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 ℚ𝑝 be the field 𝑝-adic rational numbers and let ℂ𝑝 be the completion of algebraic closure of ℚ𝑝.

Thus, ℚ𝑝=𝑥=âˆžî“ğ‘›=âˆ’ğ‘˜ğ‘Žğ‘›ğ‘ğ‘›âˆ¶0â‰¤ğ‘Žğ‘›îƒ°.<𝑝(1.1)

Then ℤ𝑝 is integral domain, which is defined by ℤ𝑝=𝑥=âˆžî“ğ‘›=0ğ‘Žğ‘›ğ‘ğ‘›âˆ¶0â‰¤ğ‘Žğ‘›îƒ°,<𝑝(1.2) or ℤ𝑝=𝑥∈ℚ𝑝∶|𝑥|𝑝.≤1(1.3)

We assume that ğ‘žâˆˆâ„‚ğ‘ with |1âˆ’ğ‘ž|𝑝<1 as an indeterminate. The 𝑝-adic absolute value |⋅|𝑝, is normally defined by |𝑥|𝑝=𝑝−𝑟,(1.4) where 𝑥=𝑝𝑟(𝑠/𝑡) with (𝑝,𝑠)=(𝑝,𝑡)=(𝑠,𝑡)=1, and 𝑟∈ℚ.

[𝑥]ğ‘ž is a ğ‘ž-extension of 𝑥, which is defined by [𝑥]ğ‘ž=1âˆ’ğ‘žğ‘¥,1âˆ’ğ‘ž(1.5) we note that limğ‘žâ†’1[𝑥]ğ‘ž=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 ğ‘Žâˆˆâ„¤ğ‘, if the difference quotient, 𝐹𝑓(𝑥,𝑦)=𝑓(𝑥)−𝑓(𝑦),𝑥−𝑦(1.6) has a limit ğ‘“î…ž(ğ‘Ž) as (𝑥,𝑦)→(ğ‘Ž,ğ‘Ž), and we denote this by 𝑓∈𝑈𝐷(ℤ𝑝). Then, for 𝑓∈𝑈𝐷(ℤ𝑝), we can start with the following expression: 1î€ºğ‘ğ‘î€»ğ‘žî“0≤𝜉<𝑝𝑁𝑓(𝜉)ğ‘žğœ‰=0≤𝜉<𝑝𝑁𝑓(𝜉)ğœ‡ğ‘žî€·ğœ‰+𝑝𝑁ℤ𝑝,(1.7) which represents a 𝑝-adic ğ‘ž-analogue of Riemann sums for 𝑓. The integral of 𝑓 on ℤ𝑝 will be defined as the limit (ğ‘â†’âˆž) of these sums, when it exists. The 𝑝-adic ğ‘ž-integral of function 𝑓∈𝑈𝐷(ℤ𝑝) is defined by Kim in [7, 12] as ğ¼ğ‘ž(𝑓)=ℤ𝑝𝑓(𝜉)ğ‘‘ğœ‡ğ‘ž(𝜉)=limğ‘â†’âˆž1î€ºğ‘ğ‘î€»ğ‘žğ‘ğ‘âˆ’1𝜉=0𝑓(𝜉)ğ‘žğœ‰.(1.8)

The bosonic integral is considered as a bosonic limit ğ‘žâ†’1, 𝐼1(𝑓)=limğ‘žâ†’1ğ¼ğ‘ž(𝑓). Similarly, the fermionic 𝑝-adic integral on ℤ𝑝 is introduced by Kim as follows: ğ¼âˆ’ğ‘ž(𝑓)=ℤ𝑝𝑓(𝜉)ğ‘‘ğœ‡âˆ’ğ‘ž(𝜉)(1.9) (for more details, see [13–16]).

From (1.9), it is well-known equality that ğ‘žğ¼âˆ’ğ‘žî€·ğ‘“1+ğ¼âˆ’ğ‘ž[2](𝑓)=ğ‘žğ‘“(0),(1.10) where 𝑓1(𝑥)=𝑓(𝑥+1) (for details, see [2, 3, 8, 9, 12, 13, 15–17]).

The ğ‘ž-Genocchi polynomials with wegiht 0 are introduced as 𝐺𝑛+1,ğ‘ž(𝑥)=𝑛+1ℤ𝑝(𝑥+𝜉)ğ‘›ğ‘‘ğœ‡âˆ’ğ‘ž(𝜉).(1.11)

From (1.11), we have 𝐺𝑛,ğ‘ž(𝑥)=𝑛𝑙=0âŽ›âŽœâŽœâŽğ‘›ğ‘™âŽžâŽŸâŽŸâŽ ğ‘¥ğ‘™î‚ğºğ‘›âˆ’ğ‘™,ğ‘ž,(1.12) where 𝐺𝑛,ğ‘žî‚ğº(0)∶=𝑛,ğ‘ž are called ğ‘ž-Genocchi numbers with weight 0. Then, ğ‘ž-Genocchi numbers are defined as 𝐺0,ğ‘žî‚€î‚ğº=0,ğ‘žğ‘žî‚+1𝑛+𝐺𝑛,ğ‘ž=[2]ğ‘ž,if𝑛=1,0,if𝑛≠1,(1.13) with the usual convention about replacing (î‚ğºğ‘ž)𝑛 by 𝐺𝑛,ğ‘ž is used (for details, see [3]).

