Some New Identities on the Bernoulli and Euler Numbers

Abstract

We give some new identities on the Bernoulli and Euler numbers by using the bosonic p-adic integral on ℤ_𝑝 and reflection symmetric properties of Bernoulli and Euler polynomials.

1. Introduction

Let 𝑝 be a fixed prime number. Throughout this paper ℤ_𝑝,ℚ_𝑝, and ℂ_𝑝 will denote the ring of 𝑝-adic rational integers, the field of 𝑝-adic rational numbers, and the completion of algebraic closure of ℚ_𝑝. 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 by 𝐼𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇(𝑥)=lim^𝑝_𝑁→∞^𝑁−1_𝑥=0𝑓(𝑥)𝜇𝑥+𝑝^𝑁ℤ_𝑝=lim_𝑁→∞1𝑝^{𝑁𝑝𝑁−1}_𝑥=0𝑓(𝑥). (1.1) From (1.1), we note that𝐼𝑓₁=𝐼𝑓+𝑓^′(0),where𝑓₁(𝑥)=𝑓(𝑥+1), (1.2) see [1]. As is well known, the ordinary Bernoulli polynomials are defined by the generating function as follows:𝐹()=𝑡𝑡,𝑥𝑒^𝑡𝑒−1^𝑥𝑡=𝑒^{𝐵(𝑥)𝑡}=^∞_𝑛=0𝐵_𝑛(𝑥)𝑡^𝑛𝑛!, (1.3) see [1–19], where we use the technical notation by replacing 𝐵^𝑛(𝑥) by 𝐵_𝑛(𝑥)(𝑛≥0), symbolically. In the special case, 𝑥=0, 𝐵_𝑛(0)=𝐵_𝑛 are called the 𝑛-th ordinary Bernoulli numbers. That is, the generating function of ordinary Bernoulli numbers is given by 𝐹(𝑡)()=𝑡=𝐹𝑡,0𝑒^𝑡=−1^∞_𝑛=0𝐵_𝑛𝑡^𝑛𝑛!, (1.4) see [1–19]. From (1.4), we can derive the following relation:𝐵₀=1,(𝐵+1)^𝑛−𝐵_𝑛=𝛿_1,𝑛, (1.5) see [1, 10], where 𝛿_1,𝑛 is the Kronecker symbol.

By (1.3) and (1.4), we easily get𝐵_𝑛(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐵_𝑙𝑥^𝑛−𝑙=^𝑛_𝑙=0𝑛𝑙𝐵_𝑛−𝑙𝑥^𝑙. (1.6) By (1.2) and (1.3), we easily get_ℤ𝑝𝑒^{(𝑥+𝑦)𝑡}𝑦=𝑡𝑑𝜇𝑒^𝑡𝑒−1^𝑥𝑡=^∞_𝑛=0𝐵_𝑛(𝑥)𝑡^𝑛𝑛!, (1.7) see [1, 10]. From (1.7), we can derive Witt’s formula for the 𝑛-th Bernoulli polynomials as follows:_ℤ𝑝𝑥+𝑦^𝑛𝑑𝜇𝑦=𝐵_𝑛(𝑥),where𝑛∈ℤ₊, (1.8) see [11]. By (1.1) and (1.8), we easily see that_ℤ𝑝𝑦+1−𝑥^𝑛𝑑𝜇𝑦=(−1)^𝑛_ℤ𝑝𝑦+𝑥^𝑛𝑑𝜇𝑦. (1.9) Thus, by (1.8) and (1.9), we get reflection symmetric relation for the Bernoulli polynomials as follows:𝐵_𝑛(1−𝑥)=(−1)^𝑛𝐵_𝑛(𝑥)where𝑛∈ℤ₊. (1.10) The ordinary Euler polynomials are defined by the generating function as follows:𝐹_𝑒()=2𝑡,𝑥𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛𝑛!. (1.11) with the usual convention about replacing 𝐸^𝑛(𝑥) by 𝐸_𝑛(𝑥) (see [8, 9]). In the special case, 𝑥=0, 𝐸_𝑛(0)=𝐸_𝑛 are called the 𝑛-th Euler numbers (see [8, 9]).

