Some Identities on Bernoulli and Euler Numbers

Abstract

Recently, Kim introduced the fermionic p-adic integral on ℤ_𝑝. By using the equations of the fermionic and bosonic p-adic integral on ℤ_𝑝, we give some interesting identities on Bernoulli and Euler numbers.

1. Introduction/Preliminaries

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 absolute value |⋅|_𝑝 is normally defined by |𝑝|_𝑝=1/𝑝.

Let UD(ℤ_𝑝) be the space of uniformly differentiable functions on ℤ_𝑝 and 𝐶(ℤ_𝑝) the space of continuous function on ℤ_𝑝. For 𝑓∈𝐶(ℤ_𝑝), the fermionic 𝑝-adic integral on ℤ_𝑝 is defined by Kim as follows:𝐼₋₁𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥)=lim^𝑝_𝑁→∞^𝑁−1_𝑥=0𝑓(𝑥)(−1)^𝑥,(see[1]). (1.1) The following fermionic 𝑝-adic integral equation on ℤ_𝑝 is well known (see [1–3]):𝐼₋₁𝑓₁+𝐼₋₁𝑓=2𝑓(0), (1.2) where 𝑓₁(𝑥)=𝑓(𝑥+1).

From (1.1) and (1.2), we can derive the generating function of Euler polynomials as follows:_ℤ𝑝𝑒^{(𝑥+𝑦)𝑡}𝑑𝜇₋₁𝑦=2𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛,𝑛! (1.3) where 𝐸_𝑛(𝑥) is the 𝑛th ordinary Euler polynomial (see [1–4]). In the special case, 𝑥=0, 𝐸_𝑛(0)=𝐸_𝑛 is called the 𝑛th ordinary Euler number.

By (1.3), we get Witt’s formula for the 𝑛th Euler polynomial as follows:_ℤ𝑝𝑥+𝑦^𝑛𝑑𝜇₋₁𝑦=𝐸_𝑛(𝑥),for𝑛∈ℤ₊. (1.4) Thus, by (1.4), we have𝐸_𝑛(𝑥)=(𝐸+𝑥)^𝑛=^𝑛_𝑙=0𝑛𝑙𝑥^𝑛−𝑙𝐸_𝑙, (1.5) with the usual convention about replacing 𝐸^𝑛 by 𝐸_𝑛 (see [5, 6]). From (1.3), we note that(𝐸+1)^𝑛+𝐸_𝑛=2𝛿_0,𝑛, (1.6) where 𝛿_𝑘,𝑛 is the Kronecker symbol (see [3]). By (1.2) and (1.4), we get_ℤ𝑝𝑥+𝑦+1^𝑛𝑑𝜇₋₁𝑦+_ℤ𝑝𝑥+𝑦^𝑛𝑑𝜇₋₁𝑦=2𝑥^𝑛. (1.7) Thus, by (1.4) and (1.7), we have𝐸_𝑛(𝑥+1)+𝐸_𝑛(𝑥)=2𝑥^𝑛,for𝑛∈ℤ₊. (1.8) Equation (1.8) is equivalent to𝑥^𝑛=𝐸_𝑛(𝑥)+12^𝑛−1_𝑙=0𝑛𝑙𝐸_𝑙(𝑥). (1.9) From (1.6), we can derive the following equation:𝐸_𝑛(2)=2−𝐸_𝑛(1)=2+𝐸_𝑛−2𝛿_0,𝑛,for𝑛∈ℤ₊. (1.10)

For 𝑓∈UD(ℤ_𝑝), the bosonic 𝑝-adic integral on ℤ_𝑝 is defined by𝐼₁𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇₁(𝑥)=lim_𝑁→∞1𝑝^{𝑁𝑝𝑁−1}_𝑥=0𝑓(𝑥),(see[4]). (1.11) From (1.11), we can easily derive the following 𝐼₁-integral equation:𝐼₁𝑓₁=𝐼𝑓+𝑓^′(0),(see[4,7,8]), (1.12) where 𝑓₁(𝑥)=𝑓(𝑥+1) and 𝑓′(0)=𝑑𝑓(𝑥)/𝑑𝑥|_𝑥=0.

