Identities Involving 𝑞-Bernoulli and 𝑞-Euler Numbers

Abstract

We give some identities on the q-Bernoulli and q-Euler numbers by using p-adic integral equations on ℤ_𝑝.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper, ℤ_𝑝, ℚ_𝑝, and ℂ_𝑝 will denote the ring of 𝑝-adic integers, the field of 𝑝-adic rational numbers, and the completion of algebraic closure of ℚ_𝑝, respectively. Let ℕ be the set of natural numbers and ℤ₊=ℕ∪{0}. The 𝑝-adic norm |⋅|_𝑝 is normally defined by |𝑝|_𝑝=1/𝑝.

As it is well known, the Euler polynomials are defined by2𝑒^𝑡𝑒+1^𝑥𝑡=𝑒^{𝐸(𝑥)𝑡}=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛,𝑛! (1.1) with the usual convention about replacing 𝐸^𝑛(𝑥) by 𝐸_𝑛(𝑥)  (see [1–14]). In the special case, 𝑥=0, 𝐸_𝑛(0)=𝐸_𝑛 is called the 𝑛th Euler number.

The ordinary Bernoulli polynomials are also defined by𝑡𝑒^𝑡𝑒−1^𝑥𝑡=𝑒^{𝐵(𝑥)𝑡}=^∞_𝑛=0𝐵_𝑛(𝑥)𝑡^𝑛,𝑛! (1.2) with the usual convention about replacing 𝐵^𝑛(𝑥) by 𝐵_𝑛(𝑥)h (see [1–14]). In the special case, 𝑥=0, 𝐵_𝑛(0)=𝐵_𝑛 is called the 𝑛th Bernoulli number.

Let 𝑈𝐷(ℤ_𝑝) be the space of uniformly differentiable functions on ℤ_𝑝. For 𝑓∈𝑈𝐷(ℤ_𝑝), the bosonic 𝑝-adic integral on ℤ_𝑝 is defined by𝐼𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇(𝑥)=lim^𝑝_𝑁→∞^𝑁−1_𝑥=0𝑓(𝑥)𝜇𝑥+𝑝^𝑁ℤ_𝑝=lim_𝑁→∞1𝑝^{𝑁𝑝𝑁−1}_𝑥=0𝑓(𝑥), (1.3) (see [1, 7]). Let 𝑓₁ be the translation of 𝑓 with 𝑓₁(𝑥)=𝑓(𝑥+1). From (1.3) we have𝐼𝑓₁−𝐼𝑓=𝑓^′(0), (1.4) (see [1, 7]).

The fermionic 𝑝-adic integral on ℤ_𝑝 is also defined by T. Kim as follows:𝐼₋₁𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥)=lim^𝑝_𝑁→∞^𝑁−1_𝑥=0𝑓(𝑥)𝜇₋₁𝑥+𝑝^𝑁ℤ_𝑝=lim^𝑝_𝑁→∞^𝑁−1_𝑥=0𝑓(𝑥)(−1)^𝑥, (1.5) (see [6, 15, 16]). By (1.5), we get𝐼₋₁𝑓₁+𝐼₋₁𝑓=2𝑓(0), (1.6) (see [6, 8]).

Let 𝑓(𝑥)=𝑒^𝑥𝑡∈𝑈𝐷(ℤ_𝑝) with |𝑡|_𝑝<|𝑝|_𝑝^1/(𝑝−1) and |𝑥|_𝑝≤1. From (1.4), we have_ℤ𝑝𝑒^{(𝑥+𝑦)𝑡}𝑦=𝑡𝑑𝜇𝑒^𝑡𝑒−1^𝑥𝑡=^∞_𝑛=0𝐵_𝑛(𝑥)𝑡^𝑛.𝑛! (1.7) Thus, by (1.7), we see that_ℤ𝑝𝑥+𝑦^𝑛𝑑𝜇𝑦=𝐵_𝑛(𝑥),𝑛∈ℤ₊. (1.8) By (1.8), we get𝐵_𝑛(𝑥)=^𝑛_𝑙=0𝑛𝑙𝑥^𝑛−𝑙_ℤ𝑝𝑦^𝑙𝑑𝜇𝑦=^𝑛_𝑙=0𝑛𝑙𝑥^𝑛−𝑙𝐵_𝑙. (1.9) As an indeterminate, let us assume that 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<1.