Let 𝑈𝐷(ℤ𝑝) be the space of continuous functions on ℤ𝑝. For 𝑓∈𝑈𝐷(ℤ𝑝), 𝑝-adic analogue of Bernstein operator for 𝑓 is defined by 𝐵𝑛(𝑓,𝑥)=𝑛𝑘=0𝑓𝑘𝑛𝐵𝑘,𝑛(𝑥)=𝑛𝑘=0ğ‘“î‚€ğ‘˜ğ‘›î‚âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ ğ‘¥ğ‘˜(1−𝑥)𝑛−𝑘,(1.14) where 𝑛,𝑘∈ℕ∗. Here, 𝐵𝑘,𝑛(𝑥) is called 𝑝-adic Bernstein polynomials, which are defined by 𝐵𝑘,ğ‘›âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ ğ‘¥(𝑥)=𝑘(1−𝑥)𝑛−𝑘[],𝑥∈0,1(1.15) (for details, see [1, 4, 5, 7]).

The ğ‘ž-Bernoulli polynomials and numbers with weight 0 are defined by Kim et al., respectively, 𝐵𝑛,ğ‘ž(𝑥)=limğ‘›â†’âˆž1[𝑝𝑛]ğ‘žğ‘ğ‘›âˆ’1𝑦=0(𝑥+𝑦)ğ‘›ğ‘žğ‘¦=ℤ𝑝(𝑥+𝜉)ğ‘›ğ‘‘ğœ‡ğ‘žî‚ğµ(𝜉),𝑛,ğ‘ž=î€œâ„¤ğ‘ğœ‰ğ‘›ğ‘‘ğœ‡ğ‘ž(𝜉)(1.16) (for more information, see [10]).

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

2. Novel Properties of ğ‘ž-Genocchi Numbers and Polynomials with Weight 0

Let 𝑓(𝑥)=𝑒𝑡(𝑥+𝑦). Then, by using (1.10), we easily procure the following: ℤ𝑝𝑒𝑡(𝑥+𝜉)ğ‘‘ğœ‡âˆ’ğ‘ž[2](𝜉)=ğ‘žğ‘žğ‘’ğ‘¡ğ‘’+1𝑥𝑡.(2.1)

From the last equality, by (1.11), we get Araci, Acikgoz, and Qi’s ğ‘ž-Genocchi polynomials with weight 0 in [3] as follows: [2]ğ‘žğ‘¡ğ‘žğ‘’ğ‘¡ğ‘’+1𝑥𝑡=âˆžî“ğ‘›=0𝐺𝑛,ğ‘ž(𝑡𝑥)𝑛,||||𝑛!logğ‘ž+𝑡<𝜋.(2.2)

Here, we assume that 𝑥 is a fixed parameter. Let î‚ğ¹ğ‘ž([2]𝑥,𝑡)=ğ‘žğ‘žğ‘’ğ‘¡ğ‘’+1𝑥𝑡=âˆžî“ğ‘›=0𝐺𝑛,ğ‘ž(𝑡𝑥)𝑛−1.𝑛!(2.3) Thus, by expression of (2.3), we can readily see the following: ğ‘žğ‘’ğ‘¡î‚ğ¹ğ‘žî‚ğ¹(𝑥,𝑡)+ğ‘ž[2](𝑥,𝑡)=ğ‘žğ‘’ğ‘¥ğ‘¡.(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 ğ‘žğ‘’ğ‘¡(𝐷+𝐼)ğ‘˜î‚ğ¹ğ‘ž(𝑥,𝑡)+ğ·ğ‘˜î‚ğ¹ğ‘ž[2](𝑥,𝑡)=ğ‘žğ‘¥ğ‘˜ğ‘’ğ‘¥ğ‘¡,(2.5) where 𝑘∈ℕ∗ and 𝐼 is identity operator. By multiplying 𝑒−𝑡 on both sides of (2.5), we get ğ‘ž(𝐷+𝐼)ğ‘˜î‚ğ¹ğ‘ž(𝑥,𝑡)+ğ‘’âˆ’ğ‘¡ğ·ğ‘˜î‚ğ¹ğ‘ž[2](𝑥,𝑡)=ğ‘žğ‘¥ğ‘˜ğ‘’(𝑥−1)𝑡.(2.6)

Let us take derivative operator 𝐷𝑚(𝑚∈ℕ) on the both sides of (2.6). Then, we get ğ‘žğ‘’ğ‘¡ğ·ğ‘š(𝐷+𝐼)ğ‘˜î‚ğ¹ğ‘ž(𝑥,𝑡)+𝐷𝑘(𝐷−𝐼)ğ‘šî‚ğ¹ğ‘ž[2](𝑥,𝑡)=ğ‘žğ‘¥ğ‘˜(𝑥−1)𝑚𝑒𝑥𝑡.(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=0âŽ›âŽœâŽœâŽğ‘˜ğ‘—âŽžâŽŸâŽŸâŽ î‚€ğ‘žğ‘’ğ‘¡ğ·ğ‘˜+ğ‘šâˆ’ğ‘—î‚ğ¹ğ‘žî‚[0]+(𝑥,𝑡)𝑚𝑗=0âŽ›âŽœâŽœâŽğ‘šğ‘—âŽžâŽŸâŽŸâŽ (−1)𝑗𝐷𝑘+ğ‘šâˆ’ğ‘—î‚ğ¹ğ‘žî‚[0]=[2](𝑥,𝑡)ğ‘žğ‘¥ğ‘˜(𝑥−1)𝑚.(2.8)

By (2.3), we easily see î‚€ğ·ğ‘î‚ğ¹ğ‘žî‚[0]=𝐺(𝑥,𝑡)𝑁+1,ğ‘ž(𝑥),𝑒𝑁+1ğ‘¡ğ·ğ‘î‚ğ¹ğ‘žî‚[0]=𝐺(𝑥,𝑡)𝑁+1,ğ‘ž(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]ğ‘ž(−1)𝑚𝐵𝑘,𝑚+𝑘(𝑥)𝑘𝑚+𝑘=max{𝑘,𝑚}𝑗=0ğ‘žî€·ğ‘˜ğ‘—î€¸+(−1)𝑗𝑚𝑗𝐺𝑘+𝑚−𝑗+1𝑘+𝑚−𝑗+1,ğ‘ž(𝑥).(2.10)