It is well known that the Bernoulli polynomial can be represented by the bosonic 𝑝-adic integral on ℤ_𝑝 as follows:_ℤ𝑝𝑒^{(𝑥+𝑦)𝑡}𝑑𝜇₁𝑦=𝑡𝑒^𝑡𝑒−1^𝑥𝑡=^∞_𝑛=0𝐵_𝑛(𝑥)𝑡^𝑛,𝑛! (1.13) where 𝐵_𝑛(𝑥) is called the 𝑛th Bernoulli polynomial (see [4, 7–13]). In the special case, 𝑥=0, 𝐵_𝑛(0)=𝐵_𝑛 is called the 𝑛th Bernoulli number. By the definition of Bernoulli numbers and polynomials, we get𝐵_𝑛(𝑥)=_ℤ𝑝𝑥+𝑦^𝑛𝑑𝜇₁𝑦=^𝑛_𝑙=0𝑛𝑙𝑥^𝑛−𝑙𝐵_𝑙. (1.14) Thus, by (1.13) and (1.14), we see that𝐵₀=1,(𝐵+1)^𝑛−𝐵_𝑛=𝛿_1,𝑛, (1.15) with the usual convention about replacing 𝐵^𝑛 by 𝐵_𝑛 (see [1–22]).

By (1.11), we easily get_ℤ𝑝1−𝑥+𝑦^𝑛𝑑𝜇₁𝑦=(−1)^𝑛_ℤ𝑝𝑥+𝑦^𝑛𝑑𝜇₁𝑦. (1.16) From (1.13), (1.14), and (1.16), we have𝐵_𝑛(1−𝑥)=(−1)^𝑛𝐵_𝑛(𝑥)for𝑛∈ℤ₊. (1.17) By (1.15), we get𝐵_𝑛(2)=𝑛+𝐵_𝑛(1)=𝑛+𝐵_𝑛+𝛿_1,𝑛. (1.18) Thus, by (1.17) and (1.18), we have(−1)^𝑛𝐵_𝑛(−1)=𝐵_𝑛(2)=𝑛+𝐵_𝑛+𝛿_1,𝑛,(see[4]). (1.19)

From (1.12) and (1.13), we get_ℤ𝑝𝑥+1+𝑦^𝑛+1𝑑𝜇₁𝑦−_ℤ𝑝𝑥+𝑦^𝑛+1𝑑𝜇₁𝑦=(𝑛+1)𝑥^𝑛. (1.20) Thus, by (1.13) and (1.20), we have𝐵_𝑛+1(𝑥+1)−𝐵_𝑛+1(𝑥)=(𝑛+1)𝑥^𝑛for𝑛∈ℤ₊. (1.21) Equation (1.21) is equivalent to the following equation:𝑥^𝑛=1𝑛+1^𝑛_𝑙=0𝑙𝐵𝑛+1_𝑙(𝑥)for𝑛∈ℤ₊. (1.22)

In this paper we derive some interesting and new identities for the Bernoulli and Euler numbers from the 𝑝-adic integral equations on ℤ_𝑝.

2. Some Identities on Bernoulli and Euler Numbers

From (1.1), we note that_ℤ𝑝1−𝑥+𝑦^𝑛𝑑𝜇₋₁𝑦=(−1)^𝑛_ℤ𝑝𝑥+𝑦^𝑛𝑑𝜇₋₁𝑦. (2.1) By (1.14) and (2.1), we get𝐸_𝑛(1−𝑥)=(−1)^𝑛𝐸_𝑛(𝑥),where𝑛∈ℤ₊. (2.2) In the special case, 𝑥=−1, we have𝐸_𝑛(2)=(−1)^𝑛𝐸_𝑛(−1)=2+𝐸_𝑛−2𝛿_0,𝑛. (2.3)