From (1.1) and (1.6), we note that the 𝑞-Euler polynomials are given by_ℤ𝑝𝑞^𝑦𝑒^{(𝑥+𝑦)𝑡}𝑑𝜇₋₁𝑦=2𝑞𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐸_𝑛,𝑞(𝑥)𝑡^𝑛,𝑛! (1.10) where 𝐸_𝑛,𝑞(𝑥) are called the 𝑛th 𝑞-Euler polynomials (see [1, 3, 6, 8]).

Thus, by (1.10), we get𝐸_𝑛,𝑞(𝑥)=_ℤ𝑝𝑞^𝑦𝑥+𝑦^𝑛𝑑𝜇₋₁𝑦,𝑛∈ℤ₊. (1.11) In the special case, 𝑥=0, 𝐸_𝑛,𝑞(0)=𝐸_𝑛,𝑞 is called the 𝑛th 𝑞-Euler number (see [8, 9]). By (1.10) and (1.11), we get the recurrence formula for the 𝑞-Euler numbers as follows:𝐸_0,𝑞=2[2]_𝑞,𝑞𝐸_𝑞+1^𝑛+𝐸_𝑛,𝑞=2𝛿_0,𝑛, (1.12) with the usual convention about replacing 𝐸^𝑛_𝑞 by 𝐸_𝑛,𝑞. Here [𝑥]_𝑞=(1−𝑞^𝑥)/(1−𝑞) is the 𝑞-number of 𝑥 and 𝛿_𝑘,𝑛 is the Kronecker symbol (see [10, 11]).

From (1.2), (1.7), and (1.8), we have𝐵₀=1,(𝐵+1)^𝑛−𝐵_𝑛=𝛿_1,𝑛, (1.13) with the usual convention about replacing 𝐵^𝑛 by 𝐵_𝑛 (see [1, 3, 14]).

From (1.11), we easily see that𝐸_𝑛,𝑞(𝑥)=^𝑛_𝑙=0𝑛𝑙𝑥^𝑛−𝑙𝐸_𝑙,𝑞, (1.14) (see [14]).

In this paper we give some interesting properties of 𝑝-adic integrals on ℤ_𝑝 associated with the 𝑞-Bernoulli and the 𝑞-Euler numbers. From those properties, we derive new identities involving the 𝑞-Bernoulli and the 𝑞-Euler numbers arising from 𝑝-adic integrals of polynomial identities.

2. Identities on 𝑞-Bernoulli and 𝑞-Euler Numbers

Let 𝐶_𝑝𝑛 be the cyclic group of order 𝑝^𝑛 with 𝐶_𝑝𝑛={𝜁∈ℂ_𝑝∣𝜁^𝑝𝑛=1}. Then 𝑇_𝑝 is defined by the direct limit as 𝑇_𝑝=lim_𝑛→∞𝐶_𝑝𝑛=𝐶_𝑝∞. In this section, we assume that 𝑞(≠1)∈𝑇_𝑝, then |𝑞−1|_𝑝<1. From (1.4), we can derive the following equation (2.1):_ℤ𝑝𝑞^𝑦𝑒^{(𝑥+𝑦)𝑡}𝑦=𝑡𝑑𝜇𝑞𝑒^𝑡𝑒−1^𝑥𝑡=^∞_𝑛=0𝐵_𝑛,𝑞(𝑥)𝑡^𝑛,𝑛! (2.1) where 𝐵_𝑛,𝑞(𝑥) is called the 𝑛th 𝑞-Bernoulli polynomial (see [7]). In the special case, 𝑥=0, 𝐵_𝑛,𝑞(0)=𝐵_𝑛,𝑞 is called the 𝑛th 𝑞-Bernoulli number.