Proof. By expressions of (2.8) and (2.9), we see that max{𝑘,𝑚}𝑗=0ğ‘žî€·ğ‘˜ğ‘—î€¸+(−1)𝑗𝑚𝑗𝐺𝑘+𝑚−𝑗+1𝑘+𝑚−𝑗+1,ğ‘ž[2](𝑥)=ğ‘žğ‘¥ğ‘˜(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]ğ‘ž(−1)𝑘𝐵𝑘,2𝑘(𝑥)𝑘2𝑘=[2]ğ‘ž[𝑘/2]𝑗=0𝑘2𝑗𝐺2𝑘−2𝑗+12𝑘−2𝑗+1,ğ‘žâŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ (𝑥)+2𝑘(ğ‘žâˆ’1)[(𝑘−1)/2]𝑗=0𝑘2𝑗+1𝐺2𝑘−2𝑗2𝑘−2𝑗,ğ‘ž(𝑥).(2.12)

Proof. When 𝑘=𝑚 into (2.10), we derive the following identity: (−1)𝑘𝐵𝑘,2𝑘(𝑥)=𝑘2𝑘1+ğ‘žğ‘˜î“ğ‘—=0âŽ¡âŽ¢âŽ¢âŽ£ğ‘žâŽ›âŽœâŽœâŽğ‘˜ğ‘—âŽžâŽŸâŽŸâŽ +(−1)ğ‘—âŽ›âŽœâŽœâŽğ‘˜ğ‘—âŽžâŽŸâŽŸâŽ âŽ¤âŽ¥âŽ¥âŽ¦î‚ğº2𝑘−𝑗+1,ğ‘ž(𝑥)=âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ 2𝑘−𝑗+12𝑘[𝑘/2]𝑗=0𝑘2𝑗𝐺2𝑘−2𝑗+12𝑘−2𝑗+1,ğ‘ž(+âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ ğ‘¥)2ğ‘˜ğ‘žâˆ’1ğ‘ž+1[(𝑘−1)/2]𝑗=0𝑘2𝑗+1𝐺2𝑘−2𝑗2𝑘−2𝑗,ğ‘ž(𝑥).(2.13) Here, [𝑥] is greatest integer ≤𝑥. Then, we complete the proof of Theorem.

From (2.2), we note that 𝑑𝐺𝑑𝑥𝑛,ğ‘žî‚(𝑥)=𝑛𝑛−1𝑙=0âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğºğ‘›âˆ’1𝑙,ğ‘žğ‘¥ğ‘›âˆ’1−𝑙𝐺=𝑛𝑛−1,ğ‘ž(𝑥).(2.14)

By (2.14) and (1.11), we easily see that 10𝐺𝑛,ğ‘žî‚ğº(𝑥)𝑑𝑥=𝑛+1,ğ‘žî‚ğº(1)−𝑛+1,ğ‘ž[2]𝑛+1=âˆ’ğ‘žâˆ’1𝐺𝑛+1,ğ‘ž[2]𝑛+1=âˆ’ğ‘žâˆ’1î€œâ„¤ğ‘ğœ‰ğ‘›ğ‘‘ğœ‡âˆ’ğ‘ž(𝜉).(2.15)

Now, let us consider definition of integral from 0 to 1 in (2.11), then we have −[2]ğ‘žâˆ’1max{𝑘,𝑚}𝑗=0ğ‘žî€·ğ‘˜ğ‘—î€¸+(−1)𝑗𝑚𝑗𝐺𝑘+𝑚−𝑗+1𝑘+𝑚−𝑗+2,ğ‘ž=[2]𝑘+𝑚−𝑗+2ğ‘ž(−1)𝑚=[2]𝐵(𝑘+1,𝑚+1)ğ‘ž(−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)=10𝑥𝑘(1−𝑥)𝑚1𝑑𝑥=(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=1ğ‘žî€·ğ‘˜ğ‘—î€¸+(−1)𝑗𝑚𝑗𝐺𝑘+𝑚−𝑗+1𝑘+𝑚−𝑗+2,ğ‘žğ‘˜+𝑚−𝑗+2=ğ‘ž(−1)𝑚+1(𝑘+𝑚+1)𝑘𝑘+𝑚−[2]ğ‘žî‚ğºğ‘˜+𝑚+1𝑘+𝑚+2,ğ‘ž.𝑘+𝑚+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=1ğ‘žî€·ğ‘˜ğ‘—î€¸+(−1)𝑗𝑗𝑘+1𝐺2𝑘−𝑗+32𝑘−𝑗+3,ğ‘ž=ğ‘ž(−1)𝑘(2𝑘+2)𝑘2𝑘+1−[2]ğ‘žî‚ğº2𝑘+22𝑘+3,ğ‘ž.2𝑘+3(2.19)

By differentiating both sides of (2.11) with respect to 𝑡, we have the following: max{𝑘,𝑚}𝑗=0âŽ§âŽªâŽ¨âŽªâŽ©ğ‘žâŽ›âŽœâŽœâŽğ‘˜ğ‘—âŽžâŽŸâŽŸâŽ +(−1)ğ‘—âŽ›âŽœâŽœâŽğ‘šğ‘—âŽžâŽŸâŽŸâŽ âŽ«âŽªâŽ¬âŽªâŽ­î‚ğºğ‘˜+𝑚−𝑗,ğ‘ž[2](𝑥)=ğ‘žğ‘¥ğ‘˜âˆ’1(𝑥−1)𝑚−1((𝑘+𝑚)𝑥−𝑘).(2.20)

We now give interesting theorem for ğ‘ž-Genocchi numbers with weight 0 as follows.