From (1.11), we note that2𝑒^𝑡𝑒+1^𝑥𝑡=21+𝑒^−𝑡𝑒^{−(1−𝑥)𝑡}=^∞_𝑛=0(−1)^𝑛𝐸_𝑛()(𝑡)1−𝑥^𝑛𝑛!, (1.12) By comparing the coefficients on both sides of (1.11) and (1.12), we obtain the following reflection symmetric relation for Euler polynomials as follows:𝐸_𝑛(𝑥)=(−1)^𝑛𝐸_𝑛(1−𝑥),where𝑛∈ℤ₊. (1.13) The equations (1.10) and (1.13) are useful in deriving our main results in this paper.

For 𝑛,𝑘∈ℤ₊, the Bernstein polynomials are defined by𝐵_𝑘,𝑛(𝑥)=𝑛𝑘𝑥^𝑘(1−𝑥)^𝑛−𝑘, (1.14) see [13]. By (1.14), we easily get 𝐵_𝑘,𝑛(𝑥)=𝐵_{𝑛−𝑘,𝑛}(1−𝑥).

In this paper we consider the 𝑝-adic integrals for the Bernoulli and Euler polynomials. From those 𝑝-adic integrals, we derive some new identities on the Bernoulli and Euler numbers.

2. Identities on the Bernoulli and Euler Numbers

First, we consider the 𝑝-adic integral on ℤ_𝑝 for the 𝑛th ordinary Bernoulli polynomials as follows: 𝐼₁=_ℤ𝑝𝐵_𝑛(𝑥)𝑑𝜇(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐵_{𝑛−𝑙ℤ𝑝}𝑥^𝑙(𝑥)=𝑑𝜇_𝑛𝑙=0𝑛𝑙𝐵_𝑛−𝑙𝐵_𝑙,where𝑛∈ℤ₊. (2.1) On the other hand, by (1.3) and (1.10), one gets𝐼₁=(−1)^𝑛_ℤ𝑝𝐵_𝑛(1−𝑥)𝑑𝜇(𝑥). (2.2) From (1.5), (1.6), (1.8), and (2.2), one notes that𝐼₁=(−1)^𝑛𝑛_𝑙=0𝑛𝑙𝐵_{𝑛−𝑙ℤ𝑝}(1−𝑥)^𝑙(𝑥)=()𝑑𝜇−1^𝑛𝑛_𝑙=0𝑛𝑙𝐵_𝑛−𝑙𝑙+𝐵_𝑙+𝛿_1,𝑙=(−1)^𝑛𝑛𝐵_𝑛−𝑙(1)+(−1)^𝑛𝑛_𝑙=0𝑛𝑙𝐵_𝑛−𝑙𝐵_𝑙+(−1)^𝑛𝑛𝐵_𝑛−𝑙. (2.3) Equating (2.1) and (2.3), one gets1+(−1)^𝑛+1𝑛_𝑙=0𝑛𝑙𝐵_𝑛−𝑙𝐵_𝑙=(−1)^𝑛𝑛𝛿_{1,𝑛−𝑙}+𝐵_𝑛−1+(−1)^𝑛𝑛𝐵_𝑛−1=2(−1)^𝑛𝑛𝐵_𝑛−𝑙+(−1)^𝑛𝑛𝛿_1,𝑛−1. (2.4) Let 𝑛∈ℕ with 𝑛≡1(mod2). Then, by (2.4), one has^{2𝑛−1𝑙=0}𝑙𝐵2𝑛−1_{2𝑛−1−𝑙}𝐵_𝑙=−(2𝑛−1)𝐵_2𝑛−2. (2.5) Therefore, by (2.4) and (2.5), we obtain the following theorem.