By (2.1), we get𝐵_0,𝑞=0,𝑞𝐵_𝑞+1^𝑛−𝐵_𝑛,𝑞=𝛿_1,𝑛, (2.2) with the usual convention about replacing 𝐵^𝑛_𝑞 by 𝐵_𝑛,𝑞 (see [7, 14]).

From (1.3), we have_ℤ𝑝𝑓−𝑦𝑑𝜇𝑦=_ℤ𝑝𝑓𝑦+1𝑑𝜇𝑦. (2.3) By (2.3), we get𝑞_ℤ𝑝1−𝑥+𝑦^𝑛𝑞^𝑦𝑑𝜇𝑦=(−1)^𝑛_ℤ𝑝𝑞^−𝑦𝑥+𝑦^𝑛𝑑𝜇𝑦. (2.4) From (2.1) and (2.4), we have𝑞𝐵_𝑛,𝑞(1−𝑥)=(−1)^𝑛𝐵_{𝑛,𝑞−1}(𝑥),𝑛∈ℤ₊. (2.5) By using (1.4), we see that𝑞_ℤ𝑝1+𝑥+𝑦^𝑛𝑞^𝑦𝑑𝜇𝑦=_ℤ𝑝𝑞^𝑦𝑥+𝑦^𝑛𝑑𝜇𝑦+𝑛𝑥^𝑛−1. (2.6) Thus, by (2.1) and (2.6), we get𝑞𝐵_𝑛,𝑞(1+𝑥)=𝐵_𝑛,𝑞(𝑥)+𝑛𝑥^𝑛−1,𝑛∈ℤ₊. (2.7) Therefore, by (2.5) and (2.7), we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℤ₊, one has (−1)^𝑛𝐵_{𝑛,𝑞−1}(−𝑥)=𝑞𝐵_𝑛,𝑞(1+𝑥)=𝐵_𝑛,𝑞(𝑥)+𝑛𝑥^𝑛−1. (2.8)

From (1.5) and (1.6), we note that_ℤ𝑝1−𝑥+𝑦^𝑛𝑞^𝑦𝑑𝜇₋₁𝑦=(−1)^𝑛_ℤ𝑝𝑥+𝑦^𝑛𝑞^−(𝑦+1)𝑑𝜇₋₁𝑦. (2.9) Therefore, by (1.11) and (2.9), we obtain the following theorem.

Theorem 2.2. For 𝑛∈ℤ₊, one has 𝑞𝐸_𝑛,𝑞(1−𝑥)=(−1)^𝑛𝐸_{𝑛,𝑞−1}(𝑥). (2.10)

By (1.6), we get𝑞_ℤ𝑝𝑞^𝑦𝑥+1+𝑦^𝑛𝑑𝜇₋₁𝑦+_ℤ𝑝𝑞^𝑦𝑥+𝑦^𝑛𝑑𝜇₋₁𝑦=2𝑥^𝑛. (2.11) Thus, by (1.11) and (2.11), we have𝑞𝐸_𝑛,𝑞(𝑥+1)=−𝐸_𝑛,𝑞(𝑥)+2𝑥^𝑛,𝑛∈ℤ₊. (2.12) Therefore, by Theorem 2.2 and (2.12), we obtain the following theorem.

Theorem 2.3. For 𝑛∈ℤ₊, one has (−1)^𝑛𝐸_{𝑛,𝑞−1}(−𝑥)=𝑞𝐸_𝑛,𝑞(1+𝑥)=−𝐸_𝑛,𝑞(𝑥)+2𝑥^𝑛. (2.13)

