𝑘/2]_𝑗=0⎡⎢⎢⎣𝑞⎛⎜⎜⎝2𝑗𝑘+1⎞⎟⎟⎠−⎛⎜⎜⎝2𝑗𝑘+1⎞⎟⎟⎠−⎛⎜⎜⎝2𝑘𝑗⎞⎟⎟⎠⎤⎥⎥⎦𝐺_{2𝑘−2𝑗,𝑞}(𝑥)=−⎧⎪⎨⎪⎩_[(𝑘+1)/2]2𝑘−2𝑗_𝑗=0⎛⎜⎜⎝2𝑘𝑗⎞⎟⎟⎠𝐺_{2𝑘−2𝑗,𝑞}(𝑥)2𝑘−2𝑗+1𝑞−+1𝑞^[𝑘/2]_𝑗=0⎛⎜⎜⎝2𝑗𝑘+1⎞⎟⎟⎠𝐺_{2𝑘−2𝑗+1,𝑞}(𝑥)2𝑘−2𝑗+1⎫⎪⎬⎪⎭+[2]^{𝑞[(𝑘+1)/2]}_𝑗=0×𝐺⎛⎜⎜⎝2𝑘𝑗_{2𝑘−2𝑗+1,𝑞}⎞⎟⎟⎠(𝑥)2𝑘−2𝑗+1+^[(𝑘+1)/2]_𝑗=1⎛⎜⎜⎝2𝑗𝑘−1⎞⎟⎟⎠𝐺_{2𝑘−2𝑗+1,𝑞}(𝑥)2𝑘−2𝑗+1−^[𝑘/2]_𝑗=0⎛⎜⎜⎝2𝑘𝑗⎞⎟⎟⎠𝐺_{2𝑘−2𝑗,𝑞}(𝑥)+(𝑞−1)_[𝑘/2]2𝑘−2𝑗_𝑗=0⎛⎜⎜⎝2𝑗𝑘+1⎞⎟⎟⎠𝐺_{2𝑘−2𝑗,𝑞}(𝑥)2𝑘−2𝑗+1𝑞−+1𝑞^[𝑘/2]_𝑗=0⎛⎜⎜⎝2𝑗𝑘+1⎞⎟⎟⎠𝐺_{2𝑘−2𝑗+1,𝑞}(𝑥)2𝑘−2𝑗+1.(2.26)
Thus, we give evidence of the theorem.

As 𝑞→1 into Theorem 2.5, it leads to the following interesting property.

Corollary 2.6. For 𝑘∈ℕ, one has ^[(𝑘^+1)/2]_𝑗=0⎛⎜⎜⎝2𝑘𝑗⎞⎟⎟⎠𝐺_{2𝑘−2𝑗+1}(𝑥)2𝑘+1−2𝑗+^[𝑘/2]_𝑗=1⎛⎜⎜⎝2𝑗𝑘−1⎞⎟⎟⎠𝐺_{2𝑘−2𝑗+1}(𝑥)4𝑘−4𝑗+2−^[𝑘/2]_𝑗=0⎛⎜⎜⎝2𝑘𝑗⎞⎟⎟⎠𝐺_{2𝑘−2𝑗,𝑞}(𝑥)=𝑥_𝑘4𝑘−4𝑗(𝑥−1)^𝑘𝑥−12,(2.27) where 𝐺_𝑛(𝑥) is ordinary Genocchi polynomials, which is defined by the means of the following generating function [9]: ^∞_𝑛=0𝐺_𝑛(𝑥)𝑡^𝑛𝑛!=𝑒^𝑡2+𝑡1𝑒^𝑥𝑡,|𝑡|<𝜋.(2.28)

3. Some Identities 𝑞-Genocchi Numbers and 𝑞-Bernoulli Numbers by Using Kim’s 𝑝-Adic 𝑞-Integrals on ℤ_𝑝

In this section, we consider 𝑞-Genocchi numbers and 𝑞-Bernoulli numbers by means of 𝑝-adic 𝑞-integral on ℤ_𝑝. Now, we start with the following theorem.