Theorem 2.4. For 𝑘∈ℕ, one has [2]ğ‘ž[𝑘/2]𝑗=0𝑘2𝑗𝐺2𝑘−2𝑗+12𝑘−2𝑗+2,ğ‘ž2𝑘−2𝑗+2+(ğ‘žâˆ’1)[(𝑘−1)/2]𝑗=0𝑘2𝑗+1𝐺2𝑘−2𝑗2𝑘−2𝑗+1,ğ‘ž=2𝑘−2𝑗+1ğ‘ž(−1)𝑘+1(2𝑘+1)𝑘2𝑘.(2.21)

Proof. It is proved by using definition of integral on the both sides in the following equality, that is, 𝑘𝑗=0ğ‘žî€·ğ‘˜ğ‘—î€¸+(−1)𝑗𝑘𝑗2𝑘−𝑗+110𝐺2𝑘−𝑗+1,ğ‘žî‚¼=[2](𝑥)ğ‘‘ğ‘¥ğ‘žî‚»î€œ10𝑥𝑘(𝑥−1)𝑘.𝑑𝑥(2.22) Last from equality, we discover the following: [2]ğ‘ž[𝑘/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ î‚†âˆ«2𝑗10𝐺2𝑘−2𝑗+1,ğ‘žî‚‡(𝑥)𝑑𝑥2𝑘−2𝑗+1+(ğ‘žâˆ’1)[(𝑘−1)/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ î‚†âˆ«2𝑗+110𝐺2𝑘−2𝑗,ğ‘žî‚‡(𝑥)𝑑𝑥=[2]2𝑘−2ğ‘—ğ‘ž(−1)𝑘10𝑥𝑘(1−𝑥)𝑘.𝑑𝑥(2.23) Then, taking integral from 0 to 1 both sides of last equality, we get −[2]ğ‘žâˆ’1[2]ğ‘ž[𝑘/2]𝑗=0𝑘2𝑗𝐺2𝑘−2𝑗+12𝑘−2𝑗+2,ğ‘ž+[2]2𝑘−2𝑗+2ğ‘žâˆ’1(1âˆ’ğ‘ž)[(𝑘−1)/2]𝑗=0𝑘2𝑗+1𝐺2𝑘−2𝑗2𝑘−2𝑗+1,ğ‘ž=[2]2𝑘−2𝑗+1ğ‘ž(−1)𝑘[2]𝐵(𝑘+1,𝑘+1)=ğ‘ž(−1)𝑘(2𝑘+1)𝑘2𝑘.(2.24)
Thus, we complete the proof of the theorem.

Theorem 2.5. For 𝑘∈ℕ, one has [2]ğ‘ž[(𝑘+1)/2]𝑗=0𝑘2𝑗𝐺2𝑘−2𝑗+12𝑘−2𝑗+1,ğ‘ž(𝑥)+[𝑘/2]𝑗=1𝑘2𝑗−1𝐺2𝑘−2𝑗+12𝑘−2𝑗+1,ğ‘ž(𝑥)−[𝑘/2]𝑗=0𝑘2𝑗𝐺2𝑘−2𝑗2𝑘−2𝑗,ğ‘ž[2](𝑥)+(ğ‘žâˆ’1)ğ‘ž[𝑘/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ îƒ¯î‚ğº2𝑗+12𝑘−2𝑗,ğ‘ž(𝑥)[2]ğ‘ž+𝐺(2𝑘−2𝑗)2𝑘−2𝑗+1,ğ‘ž(𝑥)[2]2ğ‘žîƒ°(2𝑘−2𝑗+1)=𝑥𝑘(𝑥−1)𝑘[2]ğ‘žî€¸.ğ‘¥âˆ’ğ‘ž(2.25)

Proof. In view of (2.2) and (2.23), we discover the following applications: 𝑘+1𝑗=0âŽ¡âŽ¢âŽ¢âŽ£ğ‘žâŽ›âŽœâŽœâŽğ‘˜ğ‘—âŽžâŽŸâŽŸâŽ +(−1)ğ‘—âŽ›âŽœâŽœâŽğ‘—âŽžâŽŸâŽŸâŽ âŽ¤âŽ¥âŽ¥âŽ¦î‚ğºğ‘˜+12𝑘−𝑗+1,ğ‘ž(𝑥)=[2]2𝑘−𝑗+1ğ‘žî‚ğº2𝑘+1,ğ‘ž(𝑥)+2𝑘+1[(𝑘+1)/2]𝑗=1âŽ¡âŽ¢âŽ¢âŽ£ğ‘žâŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ +âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ +âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ âŽ¤âŽ¥âŽ¥âŽ¦î‚ğº2𝑗2𝑗2𝑗−12𝑘−2𝑗+1,ğ‘ž(+2𝑘−2𝑗+1𝑥)[𝑘/2]𝑗=0âŽ¡âŽ¢âŽ¢âŽ£ğ‘žâŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ âˆ’âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ âˆ’âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ âŽ¤âŽ¥âŽ¥âŽ¦î‚ğº2𝑗+12𝑗+12𝑗2𝑘−2𝑗,ğ‘ž(𝑥)⎧⎪⎨⎪⎩2𝑘−2𝑗=−[(𝑘+1)/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ î‚ğº2𝑗2𝑘−2𝑗,ğ‘ž(𝑥)+2𝑘−2ğ‘—ğ‘žâˆ’11+ğ‘ž[𝑘/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ î‚ğº2𝑗+12𝑘−2𝑗+1,ğ‘ž(𝑥)⎫⎪⎬⎪⎭+[2]2𝑘−2𝑗+1ğ‘ž[(𝑘+1)/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ Ã—î‚ğº2𝑗2𝑘−2𝑗+1,ğ‘ž(𝑥)+2𝑘−2𝑗+1[(𝑘+1)/2]𝑗=1âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ î‚ğº2𝑗−12𝑘−2𝑗+1,ğ‘ž(𝑥)−2𝑘−2𝑗+1[𝑘/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ î‚ğº2𝑗2𝑘−2𝑗,ğ‘ž(𝑥)+2𝑘−2𝑗(ğ‘žâˆ’1)[𝑘/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ î‚ğº2𝑗+12𝑘−2𝑗,ğ‘ž(𝑥)+2𝑘−2ğ‘—ğ‘žâˆ’11+ğ‘ž[𝑘/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ î‚ğº2𝑗+12𝑘−2𝑗+1,ğ‘ž(𝑥).2𝑘−2𝑗+1(2.26)
Thus, we give evidence of the theorem.

