Explicit Formulas Involving 𝑞-Euler Numbers and Polynomials

Abstract

We deal with 𝑞-Euler numbers and 𝑞-Bernoulli numbers. We derive some interesting relations for 𝑞-Euler numbers and polynomials by using their generating function and derivative operator. Also, we derive relations between the 𝑞-Euler numbers and 𝑞-Bernoulli numbers via the 𝑝-adic 𝑞-integral in the 𝑝-adic integer ring.

1. Preliminaries

Imagine that 𝑝 is a fixed odd prime number. Throughout this paper we use the following notations, where 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=1𝑝. (1.1)

In this paper, we will assume that |𝑞−1|_𝑝<1 as an indeterminate.

[𝑥]_𝑞 is a 𝑞-extension of 𝑥, which is defined by [𝑥]_𝑞=1−𝑞^𝑥.1−𝑞 (1.2) We note that lim_𝑞→1[𝑥]_𝑞=𝑥 (see [1–12]).

We say that 𝑓 is a uniformly differentiable function at a point 𝑎∈ℤ_𝑝, if the difference quotient 𝐹_𝑓=𝑓(𝑥)𝑦𝑥,𝑦−𝑓,𝑥−𝑦 (1.3) has a limit 𝑓^′(𝑎) as (𝑥,𝑦)→(𝑎,𝑎) and denote this by 𝑓∈UD(ℤ_𝑝).

Let UD(ℤ_𝑝) be the set of uniformly differentiable function on ℤ_𝑝. For 𝑓∈UD(ℤ_𝑝), let us start with the expression 1𝑝^𝑁_{𝑞0≤𝜉<𝑝𝑁}𝑓(𝜉)𝑞^𝜉=_{0≤𝜉<𝑝𝑁}𝑓(𝜉)𝜇_𝑞𝜉+𝑝^𝑁ℤ_𝑝, (1.4) which represents 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 𝐼_𝑞𝑓=_ℤ𝑝𝑓(𝜉)𝑑𝜇_𝑞(𝜉)=lim_𝑁→∞1𝑝^𝑁_𝑞^𝑝𝑁−1_𝜉=0𝑓(𝜉)𝑞^𝜉. (1.5)

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_−𝑞𝑓=lim_{𝑞→−𝑞}𝐼_𝑞𝑓=_ℤ𝑝𝑓(𝜉)𝑑𝜇_−𝑞(𝜉) (1.6) (for more details, see [9–12]).

In [6], the 𝑞-Euler polynomials with weight 0 are introduced as 𝐸_𝑛,𝑞(𝑥)=_ℤ𝑝𝑥+𝑦^𝑛𝑑𝜇_−𝑞𝑦. (1.7)

From (1.7), we have 𝐸_𝑛,𝑞(𝑥)=^𝑛_𝑙=0𝑛𝑙𝑥^𝑙𝐸_{𝑛−𝑙,𝑞}, (1.8) where 𝐸_𝑛,𝑞(0)=𝐸_𝑛,𝑞 are called 𝑞-Euler numbers with weight 0. Then, 𝑞-Euler numbers are defined as 𝑞𝐸_𝑞+1^𝑛+𝐸_𝑛,𝑞=[2]_𝑞,if𝑛=0,0,if𝑛≠0, (1.9) where the usual convention about replacing (𝐸_𝑞)^𝑛 by 𝐸_𝑛,𝑞 is used.

Similarly, the 𝑞-Bernoulli polynomials and numbers with weight 0 are defined, respectively, as 𝐵_𝑛,𝑞(𝑥)=lim_𝑛→∞1𝑝^𝑛_𝑞^𝑝𝑛−1_𝑦=0𝑥+𝑦^𝑛𝑞^𝑦=_ℤ𝑝𝑥+𝑦^𝑛𝑑𝜇_𝑞𝑦,𝐵_𝑛,𝑞=_ℤ𝑝𝑦^𝑛𝑑𝜇_𝑞𝑦 (1.10) (for more information, see [4]).

We, by using the Kim et al. method in [2], will investigate some interesting identities on the 𝑞-Euler numbers and polynomials arising from their generating function and derivative operator. Consequently, we derive some properties on the 𝑞-Euler numbers and polynomials and 𝑞-Bernoulli numbers and polynomials by using 𝑞-Volkenborn integral and fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝.