Theorem 3.1. For 𝑚,𝑘∈ℕ, one has ^max^{𝑘,𝑚}_𝑗=0𝑞^𝑘𝑗+(−1)^𝑗^𝑚𝑗𝑘+𝑚−𝑗+1^{𝑘+𝑚−𝑗+1}_𝑙=0^{𝑙𝑘+𝑚−𝑗+1}𝑙+1𝐺^{𝑘+𝑚−𝑗−𝑙+1,𝑞}𝐺_𝑙+1,𝑞=[2]_𝑞𝑚_𝑙=0⎛⎜⎜⎝𝑚𝑙⎞⎟⎟⎠(−1)_𝑚−𝑙𝐺_{𝑙+𝑘+1,𝑞}𝑙+𝑘+1.(3.1)

Proof. For 𝑚,𝑘∈ℕ, then by (2.11), 𝐼₁=[2]_𝑞_ℤ𝑝𝑥^𝑘(𝑥−1)^𝑚𝑑𝜇_−𝑞(𝑥)=[2]_𝑞𝑚_𝑙=0⎛⎜⎜⎝𝑚𝑙⎞⎟⎟⎠(−1)_𝑚−𝑙_ℤ𝑝=[2]𝑥^𝑙+𝑘𝑑𝜇_−𝑞(𝑥)_𝑞𝑚_𝑙=0⎛⎜⎜⎝𝑚𝑙⎞⎟⎟⎠(−1)_𝑚−𝑙𝐺_{𝑙+𝑘+1,𝑞}𝑙+𝑘+1.(3.2) On the other hand, the right hand side of (2.11), 𝐼₂=^{max{𝑘,𝑚}}_𝑗=0𝑞^𝑘𝑗+(−1)^𝑗^𝑚𝑗𝑘+𝑚−𝑗+1^{𝑘+𝑚−𝑗+1}_𝑙=0⎛⎜⎜⎝𝑘+𝑚𝑙−𝑗+1⎞⎟⎟⎠𝐺_{𝑘+𝑚−𝑗−𝑙+1,𝑞}_ℤ𝑝=𝑥^𝑙𝑑𝜇_−𝑞(𝑥)_{max{𝑘,𝑚}}_𝑗=0𝑞^𝑘𝑗+(−1)^𝑗^𝑚𝑗𝑘+𝑚−𝑗+1^{𝑘+𝑚−𝑗+1}_𝑙=0^{𝑙𝑘+𝑚−𝑗+1}𝑙+1𝐺^{𝑘+𝑚−𝑗−𝑙+1,𝑞}𝐺_𝑙+1,𝑞.(3.3) Combining 𝐼₁ and 𝐼₂, we arrive to the proof of the theorem.

Theorem 3.2. For 𝑘∈ℕ, one has ^𝑘_𝑙=0⎛⎜⎜⎝𝑘𝑙⎞⎟⎟⎠(−1)_𝑘−𝑙[2]^𝑞𝐺_{𝑘+𝑙+2,𝑞}=[2]𝑘+𝑙+2−𝑞𝑘𝐺+𝑙+^{𝑘+𝑙+1,𝑞}1_𝑞[𝑘/2]_𝑗=0^𝑘2𝑗2𝑘−2𝑗+1^{2𝑘−2𝑗+1}_𝑙=0^{𝑙2𝑘−2𝑗+1}𝑙+1𝐺^{2𝑘+1−2𝑗−𝑙,𝑞}𝐺_𝑙+1,𝑞+^[𝑘/2]_𝑗=1^{𝑘2𝑗−1}2𝑘−2𝑗+1^{2𝑘−2𝑗+1}_𝑙=0^{𝑙2𝑘−2𝑗+1}𝑙+1𝐺^{2𝑘+1−2𝑗−𝑙,𝑞}𝐺_𝑙+1,𝑞+𝑞−11+𝑞^[𝑘/2]_𝑗=0⎛⎜⎜⎝2𝑗𝑘+1⎞⎟⎟⎠𝑇_{𝑞𝑘,𝑗},(3.4) here 1𝑇))^{𝑞𝑘,𝑗}(𝐺=_𝑙+1,𝑞[2]_𝑞∑^{2𝑘−2𝑗𝑙=0}(^{𝑙2𝑘−2𝑗}/(2𝑘−2𝑗))(𝐺_𝑙+1,𝑞𝐺_{2𝑘−2𝑗−𝑙,𝑞}/(𝑙+1))+∑^{2𝑘−2𝑗+1𝑙=0}(^{𝑙2𝑘−2𝑗+1}/(2𝑘−2𝑗+𝐺_{2𝑘−2𝑗−𝑙+1,𝑞}/(𝑙+1)).