As ğ‘žâ†’1 into Theorem 2.5, it leads to the following interesting property.

Corollary 2.6. For 𝑘∈ℕ, one has [(𝑘+1)/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ ğº2𝑗2𝑘−2𝑗+1(𝑥)+2𝑘+1−2𝑗[𝑘/2]𝑗=1âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ ğº2𝑗−12𝑘−2𝑗+1(𝑥)−4𝑘−4𝑗+2[𝑘/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ ğº2𝑗2𝑘−2𝑗,ğ‘ž(𝑥)4𝑘−4𝑗=𝑥𝑘(𝑥−1)𝑘1𝑥−2,(2.27) where 𝐺𝑛(𝑥) is ordinary Genocchi polynomials, which is defined by the means of the following generating function [9]: âˆžî“ğ‘›=0𝐺𝑛(𝑡𝑥)𝑛=𝑛!2𝑡𝑒𝑡𝑒+1𝑥𝑡,|𝑡|<𝜋.(2.28)

3. Some Identities ğ‘ž-Genocchi Numbers and ğ‘ž-Bernoulli Numbers by Using Kim’s 𝑝-Adic ğ‘ž-Integrals on ℤ𝑝

In this section, we consider ğ‘ž-Genocchi numbers and ğ‘ž-Bernoulli numbers by means of 𝑝-adic ğ‘ž-integral on ℤ𝑝. Now, we start with the following theorem.

Theorem 3.1. For 𝑚,𝑘∈ℕ, one has max{𝑘,𝑚}𝑗=0ğ‘žî€·ğ‘˜ğ‘—î€¸+(−1)𝑗𝑚𝑗𝑘+𝑚−𝑗+1𝑘+𝑚−𝑗+1𝑙=0𝑙𝑘+𝑚−𝑗+1𝐺𝑙+1𝑘+𝑚−𝑗−𝑙+1,ğ‘žî‚ğºğ‘™+1,ğ‘ž=[2]ğ‘žğ‘šî“ğ‘™=0âŽ›âŽœâŽœâŽğ‘šğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑚−𝑙𝐺𝑙+𝑘+1,ğ‘žğ‘™+𝑘+1.(3.1)

Proof. For 𝑚,𝑘∈ℕ, then by (2.11), 𝐼1=[2]ğ‘žî€œâ„¤ğ‘ğ‘¥ğ‘˜(𝑥−1)ğ‘šğ‘‘ğœ‡âˆ’ğ‘ž[2](𝑥)=ğ‘žğ‘šî“ğ‘™=0âŽ›âŽœâŽœâŽğ‘šğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑚−𝑙ℤ𝑝𝑥𝑙+ğ‘˜ğ‘‘ğœ‡âˆ’ğ‘ž=[2](𝑥)ğ‘žğ‘šî“ğ‘™=0âŽ›âŽœâŽœâŽğ‘šğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑚−𝑙𝐺𝑙+𝑘+1,ğ‘ž.𝑙+𝑘+1(3.2) On the other hand, the right hand side of (2.11), 𝐼2=max{𝑘,𝑚}𝑗=0ğ‘žî€·ğ‘˜ğ‘—î€¸+(−1)𝑗𝑚𝑗𝑘+𝑚−𝑗+1𝑘+𝑚−𝑗+1𝑙=0âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğºğ‘˜+𝑚−𝑗+1𝑘+𝑚−𝑗−𝑙+1,ğ‘žî€œâ„¤ğ‘ğ‘¥ğ‘™ğ‘‘ğœ‡âˆ’ğ‘ž=(𝑥)max{𝑘,𝑚}𝑗=0ğ‘žî€·ğ‘˜ğ‘—î€¸+(−1)𝑗𝑚𝑗𝑘+𝑚−𝑗+1𝑘+𝑚−𝑗+1𝑙=0𝑙𝑘+𝑚−𝑗+1𝐺𝑙+1𝑘+𝑚−𝑗−𝑙+1,ğ‘žî‚ğºğ‘™+1,ğ‘ž.(3.3) Combining 𝐼1 and 𝐼2, we arrive to the proof of the theorem.