2. On the 𝑞-Euler Numbers and Polynomials

Let us consider Kim’s 𝑞-Euler polynomials as follows: 𝐹^𝑞_𝑥=𝐹^𝑞_𝑥(𝑡)=[2]_𝑞𝑞𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐸_𝑛,𝑞(𝑥)𝑡^𝑛.𝑛! (2.1)

Here 𝑥 is a fixed parameter. Thus, by expression of (2.1), we can readily see the following: 𝑞𝑒^𝑡𝐹^𝑞_𝑥+𝐹^𝑞_𝑥=[2]_𝑞𝑒^𝑥𝑡. (2.2)

Last from equality, taking derivative operator 𝐷 as 𝐷=𝑑/𝑑𝑡 on the both sides of (2.2). Then, we easily see that 𝑞𝑒^𝑡(𝐷+𝐼)^𝑘𝐹^𝑞_𝑥+𝐷^𝑘𝐹^𝑞_𝑥=[2]_𝑞𝑥^𝑘𝑒^𝑥𝑡, (2.3) where 𝑘∈ℕ^∗ and 𝐼 is identity operator. By multiplying 𝑒^−𝑡 on both sides of (2.3), we get 𝑞(𝐷+𝐼)^𝑘𝐹^𝑞_𝑥+𝑒^−𝑡𝐷^𝑘𝐹^𝑞_𝑥=[2]_𝑞𝑥^𝑘𝑒^{(𝑥−1)𝑡}. (2.4)

Let us take derivative operator 𝐷^𝑚(𝑚∈ℕ) on both sides of (2.4). Then we get 𝑞𝑒^𝑡𝐷^𝑚(𝐷+𝐼)^𝑘𝐹^𝑞_𝑥+𝐷^𝑘(𝐷−𝐼)^𝑚𝐹^𝑞_𝑥=[2]_𝑞𝑥^𝑘(𝑥−1)^𝑚𝑒^𝑥𝑡. (2.5)

Let 𝐺[0] (not 𝐺(0)) be the constant term in a Laurent series of 𝐺(𝑡). Then, from (2.5), we get ^𝑘_𝑗=0𝑘𝑗𝑞𝑒^𝑡𝐷^{𝑘+𝑚−𝑗}𝐹^𝑞_𝑥(𝑡)[0]+^𝑚_𝑗=0𝑚𝑗(−1)^𝑗𝐷^{𝑘+𝑚−𝑗}𝐹^𝑞_𝑥(𝑡)[0]=[2]_𝑞𝑥^𝑘(𝑥−1)^𝑚. (2.6)

By (2.1), we see 𝐷^𝑁𝐹^𝑞_𝑥(𝑡)[0]=𝐸_𝑁,𝑞(𝑥),𝑒^𝑡𝐷^𝑁𝐹^𝑞_𝑥(𝑡)[0]=𝐸_𝑁,𝑞(𝑥). (2.7)

By expressions of (2.6) and (2.7), we see that ^{max{𝑘,𝑚}}_𝑗=0𝑞𝑘𝑗+(−1)^𝑗𝑚𝑗𝐸_{𝑘+𝑚−𝑗,𝑞}(𝑥)=[2]_𝑞𝑥^𝑘(𝑥−1)^𝑚. (2.8)

From (2.1), we note that 𝑑𝐸𝑑𝑥^𝑛,𝑞(𝑥)=𝑛^𝑛−1_𝑙=0𝑙𝐸𝑛−1_𝑙,𝑞𝑥^{𝑛−1−𝑙}=𝑛𝐸_{𝑛−1,𝑞}(𝑥). (2.9)

By (2.9), we easily see ¹₀𝐸_𝑛,𝑞(𝑥)𝐸𝑑𝑥=^𝑛+1,𝑞(1)−𝐸_𝑛+1,𝑞[2]𝑛+1=−^𝑞−1𝐸𝑛+1^𝑛+1,𝑞. (2.10)