Proof. Let us take fermionic 𝑝-adic 𝑞-inetgral on ℤ_𝑝 left-hand side of Theorem 2.5, we get 𝐼₃==[2]_ℤ𝑝𝑥^𝑘(𝑥−1)^𝑘[2]_𝑞𝑥−𝑞𝑑𝜇_−𝑞(𝑥)_𝑞𝑘_𝑙=0⎛⎜⎜⎝𝑘𝑙⎞⎟⎟⎠(−1)_𝑘−𝑙_ℤ𝑝𝑥^𝑘+𝑙+1𝑑𝜇_−𝑞(𝑥)−𝑞^𝑘_𝑙=0⎛⎜⎜⎝𝑘𝑙⎞⎟⎟⎠(−1)_𝑘−𝑙_ℤ𝑝=[2]𝑥^𝑘+𝑙𝑑𝜇_−𝑞(𝑥)_𝑞𝑘_𝑙=0⎛⎜⎜⎝𝑘𝑙⎞⎟⎟⎠(−1)_𝑘−𝑙𝐺_{𝑘+𝑙+2,𝑞}𝑘+𝑙+2−𝑞^𝑘_𝑙=0⎛⎜⎜⎝𝑘𝑙⎞⎟⎟⎠(−1)_𝑘−𝑙𝐺_{𝑘+𝑙+1,𝑞}𝑘+𝑙+1.(3.5) In other word, we consider the right-hand side of Theorem 2.5 as follows: 𝐼₄=[2]_𝑞[𝑘/2]_𝑗=0^𝑘2𝑗2𝑘−2𝑗+1^{2𝑘−2𝑗+1}_𝑙=0⎛⎜⎜⎝2𝑘−2𝑙𝑗+1⎞⎟⎟⎠𝐺_{2𝑘+1−2𝑗−𝑙,𝑞}_ℤ𝑝+𝑥^𝑙𝑑𝜇_−𝑞(𝑥)_[𝑘/2]_𝑗=1^{𝑘2𝑗−1}2𝑘−2𝑗+1^{2𝑘−2𝑗+1}_𝑙=0⎛⎜⎜⎝2𝑘−2𝑙𝑗+1⎞⎟⎟⎠𝐺_{2𝑘+1−2𝑗−𝑙,𝑞}_ℤ𝑝+𝑥^𝑙𝑑𝜇_−𝑞(𝑥)_[𝑘/2]_𝑗=0⎛⎜⎜⎝2𝑗𝑘+1⎞⎟⎟⎠⎧⎪⎪⎨⎪⎪⎩(𝑞−1)∑^{2𝑘−2𝑗}_𝑗=0^{𝑙2𝑘−2𝑗−𝑙}2𝑘−2𝑗𝐺^{2𝑘−2𝑗−𝑙,𝑞}+𝑞−1∫_ℤ𝑝𝑥^𝑙𝑑𝜇_−𝑞(𝑥)1+𝑞^{2𝑘−2𝑗+1}_𝑙∑₌₀^{𝑙2𝑘−2𝑗+1}2𝑘−2𝑗+1𝐺^{2𝑘−2𝑗−𝑙+1}=[2]∫_ℤ𝑝𝑥^𝑙𝑑𝜇_−𝑞(𝑥)_𝑞[𝑘/2]⎫⎪⎪⎬⎪⎪⎭_𝑗=0^𝑘2𝑗2𝑘−2𝑗+1^{2𝑘−2𝑗+1}_𝑙=0^{𝑙2𝑘−2𝑗−𝑙+1}𝑙+1𝐺^{2𝑘+1−2𝑗−𝑙,𝑞}𝐺_𝑙+1,𝑞+^[𝑘/2]_𝑗=1^{𝑘2𝑗−1}2𝑘−2𝑗+1^{2𝑘−2𝑗+1}_𝑙=0^{𝑙2𝑘−2𝑗−𝑙+1}𝑙+1𝐺^{2𝑘+1−2𝑗−𝑙,𝑞}𝐺_𝑙+1,𝑞+^[𝑘/2]_𝑗=0⎛⎜⎜⎝2𝑗𝑘+1⎞⎟⎟⎠⎧⎪⎪⎨⎪⎪⎩(𝑞−1)∑^{2𝑘−2𝑗}_𝑗=0^{𝑙2𝑘−2𝑗−𝑙}2𝑘−2𝑗𝐺^{2𝑘−2𝑗−𝑙,𝑞}𝐺_𝑙+1,𝑞+𝑞−1𝑙+11+𝑞^{2𝑘−2𝑗+1}_𝑙∑₌₀^{𝑙2𝑘−2𝑗−𝑙+1}2𝑘−2𝑗+1𝐺^{2𝑘−2𝑗−𝑙+1}𝐺_𝑙+1,𝑞𝑙+1⎫⎪⎪⎬⎪⎪⎭.(3.6)
Equating 𝐼₃ and 𝐼₄, we complete the proof of the theorem.