Theorem 2.1. For𝑛∈ℕ, one has 1+(−1)^𝑛+1𝑛_𝑙=0𝑛𝑙𝐵_𝑛−𝑙𝐵_𝑙=2(−1)^𝑛𝑛𝐵_𝑛−1+(−1)^𝑛𝑛𝛿_1,𝑛−1. (2.6) In particular, ^{2𝑛−1𝑙=0}𝑙𝐵2𝑛−1_{2𝑛−1−𝑙}𝐵_𝑙=−(2𝑛−1)𝐵_2𝑛−2. (2.7)

By the same motivation, let us also consider the 𝑝-adic integral on ℤ_𝑝 for Euler polynomials as follows:𝐼₂=_ℤ𝑝𝐸_𝑛(𝑥)𝑑𝜇(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐸_{𝑛−𝑙ℤ𝑝}𝑥^𝑙(𝑥)=𝑑𝜇_𝑛𝑙=0𝑛𝑙𝐸_𝑛−𝑙𝐵_𝑙,where𝑛∈ℤ₊. (2.8) On the other hand, by (1.12) and (1.13), one gets𝐼₂=(−1)^𝑛_ℤ𝑝𝐸_𝑛(1−𝑥)𝑑𝜇(𝑥)=(−1)^𝑛𝑛_𝑙=0𝑛𝑙𝐸_{𝑛−𝑙ℤ𝑝}(1−𝑥)^𝑙(𝑥)=()𝑑𝜇−1^𝑛𝑛_𝑙=0𝑛𝑙𝐸_𝑛−𝑙𝑙+𝐵_𝑙+𝛿_1,𝑙=𝑛(−1)^𝑛𝐸_𝑛−𝑙(1)+(−1)^𝑛𝑛_𝑙=0𝑛𝑙𝐸_𝑛−𝑙𝐵_𝑙+(−1)^𝑛𝑛𝐸_𝑛−𝑙. (2.9) From (1.12) and the definition of Euler numbers, one has𝐸_𝑛(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐸_𝑙𝑥^𝑛−𝑙=^𝑛_𝑙=0𝑛𝑙𝐸_𝑛−𝑙𝑥^𝑙=(𝐸+𝑥)^𝑛, (2.10)𝐸₀=1,(𝐸+1)^𝑛+𝐸_𝑛=2𝛿_0,𝑛, (2.11) see [8, 9] with the usual convention of replacing 𝐸^𝑛 by 𝐸_𝑛. By (2.9), (2.10), and (2.11), one gets𝐼₂=𝑛(−1)^𝑛2𝛿_0,𝑛−1−𝐸_𝑛−1+(−1)^𝑛𝑛𝐸_𝑛−1+(−1)^𝑛𝑛_𝑙=0𝑛𝑙𝐸_𝑛−𝑙𝐵_𝑙. (2.12) Equating (2.8) and (2.12), one has1+(−1)^𝑛−1𝑛_𝑙=0𝑛𝑙𝐸_𝑛−𝑙𝐵_𝑙=2𝑛(−1)^𝑛𝛿_0,𝑛−1. (2.13) Therefore, by (2.13), we obtain the following theorem.

Theorem 2.2. For 𝑛∈ℕ∪{0}, one has 1+(−1)^𝑛−1𝑛_𝑙=0𝑛𝑙𝐸_𝑛−𝑙𝐵_𝑙=2(−1)^𝑛𝑛𝛿_0,𝑛−1. (2.14) In particular, ^{2𝑛+1𝑙=0}𝑙𝐸2𝑛+1_{2𝑛+1−𝑙}𝐵_𝑙=0,for𝑛∈ℕ. (2.15)