Theorem 3.2. For 𝑘∈ℕ, one has 𝑘𝑙=0âŽ›âŽœâŽœâŽğ‘˜ğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑘−𝑙[2]ğ‘žî‚ğºğ‘˜+𝑙+2,ğ‘žî‚ğºğ‘˜+𝑙+2âˆ’ğ‘žğ‘˜+𝑙+1,ğ‘žîƒ°=[2]𝑘+𝑙+1ğ‘ž[𝑘/2]𝑗=0𝑘2𝑗2𝑘−2𝑗+12𝑘−2𝑗+1𝑙=0𝑙2𝑘−2𝑗+1𝐺𝑙+12𝑘+1−2𝑗−𝑙,ğ‘žî‚ğºğ‘™+1,ğ‘ž+[𝑘/2]𝑗=1𝑘2𝑗−12𝑘−2𝑗+12𝑘−2𝑗+1𝑙=0𝑙2𝑘−2𝑗+1𝐺𝑙+12𝑘+1−2𝑗−𝑙,ğ‘žî‚ğºğ‘™+1,ğ‘ž+ğ‘žâˆ’11+ğ‘ž[𝑘/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ ğ‘‡2𝑗+1ğ‘žğ‘˜,𝑗,(3.4) here ğ‘‡ğ‘žğ‘˜,𝑗=[2]ğ‘žâˆ‘2𝑘−2𝑗𝑙=0(𝑙2𝑘−2𝑗𝐺/(2𝑘−2𝑗))(𝑙+1,ğ‘žî‚ğº2𝑘−2𝑗−𝑙,ğ‘žâˆ‘/(𝑙+1))+2𝑘−2𝑗+1𝑙=0(𝑙2𝑘−2𝑗+1𝐺/(2𝑘−2𝑗+1))(𝑙+1,ğ‘žî‚ğº2𝑘−2𝑗−𝑙+1,ğ‘ž/(𝑙+1)).

Proof. Let us take fermionic 𝑝-adic ğ‘ž-inetgral on ℤ𝑝 left-hand side of Theorem 2.5, we get 𝐼3=ℤ𝑝𝑥𝑘(𝑥−1)𝑘[2]ğ‘žî€¸ğ‘¥âˆ’ğ‘žğ‘‘ğœ‡âˆ’ğ‘ž(=[2]𝑥)ğ‘žğ‘˜î“ğ‘™=0âŽ›âŽœâŽœâŽğ‘˜ğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑘−𝑙ℤ𝑝𝑥𝑘+𝑙+1ğ‘‘ğœ‡âˆ’ğ‘ž(𝑥)âˆ’ğ‘žğ‘˜î“ğ‘™=0âŽ›âŽœâŽœâŽğ‘˜ğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑘−𝑙ℤ𝑝𝑥𝑘+ğ‘™ğ‘‘ğœ‡âˆ’ğ‘ž=[2](𝑥)ğ‘žğ‘˜î“ğ‘™=0âŽ›âŽœâŽœâŽğ‘˜ğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑘−𝑙𝐺𝑘+𝑙+2,ğ‘žğ‘˜+𝑙+2âˆ’ğ‘žğ‘˜î“ğ‘™=0âŽ›âŽœâŽœâŽğ‘˜ğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑘−𝑙𝐺𝑘+𝑙+1,ğ‘ž.𝑘+𝑙+1(3.5) In other word, we consider the right-hand side of Theorem 2.5 as follows: 𝐼4=[2]ğ‘ž[𝑘/2]𝑗=0𝑘2𝑗2𝑘−2𝑗+12𝑘−2𝑗+1𝑙=0âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğº2𝑘−2𝑗+12𝑘+1−2𝑗−𝑙,ğ‘žî€œâ„¤ğ‘ğ‘¥ğ‘™ğ‘‘ğœ‡âˆ’ğ‘ž+(𝑥)[𝑘/2]𝑗=1𝑘2𝑗−12𝑘−2𝑗+12𝑘−2𝑗+1𝑙=0âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğº2𝑘−2𝑗+12𝑘+1−2𝑗−𝑙,ğ‘žî€œâ„¤ğ‘ğ‘¥ğ‘™ğ‘‘ğœ‡âˆ’ğ‘ž(+𝑥)[𝑘/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ âŽ§âŽªâŽªâŽ¨âŽªâŽªâŽ©2𝑗+1(ğ‘žâˆ’1)2𝑘−2𝑗∑𝑗=0𝑙2𝑘−2𝑗−𝑙𝐺2𝑘−2𝑗2𝑘−2𝑗−𝑙,ğ‘žâˆ«â„¤ğ‘ğ‘¥ğ‘™ğ‘‘ğœ‡âˆ’ğ‘ž+(𝑥)ğ‘žâˆ’11+ğ‘ž2𝑘−2𝑗+1∑𝑙=0𝑙2𝑘−2𝑗+1𝐺2𝑘−2𝑗+12𝑘−2𝑗−𝑙+1âˆ«â„¤ğ‘ğ‘¥ğ‘™ğ‘‘ğœ‡âˆ’ğ‘ž(⎫⎪⎪⎬⎪⎪⎭=[2]𝑥)ğ‘ž[𝑘/2]𝑗=0𝑘2𝑗2𝑘−2𝑗+12𝑘−2𝑗+1𝑙=0𝑙2𝑘−2𝑗−𝑙+1𝐺𝑙+12𝑘+1−2𝑗−𝑙,ğ‘žî‚ğºğ‘™+1,ğ‘ž+[𝑘/2]𝑗=1𝑘2𝑗−12𝑘−2𝑗+12𝑘−2𝑗+1𝑙=0𝑙2𝑘−2𝑗−𝑙+1𝐺𝑙+12𝑘+1−2𝑗−𝑙,ğ‘žî‚ğºğ‘™+1,ğ‘ž+[𝑘/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ âŽ§âŽªâŽªâŽ¨âŽªâŽªâŽ©(2𝑗+1ğ‘žâˆ’1)2𝑘−2𝑗∑𝑗=0𝑙2𝑘−2𝑗−𝑙𝐺2𝑘−2𝑗2𝑘−2𝑗−𝑙,ğ‘žî‚ğºğ‘™+1,ğ‘ž+𝑙+1ğ‘žâˆ’11+ğ‘ž2𝑘−2𝑗+1∑𝑙=0𝑙2𝑘−2𝑗−𝑙+1𝐺2𝑘−2𝑗+12𝑘−2𝑗−𝑙+1𝐺𝑙+1,ğ‘žâŽ«âŽªâŽªâŽ¬âŽªâŽªâŽ­.𝑙+1(3.6)
Equating 𝐼3 and 𝐼4, we complete the proof of the theorem.