Now, let us consider definition of integral from 0 to 1 in (2.8), then we have −[2]_𝑞−1^{max{𝑘,𝑚}}_𝑗=0𝑞𝑘𝑗+(−1)^𝑗𝑚𝑗𝐸_{𝑘+𝑚−𝑗+1,𝑞}=[2]𝑘+𝑚−𝑗+1_𝑞(−1)^𝑚𝐵()=[2]𝑘+1,𝑚+1_𝑞(−1)^𝑚Γ(𝑘+1)Γ(𝑚+1)Γ(),𝑘+𝑚+2 (2.11) where 𝐵(𝑚,𝑛) is beta function which is defined by 𝐵(𝑚,𝑛)=¹₀𝑥^𝑚−1(1−𝑥)^𝑛−1=Γ(𝑚)Γ(𝑛)𝑑𝑥Γ(𝑚+𝑛),𝑚>0,𝑛>0. (2.12)

As a result, we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℕ, one has ^{max{𝑘,𝑚}}_𝑗=1𝑞𝑘𝑗+(−1)^𝑗𝑚𝑗𝐸_{𝑘+𝑚−𝑗+1,𝑞}()𝑘+𝑚−𝑗+1=𝑞−1^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚−[2]_𝑞𝐸_{𝑘+𝑚+1,𝑞}.𝑘+𝑚+1 (2.13)

Substituting 𝑚=𝑘+1 into Theorem 2.1, we readily get ^𝑘+1_𝑗=1𝑞𝑘𝑗+(−1)^𝑗𝑗𝑘+1𝐸_{2𝑘+2−𝑗,𝑞}()2𝑘+2−𝑗=𝑞−1^𝑘(2𝑘+2)_𝑘^2𝑘+1−[2]_𝑞𝐸_2𝑘+2,𝑞.2𝑘+2 (2.14)

By (2.1), it follows that ^{max{𝑘,𝑚}}_𝑗=0𝑞𝑘𝑗𝑘+𝑚−𝑗+(−1)^𝑗𝑚𝑗𝐸_{𝑘+𝑚−𝑗−1,𝑞}(𝑥)=[2]_𝑞𝑥^𝑘−1(𝑥−1)^𝑚−1((𝑘+𝑚)𝑥−𝑘). (2.15)

Let 𝑚=𝑘 in (2.1), we see that ^𝑘_𝑗=0𝑞𝑘𝑗+(−1)^𝑗𝑘𝑗𝐸_{2𝑘−𝑗,𝑞}(𝑥)=[2]_𝑞𝑥^𝑘(𝑥−1)^𝑘. (2.16)

Last from equality, we discover the following: [2]_𝑞^[𝑘/2]_𝑗=0𝑘𝐸2𝑗^{2𝑘−2𝑗,𝑞}(𝑥)+𝑞−1^[𝑘/2]_𝑗=0𝑘𝐸2𝑗+1^{2𝑘−2𝑗−1,𝑞}(𝑥)=[2]_𝑞𝑥^𝑘(𝑥−1)^𝑘. (2.17)

Here [⋅] is Gauss’ symbol. Then, taking integral from 0to 1 in both sides of last equality, we get −[2]_𝑞−1[2]_𝑞^[𝑘/2]_𝑗=0𝑘𝐸2𝑗^{2𝑘−2𝑗+1,𝑞}+[2]2𝑘−2𝑗+1^𝑞−11−𝑞^[𝑘/2]_𝑗=0𝑘𝐸2𝑗+1^{2𝑘−2𝑗,𝑞}=[2]2𝑘−2𝑗_𝑞(−1)^𝑘𝐵()=[2]𝑘+1,𝑘+1_𝑞(−1)^𝑘(2𝑘+1)_𝑘^2𝑘. (2.18)

Consequently, we derive the following theorem.

Theorem 2.2. The following identity [2]_𝑞^[𝑘/2]_𝑗=0𝑘𝐸2𝑗^{2𝑘−2𝑗+1,𝑞}+2𝑘−2𝑗+1𝑞−1^[𝑘/2]_𝑗=0𝑘𝐸2𝑗+1^{2𝑘−2𝑗,𝑞}=𝑞()2𝑘−2𝑗−1^𝑘+1(2𝑘+1)_𝑘^2𝑘 (2.19) is true.