Let us consider the following 𝑝-adic integral on ℤ_𝑝 for the product of Bernoulli and Euler polynomials as follows:𝐼₃=_ℤ𝑝𝐵_𝑚(𝑥)𝐸_𝑛(𝑥)(𝑥)=𝑑𝜇_{𝑚𝑛𝑘=0ℓ=0}𝑚𝑘𝑛ℓ𝐵_𝑚−𝑘𝐸_{𝑛−ℓℤ𝑝}𝑥^𝑘+ℓ(𝑥)(𝑥)=𝑑𝜇_{𝑚𝑛𝑘=0ℓ=0}𝑚𝑘𝑛ℓ𝐵_𝑚−𝑘𝐸_𝑛−ℓ𝐵_𝑘+ℓ. (2.16) On the other hand, by (1.10) and (1.13), one gets𝐼₃=(−1)^𝑚+𝑛_ℤ𝑝𝐵_𝑚(1−𝑥)𝐸_𝑛()(𝑥)=()1−𝑥𝑑𝜇−1^{𝑚𝑚+𝑛𝑛}_𝑘=0ℓ=0𝑚𝑘𝑛ℓ𝐵_𝑚−𝑘𝐸_{𝑛−ℓℤ𝑝}(1−𝑥)^𝑘+ℓ(𝑥)=()𝑑𝜇−1^{𝑚𝑚+𝑛𝑛}_𝑘=0ℓ=0𝑚k𝑛ℓ𝐵_𝑚−𝑘𝐸_𝑛−ℓ𝑘+ℓ+𝐵_𝑘+ℓ+𝛿_1,𝑘+ℓ=(−1)^𝑚+𝑛𝑚𝐵_𝑚−1(1)𝐸_𝑛(1)+(−1)^𝑚+𝑛𝑛𝐵_𝑚(1)𝐸_𝑛−1(1)+(−1)^{𝑚𝑚+𝑛𝑛}_𝑘=0ℓ=0𝑚𝑘𝑛ℓ𝐵_𝑚−𝑘𝐸_𝑛−ℓ𝐵_𝑘+ℓ+(−1)^𝑚+𝑛𝑚𝐵_𝑚−1𝐸_𝑛+𝑛𝐵_𝑚𝐸_𝑛−1. (2.17) Equating (2.16) and (2.17), one gets(−1)^𝑚+𝑛+1+1^𝑚𝑛_𝑘=0ℓ=0𝑚𝑘𝑛ℓ𝐵_𝑚−𝑘𝐸_𝑛−ℓ𝐵_𝑘+ℓ=(−1)^𝑚+𝑛𝑚𝐵_𝑚−1+𝛿_1,𝑚−12𝛿_0,𝑛−𝐸_𝑛+(−1)^𝑚+𝑛𝑛𝐵_𝑚+𝛿_1,𝑚2𝛿_0,𝑛−1−𝐸_𝑛−1+(−1)^𝑚+𝑛𝑛𝐵_𝑚𝐸_𝑛−1+𝑚𝐵_𝑚−1𝐸_𝑛. (2.18) For 𝑛∈ℕ, by (2.18), one gets(−1)^𝑚+1+1^𝑚_𝑘=0^2𝑛_ℓ=0𝑚𝑘ℓ𝐵2𝑛_𝑚−𝑘𝐸_2𝑛−ℓ𝐵_𝑘+ℓ=(−1)^𝑚+12𝑛𝐵_𝑚+𝛿_1,𝑚𝐸_2𝑛−1+(−1)^𝑚2𝑛𝐵_𝑚𝐸_2𝑛−1=(−1)^𝑚+12𝑛𝛿_1,𝑚𝐸_2𝑛−1. (2.19) Therefore, by (2.19), one obtains the following theorem.

Theorem 2.3. For 𝑛∈ℕ, one has (−1)^𝑚+1+1^𝑚_𝑘=0^2𝑛_ℓ=0𝑚𝑘ℓ𝐵2𝑛_𝑚−𝑘𝐸_2𝑛−ℓ𝐵_𝑘+ℓ=(−1)^𝑚+12𝑛𝛿_1,𝑚𝐸_2𝑛−1. (2.20) In particular, for 𝑚∈ℕ, one has ^{2𝑚+1𝑘=02𝑛ℓ=0}𝑘ℓ𝐵2𝑚+12𝑛_{2𝑚+1−𝑘}𝐸_2𝑛−ℓ𝐵_𝑘+ℓ=0. (2.21)