By using the 𝑝-adic integrals on ℤ_𝑝, we have the following equation (2.14):_{ℤ𝑝ℤ𝑝}𝑞^(𝑥+𝑦)𝑒^{(𝑥+𝑦)𝑡}𝑑𝜇(𝑥)𝑑𝜇₋₁𝑦=_ℤ𝑝𝑞^𝑥𝑒^𝑥𝑡𝑑𝜇(𝑥)_ℤ𝑝𝑞^𝑦𝑒^𝑦𝑡𝑑𝜇₋₁𝑦=𝑡𝑞𝑒^𝑡2−1𝑞𝑒^𝑡=+12𝑡𝑞²𝑒^2𝑡=−1_ℤ𝑝𝑞^2𝑥𝑒^2𝑥𝑡𝑑𝜇(𝑥). (2.14) By (2.14), we get_{ℤ𝑝ℤ𝑝}𝑞^(𝑥+𝑦)𝑥+𝑦^𝑛𝑑𝜇(𝑥)𝑑𝜇₋₁𝑦=2^𝑛_ℤ𝑝𝑞^2𝑥𝑥^𝑛𝑑𝜇(𝑥). (2.15) It is not difficult to show that_{ℤ𝑝ℤ𝑝}𝑞^(𝑥+𝑦)𝑥+𝑦^𝑛𝑑𝜇(𝑥)𝑑𝜇₋₁𝑦=^𝑛_𝑙=0𝑛𝑙^{ℤ𝑝ℤ𝑝}𝑞^𝑥𝑥^𝑛−𝑙𝑞^𝑦𝑦^𝑙𝑑𝜇(𝑥)𝑑𝜇₋₁𝑦. (2.16) Therefore, by (2.15) and (2.16), we obtain the following theorem.

Theorem 2.4. For 𝑛∈ℤ₊, one has ^𝑛_𝑙=0𝑛𝑙𝐵_{𝑛−𝑙,𝑞}𝐸_𝑙,𝑞=2^𝑛𝐵_𝑛,𝑞2. (2.17)

By (2.5), (2.7), (2.12), Theorems 2.1, and 2.3, we get𝑞𝐵_𝑛,𝑞(𝑥)=(−1)^𝑛𝐵_{𝑛,𝑞−1}(1−𝑥)=𝐵_𝑛,𝑞(𝑥−1)+𝑛(𝑥−1)^𝑛−1,𝑞𝐸_𝑛,𝑞(𝑥)=(−1)^𝑛𝐸_{𝑛,𝑞−1}(1−𝑥)=−𝐸_𝑛,𝑞(𝑥−1)+2(𝑥−1)^𝑛. (2.18)(2.19) From (2.1), we have𝐵_𝑛,𝑞(𝑥)=_ℤ𝑝𝑞^𝑦𝑥+𝑦^𝑛𝑑𝜇𝑦=^𝑛_𝑙=0𝑛𝑙𝐵_𝑙,𝑞𝑥^𝑛−𝑙,𝑛∈ℤ₊. (2.20) Let𝐼₁=𝑞_ℤ𝑝𝑞^𝑥𝐵_𝑛,𝑞(𝑥)𝑑𝜇(𝑥)=𝑞^𝑛_𝑙=0𝑛𝑙𝐵_{𝑛−𝑙,𝑞}𝐵_𝑙,𝑞. (2.21) From (2.18), (2.20), and (2.21), we note that𝐼₁=(−1)^𝑛𝑛_𝑙=0𝑛𝑙𝐵_{𝑛−𝑙,𝑞ℤ𝑝}𝑞^𝑥(𝑥−1)^𝑙𝑑𝜇(𝑥)+𝑛_ℤ𝑝𝑞^𝑥(𝑥−1)^𝑛−1(𝑥)=()𝑑𝜇−1^𝑛𝑛_𝑙=0𝑛𝑙𝐵_{𝑛−𝑙,𝑞}𝐵_𝑙,𝑞(−1)+𝑛𝐵_{𝑛−1,𝑞}()=()−1−1^𝑛𝑛_𝑙=0𝑛𝑙𝐵_𝑙,𝑞𝐵_{𝑛−𝑙,𝑞}(−1)+𝑛𝐵_{𝑛−1,𝑞}(−1). (2.22) By (2.5), we get𝑞𝐵_𝑛,𝑞(−1)=(−1)^𝑛𝐵_{𝑛,𝑞−1}(2),𝑛∈ℤ₊. (2.23) By (2.3), we easily see that𝑞²𝐵_𝑛,𝑞(2)=𝑛𝑞+𝑞𝐵_𝑛,𝑞(1)=𝑛𝑞+𝐵_𝑛,𝑞+𝛿_1,𝑛, (2.24) where 𝛿_1,𝑛 is the Kronecker symbol.