Let us consider the following fermionic 𝑝-adic integral on ℤ_𝑝 as follows:_ℤ𝑝𝑥^𝑛𝑑𝜇₋₁(𝑥)=1𝑛+1^𝑛_𝑙=0𝑙𝑛+1_ℤ𝑝𝐵_𝑙(𝑥)𝑑𝜇₋₁(𝑥)=1𝑛+1^𝑛_𝑙=0𝑙𝑛+1^𝑙_𝑘=0𝑙𝑘𝐵_{𝑙−𝑘ℤ𝑝}𝑥^𝑘𝑑𝜇₋₁(𝑥)=1𝑛+1^𝑛_𝑙=0𝑙𝑛+1^𝑙_𝑘=0𝑙𝑘𝐵_𝑙−𝑘𝐸_𝑘. (2.4) Therefore, by (1.4) and (2.4), we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℤ₊, one has 𝐸_𝑛=1𝑛+1^𝑛_𝑙=0𝑙𝑛+1^𝑙_𝑘=0𝑙𝑘𝐵_𝑙−𝑘𝐸_𝑘. (2.5)

It is known that 𝐵_𝑛(𝑥)=(−1)^𝑛𝐵_𝑛(1−𝑥). If we take the fermionic 𝑝-adic integral on both sides of (1.22), then we have_ℤ𝑝𝑥^𝑛𝑑𝜇₋₁(𝑥)=1𝑛+1^𝑛_𝑙=0𝑙𝑛+1_ℤ𝑝𝐵_𝑙(𝑥)𝑑𝜇₋₁(𝑥)=1𝑛+1^𝑛_𝑙=0𝑙()𝑛+1−1^𝑙_ℤ𝑝𝐵_𝑙(1−𝑥)𝑑𝜇₋₁(𝑥)=1𝑛+1^𝑛_𝑙=0𝑙()𝑛+1−1^𝑙𝑙_𝑘=0𝑙𝑘𝐵_{𝑙−𝑘ℤ𝑝}(1−𝑥)^𝑘𝑑𝜇₋₁(𝑥)=1𝑛+1^𝑛_𝑙=0𝑙()𝑛+1−1^𝑙𝑙_𝑘=0𝑙𝑘𝐵_𝑙−𝑘(−1)^𝑘𝐸_𝑘(−1). (2.6) From (2.2) and (2.6), we note that_ℤ𝑝𝑥^𝑛𝑑𝜇₋₁(𝑥)=1𝑛+1^𝑛_𝑙=0𝑙()𝑛+1−1^𝑙𝑙_𝑘=0𝑙𝑘𝐵_𝑙−𝑘𝐸_𝑘(2)=1𝑛+1^𝑛_𝑙=0𝑙()𝑛+1−1^𝑙𝑙_𝑘=0𝑙𝑘𝐵_𝑙−𝑘2+𝐸_𝑘−2𝛿_0,𝑘=1𝑛+1^𝑛_𝑙=0𝑙()𝑛+1−1^𝑙2𝐵_𝑙(1)+^𝑙_𝑘=0𝑙𝑘𝐵_𝑙−𝑘𝐸_𝑘−2𝐵_𝑙=1𝑛+1^𝑛_𝑙=0𝑙()𝑛+1−1^𝑙𝑙_𝑘=0𝑙𝑘𝐵_𝑙−𝑘𝐸_𝑘+2𝛿_1,𝑙. (2.7) Therefore, by (1.4) and (2.7), we obtain the following theorem.