As 𝑞→1 in the above theorem, we reach interesting property in Analytic Numbers Theory concerning ordinary Genocchi polynomials.

Corollary 3.3. For 𝑘∈ℕ, one has ^𝑘_𝑙=0⎛⎜⎜⎝𝑘𝑙⎞⎟⎟⎠(−1)_𝑘−𝑙2𝑘𝐺+𝑙+^𝑘+𝑙+22−𝑘𝐺+𝑙+^𝑘+𝑙+11=2^[𝑘/2]_𝑗=0^𝑘2𝑗2𝑘−2𝑗+1^{2𝑘−2𝑗+1}_𝑙=0^{𝑙2𝑘−2𝑗+1}𝑙+1𝐺^{2𝑘+1−2𝑗−𝑙}𝐺_𝑙+1+^[𝑘/2]_𝑗=1^{𝑘2𝑗−1}2𝑘−2𝑗+1^{2𝑘−2𝑗+1}_𝑙=0^{𝑙2𝑘−2𝑗+1}𝑙+1𝐺^{2𝑘+1−2𝑗−𝑙}𝐺_𝑙+1.(3.7)

Theorem 3.4. For 𝑚,𝑘∈ℕ, one has [2]_𝑞𝑚_𝑙=0⎛⎜⎜⎝𝑚𝑙⎞⎟⎟⎠(−1)_𝑚−𝑙𝐵_{𝑙+𝑘,𝑞}=^{max{𝑘,𝑚}}_𝑗=0𝑞^𝑘𝑗+(−1)^𝑗^𝑚𝑗𝑘+𝑚−𝑗+1^{𝑘+𝑚−𝑗+1}_𝑙=0⎛⎜⎜⎝𝑘+𝑚𝑙−𝑗+1⎞⎟⎟⎠𝐺_{𝑘+𝑚+1−𝑗−𝑙,𝑞}𝐵_𝑙,𝑞.(3.8)

Proof. We consider (2.11) and (2.2) by means of 𝑞-Volkenborn integral. Then, by (2.11), we see [2]_𝑞_ℤ𝑝𝑥^𝑘(𝑥−1)^𝑚𝑑𝜇_𝑞(𝑥)=[2]_𝑞𝑚_𝑙=0⎛⎜⎜⎝𝑚𝑙⎞⎟⎟⎠(−1)_𝑚−𝑙_ℤ𝑝𝑥^𝑙+𝑘𝑑𝜇_𝑞(𝑥)=[2]_𝑞𝑚_𝑙=0⎛⎜⎜⎝𝑚𝑙⎞⎟⎟⎠(−1)_𝑚−𝑙𝐵_{𝑙+𝑘,𝑞}.(3.9) On the other hand, ^max^{𝑘,𝑚}_𝑗=0𝑞^𝑘𝑗+(−1)^𝑗^𝑚𝑗𝑘+𝑚−𝑗+1^{𝑘+𝑚−𝑗+1}_𝑙=0⎛⎜⎜⎝𝑘+𝑚𝑙−𝑗+1⎞⎟⎟⎠𝐺_{𝑘+𝑚+1−𝑗−𝑙,𝑞}_ℤ𝑝=𝑥^𝑙𝑑𝜇_𝑞(𝑥)_{max{𝑘,𝑚}}_𝑗=0𝑞^𝑘𝑗+(−1)^𝑗^𝑚𝑗𝑘+𝑚−𝑗+1^{𝑘+𝑚−𝑗+1}_𝑙=0⎛⎜⎜⎝𝑘+𝑚𝑙−𝑗+1⎞⎟⎟⎠𝐺_{𝑘+𝑚+1−𝑗−𝑙,𝑞}𝐵_𝑙,𝑞.(3.10) Therefore, we get the proof of theorem.