Thus, by (2.23) and (2.24), we get𝐵_𝑛,𝑞(−1)=(−1)^𝑛𝑛+𝑞𝐵_{𝑛,𝑞−1}+𝑞𝛿_1,𝑛. (2.25) By (2.22) and (2.25), we get𝐼₁=(−1)^𝑛𝑛_𝑙=0𝑛𝑙𝐵_𝑙,𝑞(−1)^𝑛−𝑙𝑛−𝑙+𝑞𝐵_{𝑛−𝑙,𝑞−1}+𝑞𝛿_{1,𝑛−𝑙}+𝑛𝐵_{𝑛−1,𝑞}(−1)=𝑛^𝑛−1_𝑙=0𝑙𝐵𝑛−1_𝑙,𝑞(−1)^𝑙+𝑞^𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝐵_𝑙,𝑞𝐵_{𝑛−𝑙,𝑞−1}+𝑞(−1)^𝑛−1𝑛𝐵_{𝑛−1,𝑞}+𝑛𝐵_{𝑛−1,𝑞}(−1)=𝑞^𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝐵_𝑙,𝑞𝐵_{𝑛−𝑙,𝑞−1}+𝑞(−1)^𝑛−1𝑛𝐵_{𝑛−1,𝑞}+1+(−1)^𝑛−1𝑛𝐵_{𝑛−1,𝑞}(−1). (2.26) Therefore, by (2.21) and (2.26), we obtain the following theorem.

Theorem 2.5. For 𝑛∈ℤ₊, one has ^2𝑛𝑙=0𝑙𝐵2𝑛_{2𝑛−𝑙,𝑞}𝐵_𝑙,𝑞−^2𝑛_𝑙=0𝑙()2𝑛−1^𝑙𝐵_𝑙,𝑞𝐵_{2𝑛−𝑙,𝑞−1}=−2𝑛𝐵_{2𝑛−1,𝑞}. (2.27)

Let us consider the following integral:𝐼₂=_ℤ𝑝𝑞^𝑥+1𝐸_𝑛,𝑞(𝑥)𝑑𝜇₋₁(𝑥)=𝑞^𝑛_𝑙=0𝑛𝑙𝐸_{𝑛−𝑙,𝑞}𝐸_𝑙,𝑞. (2.28) By (2.19), we get𝐼₂=−_ℤ𝑝𝑞^𝑥𝐸_𝑛,𝑞(𝑥−1)𝑑𝜇₋₁(𝑥)+2_ℤ𝑝𝑞^𝑥(𝑥−1)^𝑛𝑑𝜇₋₁(𝑥)=−^𝑛_𝑙=0𝑛𝑙𝐸_𝑙,𝑞𝐸_{𝑛−𝑙,𝑞}(−1)+2𝐸_𝑛,𝑞(−1)=−^𝑛_𝑙=0𝑛𝑙𝐸_{𝑛−𝑙,𝑞}𝐸_𝑙,𝑞(−1)+2𝐸_𝑛,𝑞(−1). (2.29) From Theorem 2.2, we note that𝑞𝐸_𝑛,𝑞(−1)=(−1)^𝑛𝐸_{𝑛,𝑞−1}(2),𝑛∈ℤ₊. (2.30) By (1.12), we get𝑞²𝐸_𝑛,𝑞(2)=2𝑞+𝐸_𝑛,𝑞−2𝛿_0,𝑛. (2.31) Thus, by (2.30) and (2.31), we get𝐸_𝑛,𝑞(−1)=(−1)^𝑛2+𝑞𝐸_{𝑛,𝑞−1}−2𝑞𝛿_0,𝑛. (2.32) From (2.29) and (2.32), we note that𝐼₂=−^𝑛_𝑙=0𝑛𝑙𝐸_𝑙,𝑞(−1)^𝑛−𝑙2+𝑞𝐸_{𝑛−𝑙,𝑞−1}−2𝑞𝛿_{0,𝑛−𝑙}+2(−1)^𝑛2+𝑞𝐸_{𝑛,𝑞−1}−2𝑞𝛿_0,𝑛=−𝑞^𝑛_𝑙=0𝑛𝑙(−1)^𝑛−𝑙𝐸_𝑙,𝑞𝐸_{𝑛−𝑙,𝑞−1}+2𝑞𝐸_𝑛,𝑞. (2.33) Therefore, by (2.28) and (2.33), we obtain the following theorem.