Theorem 2.2. For 𝑛∈ℤ₊, one has 𝐸_𝑛=1𝑛+1^𝑛_𝑙=0𝑙()𝑛+1−1^𝑙𝑙_𝑘=0𝑙𝑘𝐵_𝑙−𝑘𝐸_𝑘+2𝛿_1,𝑙. (2.8)

Corollary 2.3. For 𝑛∈ℕ, one has 2+𝐸_𝑛=1𝑛+1^𝑛_𝑙=0𝑙()𝑛+1−1^𝑙𝑙_𝑘=0𝑙𝑘𝐵_𝑙−𝑘𝐸_𝑘. (2.9)

Let us take the bosonic 𝑝-adic integral on both sides of (1.9) as follows:_ℤ𝑝𝑥^𝑛𝑑𝜇₁(𝑥)=_ℤ𝑝𝐸_𝑛(𝑥)+12^𝑛−1_𝑙=0𝑛𝑙𝐸_𝑙(𝑥)𝑑𝜇₁(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐸_{𝑛−𝑙ℤ𝑝}𝑥^𝑙𝑑𝜇₁(𝑥)+12^𝑛−1_𝑙=0𝑛𝑙^𝑙𝑘=0𝑙𝑘𝐸_{𝑙−𝑘ℤ𝑝}𝑥^𝑘𝑑𝜇₁(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐸_𝑛−𝑙𝐵_𝑙+12^𝑛−1_𝑙=0𝑛𝑙^𝑙𝑘=0𝑙𝑘𝐸_𝑙−𝑘𝐵_𝑘. (2.10) Thus, by (1.14) and (2.10), we obtain the following theorem.

Theorem 2.4. For 𝑛∈ℤ₊, one has 𝐵_𝑛=^𝑛_𝑙=0𝑛𝑙𝐸_𝑛−𝑙𝐵_𝑙+12^𝑛−1_𝑙=0𝑛𝑙^𝑙𝑘=0𝑙𝑘𝐸_𝑙−𝑘𝐵_𝑘. (2.11)

On the other hand, by (2.2) and (2.10), we get_ℤ𝑝𝑥^𝑛𝑑𝜇₁(𝑥)=(−1)^𝑛_ℤ𝑝𝐸_𝑛(1−𝑥)𝑑𝜇₁(𝑥)+12^𝑛−1_𝑙=0𝑛𝑙(−1)^𝑙_ℤ𝑝𝐸_𝑙(1−𝑥)𝑑𝜇₁(𝑥)=(−1)^𝑛𝑛_𝑙=0𝑛𝑙𝐸_{𝑛−𝑙ℤ𝑝}(1−𝑥)^𝑙𝑑𝜇₁(𝑥)+12^𝑛−1_𝑙=0𝑛𝑙(−1)^𝑙𝑙_𝑘=0𝑙𝑘𝐸_{𝑙−𝑘ℤ𝑝}(1−𝑥)^𝑘𝑑𝜇₁(𝑥)=(−1)^𝑛𝑛_𝑙=0𝑛𝑙𝐸_𝑛−𝑙(−1)^𝑙𝐵_𝑙()+1−12^𝑛−1_𝑙=0𝑛𝑙(−1)^𝑙𝑙_𝑘=0𝑙𝑘𝐸_𝑙−𝑘(−1)^𝑘𝐵_𝑘()=()−1−1^𝑛𝑛_𝑙=0𝑛𝑙𝐸_𝑛−𝑙𝐵_𝑙(2)+12^𝑛−1_𝑙=0𝑛𝑙(−1)^𝑙𝑙_𝑘=0𝑙𝑘𝐸_𝑙−𝑘𝐵_𝑘(2)=(−1)^𝑛𝑛_𝑙=0𝑛𝑙𝐸_𝑛−𝑙𝑙+𝐵_𝑙+𝛿_1,𝑙+12^𝑛−1_𝑙=0𝑛𝑙(−1)^𝑙𝑙_𝑘=0𝑙𝑘𝐸_𝑙−𝑘𝑘+𝐵_𝑘+𝛿_1,𝑘=(−1)^𝑛𝑛𝐸_𝑛−1(1)+(−1)^𝑛𝑛_𝑙=0𝑛𝑙𝐸_𝑛−𝑙𝐵_𝑙+(−1)^𝑛𝑛𝐸_𝑛−1+12^𝑛−1_𝑙=0𝑛𝑙(−1)^𝑙𝑙𝐸_𝑙−1(1)+12^𝑛−1_𝑙=0𝑛𝑙(−1)^𝑙𝑙_𝑘=0𝑙𝑘𝐸_𝑙−𝑘𝐵_𝑘+12^𝑛−1_𝑙=1𝑛𝑙(−1)^𝑙𝑙𝐸_𝑙−1=(−1)^𝑛𝑛2+𝐸_𝑛−1−2𝛿_0,𝑛−1+(−1)^𝑛𝑛_𝑙=0𝑛𝑙𝐸_𝑛−𝑙𝐵_𝑙+(−1)^𝑛𝑛𝐸_𝑛−1+12^𝑛−1_𝑙=0𝑛𝑙(−1)^𝑙𝑙2+𝐸_𝑙−1−𝛿_0,𝑙−1+12^𝑛−1_𝑙=0𝑛𝑙(−1)^𝑙𝑙_𝑘=0𝑙𝑘𝐸_𝑙−𝑘𝐵_𝑘+12^𝑛−1_𝑙=1𝑛𝑙(−1)^𝑙𝑙𝐸_𝑙−1, (2.12) where 𝑛∈ℕ with 𝑛≥2. Therefore, by (2.12), we obtain the following theorem.