Corollary 3.5. For 𝑘∈ℕ, one gets ^𝑘_𝑙=0⎛⎜⎜⎝𝑘𝑙⎞⎟⎟⎠(−1)_𝑘−𝑙[2]^𝑞𝐵_{𝑘+𝑙+1,𝑞}−𝑞𝐵_{𝑘+𝑙,𝑞}=[2]_𝑞[𝑘/2]_𝑗=0^𝑘2𝑗2𝑘−2𝑗+1^{2𝑘−2𝑗+1}_𝑙=0⎛⎜⎜⎝2𝑘−2𝑙𝑗+1⎞⎟⎟⎠𝐺_{2𝑘+1−2𝑗−𝑙,𝑞}𝐵_𝑙,𝑞+^[𝑘/2]_𝑗=1^{𝑘2𝑗−1}×2𝑘−2𝑗+1_{2𝑘−2𝑗+1}_𝑙=0⎛⎜⎜⎝2k−2𝑙𝑗+1⎞⎟⎟⎠𝐺_{2𝑘+1−2𝑗−𝑙,𝑞}𝐵_𝑙,𝑞+𝑞−1𝑞+1^[𝑘/2]_𝑗=0⎛⎜⎜⎝2𝑗𝑘+1⎞⎟⎟⎠𝑆_{𝑞𝑘,𝑗},(3.11) where 𝑆^{𝑞𝑘,𝑗}=[2]_𝑞∑^{2𝑘−2𝑗𝑗=0}(1/(2𝑘−2𝑗))^{𝑙2𝑘−2𝑗}𝐺_{2𝑘−2𝑗−𝑙,𝑞}𝐵_𝑙,𝑞+∑^{2𝑘−2𝑗+1𝑙=0}(1/(2𝑘−2𝑗+1))^{𝑙2𝑘−2𝑗+1}𝐺_{2𝑘−2𝑗−𝑙+1}𝐵_𝑙,𝑞.