In view of (2.1) and (2.17), we discover the following applications: =^𝑘+1_𝑗=0𝑞𝑘𝑗+(−1)^𝑗𝑗𝐸𝑘+1_{2𝑘+1−𝑗,𝑞}(𝑥)=[2]_𝑞𝐸_2𝑘+1,𝑞(𝑥)+^[(𝑘+1)/2]_𝑗=1𝑞𝑘+𝑘+𝑘𝐸2𝑗2𝑗2𝑗−1^{2𝑘+1−2𝑗,𝑞}(𝑥)+^[(𝑘+1)/2]_𝑗=0𝑞𝑘−𝑘−𝑘𝐸2𝑗+12𝑗+12𝑗^{2𝑘−2𝑗,𝑞}(𝑥)=−^[𝑘/2]_𝑗=0𝑘𝐸2𝑗^{2𝑘−2𝑗,𝑞}(𝑥)+𝑞−11+𝑞^[𝑘/2]_𝑗=0𝑘𝐸2𝑗+1^{2𝑘−2𝑗+1}(𝑥)+[2]_𝑞^[𝑘/2]_𝑗=0𝑘𝐸2𝑗^{2𝑘+1−2𝑗,𝑞}(𝑥)+^[𝑘/2]_𝑗=1𝑘𝐸2𝑗−1^{2𝑘+1−2𝑗,𝑞}(𝑥)+𝑞−1^[𝑘/2]_𝑗=0𝑘𝐸2𝑗+1^{2𝑘−2𝑗,𝑞}(𝑥)+𝑞−11+𝑞^[𝑘/2]_𝑗=0𝑘𝐸2𝑗+1^{2𝑘−2𝑗+1}(𝑥). (2.20)

By expressions (2.17) and (2.20), we have the following theorem.

Theorem 2.3. For 𝑘∈ℕ, one has [2]_𝑞^[𝑘/2]_𝑗=0𝑘𝐸2𝑗^{2𝑘+1−2𝑗,𝑞}(𝑥)+^[𝑘/2]_𝑗=1𝑘𝐸2𝑗−1^{2𝑘+1−2𝑗,𝑞}(𝑥)+𝑞−1^[𝑘/2]_𝑗=0𝑘𝐸2𝑗+1^{2𝑘−2𝑗,𝑞}(𝑥)+1𝐸1+𝑞^{2𝑘−2𝑗+1}(𝑥)=𝑥^𝑘(𝑥−1)^𝑘[2]_𝑞𝑥−𝑞. (2.21)

3. 𝑝-adic Integral on ℤ_𝑝 Associated with Kim’s 𝑞-Euler Polynomials

In this section, we consider Kim’s 𝑞-Euler polynomials by means of 𝑝-adic 𝑞-integral on ℤ_𝑝. Now we start with the following assertion.

Let 𝑚,𝑘∈ℕ. Then by (2.8), 𝐼₁=[2]_𝑞ℤ𝑝𝑥^𝑘(𝑥−1)^𝑚𝑑𝜇_−𝑞(𝑥)=[2]_𝑞^𝑚_𝑙=0𝑚𝑙(−1)^𝑚−𝑙_ℤ𝑝𝑥^𝑙+𝑘𝑑𝜇_−𝑞(𝑥)=[2]_𝑞^𝑚_𝑙=0𝑚𝑙(−1)^𝑚−𝑙𝐸_{𝑙+𝑘,𝑞}. (3.1)

On the other hand, in right hand side of (2.8), 𝐼₂=^{max{𝑘,𝑚}}_𝑗=0𝑞𝑘𝑗+(−1)^𝑗𝑚𝑗^{𝑘+𝑚−𝑗𝑙=0}𝑙𝐸𝑘+𝑚−𝑗_{𝑘+𝑚−𝑗−𝑙,𝑞ℤ𝑝}𝑥^𝑙𝑑𝜇_−𝑞(𝑥)=^{max{𝑘,𝑚}}_𝑗=0𝑞𝑘𝑗+(−1)^𝑗𝑚𝑗^{𝑘+𝑚−𝑗𝑙=0}𝑙𝐸𝑘+𝑚−𝑗_{𝑘+𝑚−𝑗−𝑙,𝑞}𝐸_𝑙,𝑞. (3.2)

Equating 𝐼₁ and 𝐼₂, we get the following theorem.