Theorem 2.6. For 𝑛∈ℤ₊, one has ^𝑛_𝑙=0𝑛𝑙𝐸_{𝑛−𝑙,𝑞}𝐸_𝑙,𝑞+^𝑛_𝑙=0𝑛𝑙(−1)^𝑛−𝑙𝐸_𝑙,𝑞𝐸_{𝑛−𝑙,𝑞−1}=2𝐸_𝑛,𝑞. (2.34)

Now we consider the fermionic 𝑝-adic integral on ℤ_𝑝 for the 𝑛th 𝑞-Euler polynomials as follows:𝐼₃=_ℤ𝑝𝑞^𝑥𝐸_𝑛,𝑞(𝑥)𝑑𝜇₋₁(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐸_{𝑙,𝑞ℤ𝑝}𝑞^𝑥𝑥^𝑛−𝑙𝑑𝜇₋₁(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐸_𝑙,𝑞𝐸_{𝑛−𝑙,𝑞},𝑛∈ℤ₊. (2.35) On the other hand, by Theorem 2.2, we get𝐼₃=(−1)^𝑛𝑞⁻¹_ℤ𝑝𝐸_{𝑛,𝑞−1}(1−𝑥)𝑞^𝑥𝑑𝜇₋₁(𝑥)=(−1)^𝑛𝑞^𝑛−1_𝑙=0𝑛𝑙𝐸_{𝑛−𝑙,𝑞−1ℤ𝑝}𝑞^𝑥(1−𝑥)^𝑙𝑑𝜇₋₁(𝑥)=𝑞^𝑛−1_𝑙=0𝑛𝑙𝐸_{𝑛−𝑙,𝑞−1}(−1)^𝑛−𝑙𝐸_𝑙,𝑞(−1). (2.36) From (2.32) and (2.36), we note that𝐼₃=(−1)^𝑛𝑞^𝑛−1_𝑙=0𝑛𝑙𝐸_{𝑛−𝑙,𝑞−1}2+𝑞𝐸_{𝑙,𝑞−1}−2𝑞𝛿_0,𝑙=2(−1)^𝑛𝑞⁻¹𝐸_{𝑛,𝑞−1}(1)+(−1)^𝑛𝑛_𝑙=0𝑛𝑙𝐸_{𝑛−𝑙,𝑞−1}𝐸_{𝑙,𝑞−1}−2(−1)^𝑛𝐸_{𝑛,𝑞−1}=−2(−1)^𝑛𝐸_{𝑛,𝑞−1}+4(−1)^𝑛𝛿_0,𝑛+(−1)^𝑛𝑛_𝑙=0𝑛𝑙𝐸_{𝑛−𝑙,𝑞−1}𝐸_{𝑙,𝑞−1}−2(−1)^𝑛𝐸_{𝑛,𝑞−1}. (2.37) Therefore, by (2.35) and (2.37), we obtain the following theorem.