Proof. By using 𝑝-adic 𝑞-integral on ℤ_𝑝 left-hand side of Theorem 2.5, we get 𝐼₅=[2]_𝑞_ℤ𝑝=[2]𝑥^𝑘(𝑥−1)^𝑘([2]𝑥−𝑞)𝑑𝜇_𝑞(𝑥)_𝑞𝑘_𝑙=0⎛⎜⎜⎝𝑘𝑙⎞⎟⎟⎠(−1)_𝑘−𝑙_ℤ𝑝𝑥^𝑘+𝑙+1𝑑𝜇_𝑞(𝑥)−𝑞^𝑘_𝑙=0⎛⎜⎜⎝𝑘𝑙⎞⎟⎟⎠(−1)_𝑘−𝑙_ℤ𝑝=[2]𝑥^𝑘+𝑙𝑑𝜇_𝑞(𝑥)_𝑞𝑘_𝑙=0⎛⎜⎜⎝𝑘𝑙⎞⎟⎟⎠(−1)_𝑘−𝑙𝐵_{𝑘+𝑙+1,𝑞}−𝑞^𝑘_𝑙=0⎛⎜⎜⎝𝑘𝑙⎞⎟⎟⎠(−1)_𝑘−𝑙𝐵_{𝑘+𝑙,𝑞}.(3.12) Also, we compute the right-hand side of Theorem 2.5 as follows: 𝐼₆=[2]_𝑞[𝑘/2]_𝑗=012𝑘−2𝑗+1⎛⎜⎜⎝2𝑘𝑗⎞⎟⎟⎠^{2𝑘−2𝑗+1}_𝑙=0⎛⎜⎜⎝2𝑘−2𝑙𝑗+1⎞⎟⎟⎠𝐺_{2𝑘+1−2𝑗−𝑙,𝑞}_ℤ𝑝+𝑥^𝑙𝑑𝜇_𝑞(𝑥)_[𝑘/2]_𝑗=112𝑘−2𝑗+1⎛⎜⎜⎝2𝑗𝑘−1⎞⎟⎟⎠^{2𝑘−2𝑗+1}_𝑙=0⎛⎜⎜⎝2𝑘−2𝑙𝑗+1⎞⎟⎟⎠𝐺_{2𝑘+1−2𝑗−𝑙,𝑞}_ℤ𝑝+𝑥^𝑙𝑑𝜇_𝑞(𝑥)_[𝑘/2]_𝑗=0⎛⎜⎜⎝2𝑗𝑘+1⎞⎟⎟⎠⎧⎪⎪⎪⎨⎪⎪⎪⎩(𝑞−1)^{2𝑘−2𝑗}_𝑗∑₌₀12𝑘−2𝑗⎛⎜⎜⎝2𝑘−𝑙2𝑗⎞⎟⎟⎠𝐺^{2𝑘−2𝑗−𝑙,𝑞}+𝑞−1∫_ℤ𝑝𝑥^𝑙𝑑𝜇_𝑞(𝑥)1+𝑞^{2𝑘−2𝑗+1}_𝑙∑₌₀12𝑘−2𝑗+1⎛⎜⎜⎝2𝑘−2𝑙𝑗+1⎞⎟⎟⎠𝐺^{2𝑘−2𝑗−𝑙+1}=[2]∫_ℤ𝑝𝑥^𝑙𝑑𝜇_𝑞(𝑥)_𝑞[𝑘/2]⎫⎪⎪⎪⎬⎪⎪⎪⎭_𝑗=012𝑘−2𝑗+1⎛⎜⎜⎝2𝑘𝑗⎞⎟⎟⎠^{2𝑘−2𝑗+1}_𝑙=0⎛⎜⎜⎝2𝑘−2𝑙𝑗+1⎞⎟⎟⎠𝐺_{2𝑘+1−2𝑗−𝑙,𝑞}𝐵_𝑙,𝑞+^[𝑘/2]_𝑗=112𝑘−2𝑗+1⎛⎜⎜⎝2𝑗𝑘−1⎞⎟⎟⎠^{2𝑘−2𝑗+1}_𝑙=0⎛⎜⎜⎝2𝑘−2𝑙𝑗+1⎞⎟⎟⎠𝐺_{2𝑘+1−2𝑗−𝑙,𝑞}𝐵_𝑙,𝑞+^[𝑘/2]_𝑗=0⎛⎜⎜⎝2𝑗𝑘+1⎞⎟⎟⎠⎧⎪⎪⎪⎨⎪⎪⎪⎩(𝑞−1)^{2𝑘−2𝑗}_𝑗∑₌₀12𝑘−2𝑗⎛⎜⎜⎝2𝑘−𝑙2𝑗⎞⎟⎟⎠𝐺^{2𝑘−2𝑗−𝑙,𝑞}𝐵_𝑙,𝑞+𝑞−11+𝑞^{2𝑘−2𝑗+1}_𝑙∑₌₀12𝑘−2𝑗+1⎛⎜⎜⎝2𝑘−2𝑙𝑗+1⎞⎟⎟⎠𝐺^{2𝑘−2𝑗−𝑙+1}𝐵_𝑙,𝑞⎫⎪⎪⎪⎬⎪⎪⎪⎭.(3.13)
Equating 𝐼₅ and 𝐼₆, we get the proof of Corollary.

As 𝑞→1 in the above theorem, we easily derive the following corollary.

Corollary 3.6. For 𝑘∈ℕ, one has ^𝑘_𝑙=0⎛⎜⎜⎝𝑘𝑙⎞⎟⎟⎠(−1)_𝑘−𝑙2𝐵_𝑘+𝑙+1−𝐵_𝑘+𝑙=2^[𝑘/2]_𝑗=0^𝑘2𝑗2𝑘−2𝑗+1^{2𝑘−2𝑗+1}_𝑙=0⎛⎜⎜⎝2𝑘−2𝑙𝑗+1⎞⎟⎟⎠𝐺_{2𝑘+1−2𝑗−𝑙}𝐵_𝑙+^[𝑘/2]_𝑗=1^{𝑘2𝑗−1}2𝑘−2𝑗+1^{2𝑘−2𝑗+1}_𝑙=0⎛⎜⎜⎝2𝑘−2𝑙𝑗+1⎞⎟⎟⎠𝐺_{2𝑘+1−2𝑗−𝑙}𝐵_𝑙.(3.14)

Acknowledgment

The author would like to express his sincere gratitude to the referee for his/her valuable comments and suggestions which have improved the presentation of the paper.