By the same motivation, we consider the 𝑝-adic integral on ℤ_𝑝 for the product of Bernoulli and Bernstein polynomials as follows:𝐼₄=_ℤ𝑝𝐵_𝑚(𝑥)𝐵_𝑘,𝑛(𝑥)𝑑𝜇(𝑥)where𝑚,𝑛,𝑘∈ℕ∪{0}. (2.22) From (1.6) and (1.14), one gets𝐼₄=^𝑚_ℓ=0𝑚ℓ𝐵_{𝑚−ℓℤ𝑝}𝑥^ℓ𝐵_𝑘,𝑛(𝑥)(𝑥)=𝑛𝑘𝑑𝜇_𝑚ℓ=0𝑚ℓ𝐵_{𝑚−ℓℤ𝑝}𝑥^𝑘+ℓ(1−𝑥)^𝑛−𝑘(𝑥)=𝑛𝑘𝑑𝜇_{𝑚ℓ=0𝑛−𝑘𝑗=0}(−1)^𝑗𝑚ℓ𝑗𝐵𝑛−𝑘_𝑚−ℓ𝐵_{𝑘+ℓ+𝑗}. (2.23) On the other hand, 𝐼₄=(−1)^𝑚_ℤ𝑝𝐵_𝑚(1−𝑥)𝐵_{𝑛−𝑘,𝑛}()(𝑥)=()1−𝑥𝑑𝜇−1^𝑚𝑛𝑘^{𝑚𝑘ℓ=0𝑗=0}(−1)^𝑗𝑚ℓ𝑘𝑗𝐵_𝑚−ℓ𝑛−𝑘+𝑗+ℓ+𝐵_{𝑛−𝑘+ℓ+𝑗}+𝛿_{1,𝑛−𝑘+ℓ+𝑗}=(−1)^𝑚𝑛𝑘(𝑛−𝑘)𝐵_𝑚(1)𝛿_0,𝑘+(−1)^𝑚𝑛𝑘𝑚𝐵_𝑚−1(1)𝛿_0,𝑘−(−1)^𝑚𝑛𝑘𝑚𝐵_𝑚(1)𝑘𝛿_0,𝑘−1+(−1)^𝑚𝑛𝑘^{𝑚𝑘ℓ=0𝑗=0}(−1)^𝑗𝑚ℓ𝑘𝑗𝐵_𝑚−ℓ𝐵_{𝑛−𝑘+ℓ+𝑗}+(−1)^𝑚𝑛𝑘𝑚𝐵_𝑚−1−𝑘𝐵_𝑚𝛿_𝑛,𝑘+(−1)^𝑚𝑛𝑘𝐵_𝑚𝛿_𝑛,𝑘+1. (2.24) Equating (2.23) and (2.24), one gets(−1)^𝑚𝑚_ℓ=0^𝑛−𝑘_𝑗=0(−1)^𝑗𝑚ℓ𝑗𝐵𝑛−𝑘_𝑚−ℓ𝐵_{𝑘+ℓ+𝑗}=(𝑛−𝑘)𝐵_𝑚(1)+𝑚𝐵_𝑚−1(1)𝛿_0,𝑘−𝑘𝐵_𝑚(1)𝛿_0,𝑘−1+𝑚𝐵_𝑚−1−𝑘𝐵_𝑚𝛿_𝑛,𝑘+𝐵_𝑚𝛿_𝑛,𝑘+1+^𝑚𝑘_ℓ=0𝑗=0(−1)^𝑗𝑚ℓ𝑘𝑗𝐵_𝑚−ℓ𝐵_{𝑛−𝑘+ℓ+𝑗}. (2.25) By (2.25), we obtain the following theorem.

Theorem 2.4. For 𝑛,𝑚∈ℕ, one has ^2𝑚_ℓ=0^2𝑛_𝑗=0(−1)^𝑗ℓ𝑗𝐵2𝑚2𝑛_2𝑚−ℓ𝐵_ℓ+𝑗=2𝑛𝐵_2𝑚+^2𝑚_ℓ=0ℓ𝐵2𝑚_2𝑚−ℓ𝐵_2𝑛+ℓ. (2.26)