Theorem 2.5. For 𝑛∈ℕ with 𝑛≥2, one has 𝐵_2𝑛−1=−2𝑛−12−(2𝑛−1)𝐸_2𝑛−2(−1)−^2𝑛−1_𝑙=0𝑙𝐸2𝑛−1_{2𝑛−1−𝑙}𝐵_𝑙+12^2𝑛−2_𝑙=0𝑙()2𝑛−1−1^𝑙𝑙_𝑘=0𝑙𝑘𝐸_𝑙−𝑘𝐵_𝑘. (2.13)

By (1.9) and (1.22), we get_ℤ𝑝𝑥^𝑚𝑦^𝑛𝑑𝜇₋₁(𝑥)𝑑𝜇₁𝑦=_ℤ𝑝1𝑚+1^𝑚_𝑘=0𝑘𝐵𝑚+1_𝑘(𝑥)𝐸_𝑛𝑦+12^𝑛−1_𝑙=0𝑛𝑙𝐸_𝑙𝑦𝑑𝜇₋₁(𝑥)𝑑𝜇₁𝑦=1𝑚+1^𝑚_𝑘=0𝑘𝑚+1_ℤ𝑝𝐵_𝑘(𝑥)𝐸_𝑛𝑦𝑑𝜇₋₁(𝑥)𝑑𝜇₁𝑦+12(𝑚+1)^𝑚_𝑘=0^𝑛−1_𝑙=0𝑘𝑛𝑙𝑚+1_ℤ𝑝𝐵_𝑘(𝑥)𝐸_𝑙𝑦𝑑𝜇₋₁(𝑥)𝑑𝜇₁𝑦=1𝑚+1^𝑚𝑘_𝑘=0^𝑛_𝑙=0𝑝=0𝑘𝑘𝑙𝑛𝑝𝐵𝑚+1_𝑘−𝑙𝐸_𝑛−𝑝𝐵_𝑝𝐸_𝑙+12(𝑚+1)^𝑚_𝑘=0^𝑛−1𝑘_𝑙=0^𝑙_𝑠=0𝑝=0𝑘𝑛𝑙𝑘𝑠𝑙𝑝𝐵𝑚+1_𝑘−𝑠𝐸_𝑙−𝑝𝐸_𝑠𝐵_𝑝. (2.14) Therefore, by (1.4), (1.14), and (2.14), we obtain the following theorem.