As ğ‘žâ†’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=0âŽ›âŽœâŽœâŽğ‘˜ğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑘−𝑙2𝐺𝑘+𝑙+2−𝐺𝑘+𝑙+2𝑘+𝑙+1𝑘+𝑙+1=2[𝑘/2]𝑗=0𝑘2𝑗2𝑘−2𝑗+12𝑘−2𝑗+1𝑙=0𝑙2𝑘−2𝑗+1𝐺𝑙+12𝑘+1−2𝑗−𝑙𝐺𝑙+1+[𝑘/2]𝑗=1𝑘2𝑗−12𝑘−2𝑗+12𝑘−2𝑗+1𝑙=0𝑙2𝑘−2𝑗+1𝐺𝑙+12𝑘+1−2𝑗−𝑙𝐺𝑙+1.(3.7)

Theorem 3.4. For 𝑚,𝑘∈ℕ, one has [2]ğ‘žğ‘šî“ğ‘™=0âŽ›âŽœâŽœâŽğ‘šğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑚−𝑙𝐵𝑙+𝑘,ğ‘ž=max{𝑘,𝑚}𝑗=0ğ‘žî€·ğ‘˜ğ‘—î€¸+(−1)𝑗𝑚𝑗𝑘+𝑚−𝑗+1𝑘+𝑚−𝑗+1𝑙=0âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğºğ‘˜+𝑚−𝑗+1𝑘+𝑚+1−𝑗−𝑙,ğ‘žî‚ğµğ‘™,ğ‘ž.(3.8)

Proof. We consider (2.11) and (2.2) by means of ğ‘ž-Volkenborn integral. Then, by (2.11), we see [2]ğ‘žî€œâ„¤ğ‘ğ‘¥ğ‘˜(𝑥−1)ğ‘šğ‘‘ğœ‡ğ‘ž[2](𝑥)=ğ‘žğ‘šî“ğ‘™=0âŽ›âŽœâŽœâŽğ‘šğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑚−𝑙ℤ𝑝𝑥𝑙+ğ‘˜ğ‘‘ğœ‡ğ‘ž[2](𝑥)=ğ‘žğ‘šî“ğ‘™=0âŽ›âŽœâŽœâŽğ‘šğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑚−𝑙𝐵𝑙+𝑘,ğ‘ž.(3.9) On the other hand, max{𝑘,𝑚}𝑗=0ğ‘žî€·ğ‘˜ğ‘—î€¸+(−1)𝑗𝑚𝑗𝑘+𝑚−𝑗+1𝑘+𝑚−𝑗+1𝑙=0âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğºğ‘˜+𝑚−𝑗+1𝑘+𝑚+1−𝑗−𝑙,ğ‘žî€œâ„¤ğ‘ğ‘¥ğ‘™ğ‘‘ğœ‡ğ‘ž=(𝑥)max{𝑘,𝑚}𝑗=0ğ‘žî€·ğ‘˜ğ‘—î€¸+(−1)𝑗𝑚𝑗𝑘+𝑚−𝑗+1𝑘+𝑚−𝑗+1𝑙=0âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğºğ‘˜+𝑚−𝑗+1𝑘+𝑚+1−𝑗−𝑙,ğ‘žî‚ğµğ‘™,ğ‘ž.(3.10) Therefore, we get the proof of theorem.

Corollary 3.5. For 𝑘∈ℕ, one gets 𝑘𝑙=0âŽ›âŽœâŽœâŽğ‘˜ğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑘−𝑙[2]ğ‘žî‚ğµğ‘˜+𝑙+1,ğ‘žî‚ğµâˆ’ğ‘žğ‘˜+𝑙,ğ‘žî‚‡=[2]ğ‘ž[𝑘/2]𝑗=0𝑘2𝑗2𝑘−2𝑗+12𝑘−2𝑗+1𝑙=0âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğº2𝑘−2𝑗+12𝑘+1−2𝑗−𝑙,ğ‘žî‚ğµğ‘™,ğ‘ž+[𝑘/2]𝑗=1𝑘2𝑗−1×2𝑘−2𝑗+12𝑘−2𝑗+1𝑙=0âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğº2k−2𝑗+12𝑘+1−2𝑗−𝑙,ğ‘žî‚ğµğ‘™,ğ‘ž+î‚µğ‘žâˆ’1î‚¶ğ‘ž+1[𝑘/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ ğ‘†2𝑗+1ğ‘žğ‘˜,𝑗,(3.11) where ğ‘†ğ‘žğ‘˜,𝑗=[2]ğ‘žâˆ‘2𝑘−2𝑗𝑗=0(1/(2𝑘−2𝑗))𝑙2𝑘−2𝑗𝐺2𝑘−2𝑗−𝑙,ğ‘žî‚ğµğ‘™,ğ‘ž+∑2𝑘−2𝑗+1𝑙=0(1/(2𝑘−2𝑗+1))𝑙2𝑘−2𝑗+1𝐺2𝑘−2𝑗−𝑙+1𝐵𝑙,ğ‘ž.