Now, we consider the 𝑝-adic integral on ℤ_𝑝 for the product of Euler and Bernstein polynomials as follows:𝐼₅=_ℤ𝑝𝐸_𝑚(𝑥)𝐵_𝑘,𝑛(𝑥)(𝑥)=𝑑𝜇_𝑚ℓ=0𝑚ℓ𝐸_{𝑚−ℓℤ𝑝}𝑥^ℓ𝐵_𝑘,𝑛(𝑥)(𝑥)=𝑛𝑘𝑑𝜇_{𝑚ℓ=0𝑛−𝑘𝑗=0}(−1)^𝑗𝑚ℓ𝑗𝐸𝑛−𝑘_𝑚−ℓ𝐵_{𝑘+ℓ+𝑗}. (2.27) On the other hand, by (1.13) and (1.14), one gets 𝐼₅=(−1)^𝑚_ℤ𝑝𝐵_{𝑛−𝑘,𝑛}(1−𝑥)𝐸_𝑚()(𝑥)=()1−𝑥𝑑𝜇−1^𝑚𝑛𝑘^{𝑚𝑘ℓ=0𝑗=0}(−1)^𝑗𝑚ℓ𝑘𝑗𝐸_{𝑚−ℓℤ𝑝}(1−𝑥)^{𝑛−𝑘+ℓ+𝑗}(𝑥)=()𝑑𝜇−1^𝑚𝑛𝑘^{𝑚𝑘ℓ=0𝑗=0}(−1)^𝑗𝑚ℓ𝑘𝑗𝑛−𝑘+ℓ+𝑗+𝐵_{𝑛−𝑘+ℓ+j}+𝛿_{1,𝑛−𝑘+ℓ+𝑗}𝐸_𝑚−ℓ=(−1)^𝑚(𝑛𝑘𝑛−𝑘)𝐸_𝑚(1)𝛿_0,𝑘+(−1)^𝑚𝑛𝑘𝑚𝐸_𝑚−1(1)𝛿_0,𝑘−(−1)^𝑚𝑛𝑘𝐸_𝑚(1)𝑘𝛿_0,𝑘−1+(−1)^𝑚𝑛𝑘^{𝑚𝑘ℓ=0𝑗=0}(−1)^𝑗𝑚ℓ𝑘𝑗𝐸_𝑚−ℓ𝐵_{𝑛−𝑘+ℓ+𝑗}+(−1)^𝑚𝑛𝑘𝛿_𝑛,𝑘+1𝐸_𝑚+𝛿_𝑛,𝑘𝑚𝐸_𝑚−1−𝑘𝐸_𝑚. (2.28) Equating (2.27) and (2.28), one gets(−1)^𝑚𝑚_ℓ=0^𝑛−𝑘_𝑗=0(−1)^𝑗𝑚ℓ𝑗𝐸𝑛−𝑘_𝑚−ℓ𝐵_{𝑘+ℓ+𝑗}=(𝑛−𝑘)𝐸_𝑚(1)𝛿_0,𝑘+𝑚𝛿_0,𝑘𝐸_𝑚−1(1)−𝑘𝐸_𝑚(1)𝛿_0,𝑘−1+^𝑚𝑘_ℓ=0𝑗=0(−1)^𝑗𝑚ℓ𝑘𝑗𝐸_𝑚−ℓ𝐵_{𝑛−𝑘+ℓ+𝑗}+𝛿_𝑛,𝑘+1𝐸_𝑚+𝑚𝐸_𝑚−1−𝑘𝐸_𝑚𝛿_𝑛,𝑘. (2.29) Therefore, by (2.11) and (2.29), we obtain the following theorem.

Theorem 2.5. For 𝑛,𝑚∈ℕ, one has ^2𝑚_ℓ=0^2𝑛_𝑗=0(−1)^𝑗ℓ𝑗𝐸2𝑚2𝑛_2𝑚−ℓ𝐵_ℓ+𝑗=−2𝑚𝐸_2𝑚−1+𝐵_2𝑚+2𝑛. (2.30)

Finally, we consider the 𝑝-adic integral on ℤ_𝑝 for the product of Euler, Bernoulli, and Bernstein polynomials as follows:𝐼₆=_ℤ𝑝𝐵_𝑟(𝑥)𝐸_𝑠(𝑥)𝐵_𝑘,𝑛(𝑥)(𝑥)=𝑛𝑘𝑑𝜇_{𝑟𝑠ℓ=0𝑗=0}𝑟ℓ𝑠𝑗𝐵_𝑟−ℓ𝐸_{𝑠−𝑗ℤ𝑝}𝑥^{𝑘+ℓ+𝑗}(1−𝑥)^𝑛−𝑘(𝑥)=𝑛𝑘𝑑𝜇_{𝑟𝑠ℓ=0𝑗=0𝑛−𝑘𝑖=0}(−1)^𝑖𝑟ℓ𝑠𝑗𝑖𝐵𝑛−𝑘_𝑟−ℓ𝐸_𝑠−𝑗𝐵_{𝑘+ℓ+𝑖+𝑗}. (2.31)