Theorem 3.1. For 𝑚,𝑘∈ℕ, one has ^{max{𝑘,𝑚}}_𝑗=0𝑞𝑘𝑗+(−1)^𝑗𝑚𝑗^{𝑘+𝑚−𝑗𝑙=0}𝑙𝐸𝑘+𝑚−𝑗_{𝑘+𝑚−𝑗−𝑙,𝑞}𝐸_𝑙,𝑞=[2]_𝑞^𝑚_𝑙=0𝑚𝑙(−1)^𝑚−𝑙𝐸_{𝑙+𝑘,𝑞}. (3.3)

Let us take fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 in left hand side of (2.21), we get 𝐼₃=_ℤ𝑝𝑥^𝑘(𝑥−1)^𝑘[2]_𝑞𝑥−𝑞𝑑𝜇_−𝑞(𝑥)=[2]_𝑞^𝑘_𝑙=0𝑘𝑙(−1)^𝑘−𝑙_ℤ𝑝𝑥^𝑘+𝑙+1𝑑𝜇_−𝑞(𝑥)−𝑞^𝑘_𝑙=0𝑘𝑙(−1)^𝑘−𝑙_ℤ𝑝𝑥^𝑘+𝑙𝑑𝜇_−𝑞(𝑥)=[2]_𝑞^𝑘_𝑙=0𝑘𝑙(−1)^𝑘−𝑙𝐸_{𝑘+𝑙+1,𝑞}−𝑞^𝑘_𝑙=0𝑘𝑙(−1)^𝑘−𝑙𝐸_{𝑘+𝑙,𝑞}. (3.4)

In other words, we consider right hand side of (2.21) as follows: 𝐼₄=[2]_𝑞^[𝑘/2]_𝑗=0𝑘2𝑗^{2𝑘−2𝑗+1𝑙=0}𝑙𝐸2𝑘−2𝑗+1_{2𝑘+1−2𝑗−𝑙,𝑞ℤ𝑝}𝑥^𝑙𝑑𝜇_−𝑞(𝑥)+^[𝑘/2]_𝑗=1𝑘2𝑗−1^{2𝑘−2𝑗+1𝑙=0}𝑙𝐸2𝑘−2𝑗+1_{2𝑘+1−2𝑗−𝑙,𝑞ℤ𝑝}𝑥^𝑙𝑑𝜇_−𝑞(𝑥)+^[𝑘/2]_𝑗=0𝑘2𝑗+1𝑞−1∑^{2𝑘−2𝑗}_𝑗=0𝑙𝐸2𝑘−2𝑗_{2𝑘−2𝑗−𝑙,𝑞}∫_ℤ𝑝𝑥^𝑙𝑑𝜇_−𝑞(𝑥)+𝑞−11+𝑞^{2𝑘−2𝑗+1}_𝑙∑₌₀𝑙𝐸2𝑘−2𝑗+1_{2𝑘−2𝑗−𝑙+1}∫_ℤ𝑝𝑥^𝑙𝑑𝜇_−𝑞(𝑥)=[2]_𝑞^[𝑘/2]_𝑗=0𝑘2𝑗^{2𝑘−2𝑗+1𝑙=0}𝑙𝐸2𝑘−2𝑗+1_{2𝑘+1−2𝑗−𝑙,𝑞}𝐸_𝑙,𝑞+^[𝑘/2]_𝑗=1𝑘2𝑗−1^{2𝑘−2𝑗+1𝑙=0}𝑙𝐸2𝑘−2𝑗+1_{2𝑘+1−2𝑗−𝑙,𝑞}𝐸_𝑙,𝑞+^[𝑘/2]_𝑗=0𝑘2𝑗+1𝑞−1∑^{2𝑘−2𝑗}_𝑗=0𝑙𝐸2𝑘−2𝑗_{2𝑘−2𝑗−𝑙,𝑞}𝐸_𝑙,𝑞+𝑞−11+𝑞^{2𝑘−2𝑗+1}_𝑙∑₌₀𝑙𝐸2𝑘−2𝑗+1_{2𝑘−2𝑗−𝑙+1}𝐸_𝑙,𝑞. (3.5)

Equating 𝐼₃ and 𝐼₄, we get the following theorem.