Proof. By using 𝑝-adic ğ‘ž-integral on ℤ𝑝 left-hand side of Theorem 2.5, we get 𝐼5=[2]ğ‘žî€œâ„¤ğ‘ğ‘¥ğ‘˜(𝑥−1)𝑘([2]ğ‘¥âˆ’ğ‘ž)ğ‘‘ğœ‡ğ‘ž(=[2]𝑥)ğ‘žğ‘˜î“ğ‘™=0âŽ›âŽœâŽœâŽğ‘˜ğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑘−𝑙ℤ𝑝𝑥𝑘+𝑙+1ğ‘‘ğœ‡ğ‘ž(𝑥)âˆ’ğ‘žğ‘˜î“ğ‘™=0âŽ›âŽœâŽœâŽğ‘˜ğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑘−𝑙ℤ𝑝𝑥𝑘+ğ‘™ğ‘‘ğœ‡ğ‘ž=[2](𝑥)ğ‘žğ‘˜î“ğ‘™=0âŽ›âŽœâŽœâŽğ‘˜ğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑘−𝑙𝐵𝑘+𝑙+1,ğ‘žâˆ’ğ‘žğ‘˜î“ğ‘™=0âŽ›âŽœâŽœâŽğ‘˜ğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑘−𝑙𝐵𝑘+𝑙,ğ‘ž.(3.12) Also, we compute the right-hand side of Theorem 2.5 as follows: 𝐼6=[2]ğ‘ž[𝑘/2]𝑗=01âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ 2𝑘−2𝑗+12𝑗2𝑘−2𝑗+1𝑙=0âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğº2𝑘−2𝑗+12𝑘+1−2𝑗−𝑙,ğ‘žî€œâ„¤ğ‘ğ‘¥ğ‘™ğ‘‘ğœ‡ğ‘ž+(𝑥)[𝑘/2]𝑗=11âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ 2𝑘−2𝑗+12𝑗−12𝑘−2𝑗+1𝑙=0âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğº2𝑘−2𝑗+12𝑘+1−2𝑗−𝑙,ğ‘žî€œâ„¤ğ‘ğ‘¥ğ‘™ğ‘‘ğœ‡ğ‘ž(+𝑥)[𝑘/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ âŽ§âŽªâŽªâŽªâŽ¨âŽªâŽªâŽªâŽ©2𝑗+1(ğ‘žâˆ’1)2𝑘−2𝑗∑𝑗=01âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğº2𝑘−2𝑗2𝑘−2𝑗2𝑘−2𝑗−𝑙,ğ‘žâˆ«â„¤ğ‘ğ‘¥ğ‘™ğ‘‘ğœ‡ğ‘ž+(𝑥)ğ‘žâˆ’11+ğ‘ž2𝑘−2𝑗+1∑𝑙=01âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğº2𝑘−2𝑗+12𝑘−2𝑗+12𝑘−2𝑗−𝑙+1âˆ«â„¤ğ‘ğ‘¥ğ‘™ğ‘‘ğœ‡ğ‘žâŽ«âŽªâŽªâŽªâŽ¬âŽªâŽªâŽªâŽ­=[2](𝑥)ğ‘ž[𝑘/2]𝑗=01âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ 2𝑘−2𝑗+12𝑗2𝑘−2𝑗+1𝑙=0âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğº2𝑘−2𝑗+12𝑘+1−2𝑗−𝑙,ğ‘žî‚ğµğ‘™,ğ‘ž+[𝑘/2]𝑗=11âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ 2𝑘−2𝑗+12𝑗−12𝑘−2𝑗+1𝑙=0âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğº2𝑘−2𝑗+12𝑘+1−2𝑗−𝑙,ğ‘žî‚ğµğ‘™,ğ‘ž+[𝑘/2]𝑗=0âŽ›âŽœâŽœâŽğ‘˜âŽžâŽŸâŽŸâŽ âŽ§âŽªâŽªâŽªâŽ¨âŽªâŽªâŽªâŽ©2𝑗+1(ğ‘žâˆ’1)2𝑘−2𝑗∑𝑗=01âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğº2𝑘−2𝑗2𝑘−2𝑗2𝑘−2𝑗−𝑙,ğ‘žî‚ğµğ‘™,ğ‘ž+ğ‘žâˆ’11+ğ‘ž2𝑘−2𝑗+1∑𝑙=01âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ î‚ğº2𝑘−2𝑗+12𝑘−2𝑗+12𝑘−2𝑗−𝑙+1𝐵𝑙,ğ‘žâŽ«âŽªâŽªâŽªâŽ¬âŽªâŽªâŽªâŽ­.(3.13)
Equating 𝐼5 and 𝐼6, we get the proof of Corollary.

As ğ‘žâ†’1 in the above theorem, we easily derive the following corollary.

Corollary 3.6. For 𝑘∈ℕ, one has 𝑘𝑙=0âŽ›âŽœâŽœâŽğ‘˜ğ‘™âŽžâŽŸâŽŸâŽ (−1)𝑘−𝑙2𝐵𝑘+𝑙+1−𝐵𝑘+𝑙=2[𝑘/2]𝑗=0𝑘2𝑗2𝑘−2𝑗+12𝑘−2𝑗+1𝑙=0âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ ğº2𝑘−2𝑗+12𝑘+1−2𝑗−𝑙𝐵𝑙+[𝑘/2]𝑗=1𝑘2𝑗−12𝑘−2𝑗+12𝑘−2𝑗+1𝑙=0âŽ›âŽœâŽœâŽğ‘™âŽžâŽŸâŽŸâŽ ğº2𝑘−2𝑗+12𝑘+1−2𝑗−𝑙𝐵𝑙.(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.