On the other hand, by (1.10), (1.13), and (1.14), one gets𝐼₆=(−1)^𝑟+𝑠_ℤ𝑝𝐵_𝑟(1−𝑥)𝐸_𝑠(1−𝑥)𝐵_{𝑛−𝑘,𝑛}()(𝑥)=()1−𝑥𝑑𝜇−1^𝑟+𝑠𝑛𝑘^{𝑟𝑠ℓ=0𝑘𝑗=0𝑖=0}(−1)^𝑖𝑟ℓ𝑠𝑗𝑘𝑖𝐵_𝑟−ℓ𝐸_{𝑠−𝑗ℤ𝑝}(1−𝑥)^{𝑛−𝑘+ℓ+𝑖+𝑗}𝑑𝜇(𝑥). (2.32) Equating (2.31) and (2.32), we easily see that(−1)^{𝑟𝑟+𝑠𝑠}_ℓ=0𝑗=0^𝑛−𝑘_𝑖=0(−1)^𝑖𝑟ℓ𝑠𝑗𝑖𝐵𝑛−𝑘_𝑟−ℓ𝐸_𝑠−𝑗𝐵_{𝑘+ℓ+𝑖+𝑗}=^𝑟𝑠_ℓ=0^𝑘_𝑗=0𝑖=0(−1)^𝑖𝑟ℓ𝑠𝑗𝑘𝑖𝑛−𝑘+ℓ+𝑖+𝑗+𝐵_{𝑛−𝑘+ℓ+𝑖+𝑗}+𝛿_{1,𝑛−𝑘+ℓ+𝑖+𝑗}𝐵_𝑟−ℓ𝐸_𝑠−𝑗=(𝑛−𝑘)𝐵_𝑟(1)𝐸_𝑠(1)𝛿_0,𝑘+𝑟𝐵_𝑟−1(1)𝛿_0,𝑘𝐸_𝑠(1)+𝑠𝐵_𝑟(1)𝐸_𝑠−1(1)𝛿_0,𝑘−𝑘𝐵_𝑟(1)𝐸_𝑠(1)𝛿_0,𝑘−1+^r𝑠_ℓ=0^𝑘_𝑗=0𝑖=0(−1)^𝑖𝑟ℓ𝑠𝑗𝑘𝑖𝐵_𝑟−ℓ𝐸_𝑠−𝑗𝐵_{𝑛−𝑘+ℓ+𝑖+𝑗}+𝛿_𝑛,𝑘+1𝐵_𝑟𝐸_𝑠+𝑟𝐵_𝑟−1𝐸_𝑠+𝑠𝐵_𝑟𝐸_𝑠−1−𝑘𝐵_𝑟𝐸_𝑠𝛿_𝑛,𝑘. (2.33) Therefore, by (1.5) and (2.11), we obtain the following theorem.

Theorem 2.6. For 𝑟,𝑛,𝑠∈ℕ, one has ^2𝑟_ℓ=0^2𝑠_𝑗=0^2𝑛_𝑖=0(−1)^𝑖ℓ𝑗𝑖𝐵2𝑟2𝑠2𝑛_2𝑟−ℓ𝐸_2𝑠−𝑗𝐵_{ℓ+𝑖+𝑗}=−2𝑠𝐵_2𝑟𝐸_2𝑠−1+^𝑟_ℓ=0𝐵2𝑟2𝑙^{2𝑟−2𝑙}𝐵_{2𝑛+2𝑙+2𝑠}−𝑟^𝑠_𝑗=1𝐸2𝑠2𝑗−1^{2𝑠−2𝑗+1}𝐵_{2𝑛+2𝑟+2𝑗−2}. (2.34)

Acknowledgments

The authors express their sincere gratitude to the referees for their valuable suggestions and comments. This paper is supported in part by the Research Grant of Kwangwoon University in 2011.