Theorem 3.2. For 𝑘∈ℕ, one has ^𝑘_𝑙=0𝑘𝑙(−1)^𝑘−𝑙[2]_𝑞𝐸_{𝑘+𝑙+1,𝑞}−𝑞𝐸_{𝑘+𝑙,𝑞}=[2]_𝑞^[𝑘/2]_𝑗=0𝑘2𝑗^{2𝑘−2𝑗+1𝑙=0}𝑙𝐸2𝑘−2𝑗+1_{2𝑘+1−2𝑗−𝑙,𝑞}𝐸_𝑙,𝑞+^[𝑘/2]_𝑗=1𝑘2𝑗−1^{2𝑘−2𝑗+1𝑙=0}𝑙𝐸2𝑘−2𝑗+1_{2𝑘+1−2𝑗−𝑙,𝑞}𝐸_𝑙,𝑞+^[𝑘/2]_𝑗=0𝑘2𝑗+1𝑞−1∑^{2𝑘−2𝑗}_𝑗=0𝑙𝐸2𝑘−2𝑗_{2𝑘−2𝑗−𝑙,𝑞}𝐸_𝑙,𝑞+𝑞−11+𝑞^{2𝑘−2𝑗+1}_𝑙∑₌₀𝑙𝐸2𝑘−2𝑗+1_{2𝑘−2𝑗−𝑙+1}𝐸_𝑙,𝑞. (3.6)

Now, we consider (2.8) and (2.1) by means of 𝑞-Volkenborn integral. Then, by (2.8), we see [2]_𝑞ℤ𝑝𝑥^𝑘(𝑥−1)^𝑚𝑑𝜇_𝑞(𝑥)=[2]_𝑞^𝑚_𝑙=0𝑚𝑙(−1)^𝑚−𝑙_ℤ𝑝𝑥^𝑙+𝑘𝑑𝜇_𝑞(𝑥)=[2]_𝑞^𝑚_𝑙=0𝑚𝑙(−1)^𝑚−𝑙𝐵_{𝑙+𝑘,𝑞}. (3.7)

On the other hand, ^{max{𝑘,𝑚}}_𝑗=0𝑞𝑘𝑗+(−1)^𝑗𝑚𝑗^{𝑘+𝑚−𝑗𝑙=0}𝑙𝐸𝑘+𝑚−𝑗_{𝑘+𝑚−𝑗−𝑙,𝑞ℤ𝑝}𝑥^𝑙𝑑𝜇_𝑞(𝑥)=^{max{𝑘,𝑚}}_𝑗=0𝑞𝑘𝑗+(−1)^𝑗𝑚𝑗^{𝑘+𝑚−𝑗𝑙=0}𝑙𝐸𝑘+𝑚−𝑗_{𝑘+𝑚−𝑗−𝑙,𝑞}𝐵_𝑙,𝑞. (3.8)

Therefore, we get the following theorem.

Theorem 3.3. For 𝑚,𝑘∈ℕ, one has [2]_𝑞^𝑚_𝑙=0𝑚𝑙(−1)^𝑚−𝑙𝐵_{𝑙+𝑘,𝑞}=^{max{𝑘,𝑚}}_𝑗=0𝑞𝑘𝑗+(−1)^𝑗𝑚𝑗^{𝑘+𝑚−𝑗𝑙=0}𝑙𝐸𝑘+𝑚−𝑗_{𝑘+𝑚−𝑗−𝑙,𝑞}𝐵_𝑙,𝑞. (3.9)

By using fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 in left hand side of (2.21), we get 𝐼₅=[2]_𝑞ℤ𝑝𝑥^𝑘(𝑥−1)^𝑘[2]𝑥−𝑞𝑑𝜇_𝑞(𝑥)=[2]_𝑞^𝑘_𝑙=0𝑘𝑙(−1)^𝑘−𝑙_ℤ𝑝𝑥^𝑘+𝑙+1𝑑𝜇_𝑞(𝑥)−𝑞^𝑘_𝑙=0𝑘𝑙(−1)^𝑘−𝑙_ℤ𝑝𝑥^𝑘+𝑙𝑑𝜇_𝑞(𝑥)=[2]_𝑞^𝑘_𝑙=0𝑘𝑙(−1)^𝑘−𝑙𝐵_{𝑘+𝑙+1,𝑞}−𝑞^𝑘_𝑙=0𝑘𝑙(−1)^𝑘−𝑙𝐵_{𝑘+𝑙,𝑞}. (3.10)