Theorem 2.7. For 𝑛∈ℤ₊, one has ^{2𝑛+1𝑙=0}𝑙𝐸2𝑛+1_𝑙,𝑞𝐸_{2𝑛+1−𝑙,𝑞}+^2𝑛+1_𝑙=0𝑙𝐸2𝑛+1_{𝑙,𝑞−1}𝐸_{2𝑛+1−𝑙,𝑞−1}=4𝐸_{2𝑛+1,𝑞−1}. (2.38)

From (2.1) and (2.7), we note that𝑥^𝑛=𝑞𝑛+1^𝑛_𝑙=0𝑙𝐵𝑛+1_𝑙,𝑞(𝑥)+𝑞−1𝐵𝑛+1^𝑛+1,𝑞(𝑥). (2.39) Let us consider the following fermionic 𝑝-adic integral on ℤ_𝑝:_ℤ𝑝𝑥^𝑛𝑞^𝑥𝑑𝜇₋₁(𝑥)=𝑞𝑛+1^𝑛_𝑙=0𝑙𝑛+1_ℤ𝑝𝑞^𝑥𝐵_𝑙,𝑞(𝑥)𝑑𝜇₋₁(𝑥)+𝑞−1𝑛+1_ℤ𝑝𝑞^𝑥𝐵_𝑛+1,𝑞(𝑥)𝑑𝜇₋₁(𝑥)=𝑞𝑛+1^𝑛_𝑙=0𝑙𝑛+1^𝑙_𝑘=0𝑙𝑘𝐵_{𝑙−𝑘,𝑞}𝐸_𝑘,𝑞+𝑞−1𝑛+1^𝑛+1_𝑙=0𝑙𝐵𝑛+1_{𝑛+1−𝑙,𝑞}𝐸_𝑙,𝑞. (2.40) Therefore, by (2.40), we obtain the following theorem.

Theorem 2.8. For 𝑛∈ℤ₊, one has 𝐸_𝑛,𝑞=𝑞𝑛+1^𝑛_𝑙=0𝑙𝑛+1^𝑙_𝑘=0𝑙𝑘𝐵_{𝑙−𝑘,𝑞}𝐸_𝑘,𝑞+𝑞−1𝑛+1^𝑛+1_𝑙=0𝑙𝐵𝑛+1_{𝑛+1−𝑙,𝑞}𝐸_𝑙,𝑞. (2.41)

From (1.10) and (2.12), we note that𝑥^𝑛=[2]_𝑞2𝐸_𝑛,𝑞(𝑥)+𝑞2^𝑛−1_𝑙=0𝑛𝑙𝐸_𝑙,𝑞(𝑥). (2.42) Thus, by (2.42), we get_ℤ𝑝𝑞^𝑥𝑥^𝑛𝑑𝜇₋₁(𝑥)=[2]_𝑞2_ℤ𝑝𝐸_𝑛,𝑞(𝑥)𝑞^𝑥𝑑𝜇₋₁(𝑥)+𝑞2^𝑛−1_𝑙=0𝑛𝑙^ℤ𝑝𝑞^𝑥𝐸_𝑙,𝑞(𝑥)𝑑𝜇₋₁(𝑥)=[2]_𝑞2^𝑛_𝑙=0𝑛𝑙𝐸_{𝑛−𝑙,𝑞}𝐸_𝑙,𝑞+𝑞2^𝑛−1_𝑙=0𝑛𝑙^𝑙𝑘=0𝑙𝑘𝐸_{𝑙−𝑘,𝑞}𝐸_𝑘,𝑞. (2.43) Thus, by (2.43), we have𝐸_𝑛,𝑞=[2]_𝑞2^𝑛_𝑙=0𝑛𝑙𝐸_{𝑛−𝑙,𝑞}𝐸_𝑙,𝑞+𝑞2^𝑛−1_𝑙=0𝑛𝑙^𝑙𝑘=0𝑙𝑘𝐸_{𝑙−𝑘,𝑞}𝐸_𝑘,𝑞. (2.44)

Acknowledgments

The authors would like to express their sincere gratitude to referees for their valuable suggestions and comments. The first author was supported by National Research Foundation of Korea Grant funded by the Korean Government 2011-0002486.