Also, we consider right hand side of (2.21) as follows: 𝐼₆=[2]_𝑞^[𝑘/2]_𝑗=0𝑘2𝑗^{2𝑘−2𝑗+1𝑙=0}𝑙𝐸2𝑘−2𝑗+1_{2𝑘+1−2𝑗−𝑙,𝑞ℤ𝑝}𝑥^𝑙𝑑𝜇_𝑞(𝑥)+^[𝑘/2]_𝑗=1𝑘2𝑗−1^{2𝑘−2𝑗+1𝑙=0}𝑙𝐸2𝑘−2𝑗+1_{2𝑘+1−2𝑗−𝑙,𝑞ℤ𝑝}𝑥^𝑙𝑑𝜇_𝑞(𝑥)+^[𝑘/2]_𝑗=0𝑘2𝑗+1𝑞−1∑^{2𝑘−2𝑗}_𝑗=0𝑙𝐸2𝑘−2𝑗_{2𝑘−2𝑗−𝑙,𝑞}∫_ℤ𝑝𝑥^𝑙𝑑𝜇_𝑞(𝑥)+𝑞−11+𝑞^{2𝑘−2𝑗+1}_𝑙∑₌₀𝑙𝐸2𝑘−2𝑗+1_{2𝑘−2𝑗−𝑙+1}∫_ℤ𝑝𝑥^𝑙𝑑𝜇_𝑞(𝑥)=[2]_𝑞^[𝑘/2]_𝑗=0𝑘2𝑗^{2𝑘−2𝑗+1𝑙=0}𝑙𝐸2𝑘−2𝑗+1_{2𝑘+1−2𝑗−𝑙,𝑞}𝐵_𝑙,𝑞+^[𝑘/2]_𝑗=1𝑘2𝑗−1^{2𝑘−2𝑗+1𝑙=0}𝑙𝐸2𝑘−2𝑗+1_{2𝑘+1−2𝑗−𝑙,𝑞}𝐵_𝑙,𝑞+^[𝑘/2]_𝑗=0𝑘2𝑗+1𝑞−1∑^{2𝑘−2𝑗}_𝑗=0𝑙𝐸2𝑘−2𝑗_{2𝑘−2𝑗−𝑙,𝑞}𝐵_𝑙,𝑞+𝑞−11+𝑞^{2𝑘−2𝑗+1}_𝑙∑₌₀𝑙𝐸2𝑘−2𝑗+1_{2𝑘−2𝑗−𝑙+1}𝐵_𝑙,𝑞. (3.11)

Equating 𝐼₅ and 𝐼₆, we get the following corollary.

Corollary 3.4. For 𝑘∈ℕ, one gets ^𝑘_𝑙=0𝑘𝑙(−1)^𝑘−𝑙[2]_𝑞𝐵_{𝑘+𝑙+1,𝑞}−𝑞𝐵_{𝑘+𝑙,𝑞}=[2]_𝑞^[𝑘/2]_𝑗=0𝑘2𝑗^{2𝑘−2𝑗+1𝑙=0}𝑙𝐸2𝑘−2𝑗+1_{2𝑘+1−2𝑗−𝑙,𝑞}𝐵_𝑙,𝑞+^[𝑘/2]_𝑗=1𝑘2𝑗−1^{2𝑘−2𝑗+1𝑙=0}𝑙𝐸2𝑘−2𝑗+1_{2𝑘+1−2𝑗−𝑙,𝑞}𝐵_𝑙,𝑞+^[𝑘/2]_𝑗=0𝑘2𝑗+1𝑞−1∑^{2𝑘−2𝑗}_𝑗=0𝑙𝐸2𝑘−2𝑗_{2𝑘−2𝑗−𝑙,𝑞}𝐵_𝑙,𝑞+𝑞−11+𝑞^{2𝑘−2𝑗+1}_𝑙∑₌₀𝑙𝐸2𝑘−2𝑗+1_{2𝑘−2𝑗−𝑙+1}𝐵_𝑙,𝑞. (3.12)

Acknowledgment

The authors would like to thank the referee for his/her valuable comments on this work.