Theorem 2.6. For 𝑚∈ℤ₊ and 𝑛∈ℕ, one has 𝐸_𝑚𝐵_𝑛=1𝑚+1^𝑚𝑘_𝑘=0^𝑛_𝑙=0𝑝=0𝑘𝑘𝑙𝑛𝑝𝐵𝑚+1_𝑘−𝑙𝐸_𝑛−𝑝𝐵_𝑝𝐸_𝑙+12(𝑚+1)^𝑚_𝑘=0^𝑛−1𝑘_𝑙=0^𝑙_𝑠=0𝑝=0𝑘𝑛𝑙𝑘𝑠𝑙𝑝𝐵𝑚+1_𝑘−𝑠𝐸_𝑙−𝑝𝐸_𝑠𝐵_𝑝. (2.15)

It is easy to show that_ℤ𝑝𝑥^𝑚+𝑛𝑑𝜇₋₁(𝑥)=_ℤ𝑝1𝑚+1^𝑚_𝑘=0𝑘𝐵𝑚+1_𝑘(𝑥)𝐸_𝑛(𝑥)+12^𝑛−1_𝑙=0𝑛𝑙𝐸_𝑙(𝑥)𝑑𝜇₋₁(𝑥)=1𝑚+1^𝑚𝑘_𝑘=0^𝑛_𝑖=0𝑗=0𝑘𝑘𝑖𝑛𝑗𝐵𝑚+1_𝑘−𝑖𝐸_{𝑛−𝑗ℤ𝑝}𝑥^𝑖+𝑗𝑑𝜇₋₁(𝑥)+12(𝑚+1)^𝑚_𝑘=0^𝑛−1𝑘_𝑙=0^𝑙_𝑖=0𝑗=0𝑘𝑛𝑙𝑘𝑖𝑙𝑗𝐵𝑚+1_𝑘−𝑖𝐸_{𝑙−𝑗ℤ𝑝}𝑥^𝑖+𝑗𝑑𝜇₋₁(𝑥)=1𝑚+1^𝑚𝑘_𝑘=0^𝑛_𝑖=0𝑗=0𝑘𝑘𝑖𝑛𝑗𝐵𝑚+1_𝑘−𝑖𝐸_𝑛−𝑗𝐸_𝑖+𝑗+12(𝑚+1)^𝑚_𝑘=0^𝑛−1𝑘_𝑙=0^𝑙_𝑖=0𝑗=0𝑘𝑛𝑙𝑘𝑖𝑙𝑗𝐵𝑚+1_𝑘−𝑖𝐸_𝑙−𝑗𝐸_𝑖+𝑗. (2.16) Therefore, by (2.16), we obtain the following corollay.

Corollary 2.7. For 𝑚∈ℤ₊ and 𝑛∈ℕ, one has 𝐸_𝑚+𝑛=1𝑚+1^𝑚𝑘_𝑘=0^𝑛_𝑖=0𝑗=0𝑘𝑘𝑖𝑛𝑗𝐵𝑚+1_𝑘−𝑖𝐸_𝑛−𝑗𝐸_𝑖+𝑗+12(𝑚+1)^𝑚_𝑘=0^𝑛−1𝑘_𝑙=0^𝑙_𝑖=0𝑗=0𝑘𝑛𝑙𝑘𝑖𝑙𝑗𝐵𝑚+1_𝑘−𝑖𝐸_𝑙−𝑗𝐸_𝑖+𝑗. (2.17)

For 𝑓∈𝐶(ℤ_𝑝),  𝑝-adic analogue of Bernstein operator of order 𝑛 for 𝑓 is given by𝔹_𝑛𝑓∣𝑥=^𝑛_𝑘=0𝑓k𝑛𝑛𝑘𝑥^𝑘(1−𝑥)^𝑛−𝑘=^𝑛_𝑘=0𝑓𝑘𝑛𝐵_𝑘,𝑛(𝑥), (2.18) where 𝐵_𝑘,𝑛(𝑥)=(^𝑛_𝑘)𝑥^𝑘(1−𝑥)^𝑛−𝑘 for 𝑛,𝑘∈ℤ₊ is called the Bernstein polynomial of degree 𝑛 (see [8]). From the definition of 𝐵_𝑘,𝑛(𝑥), we note that 𝐵_{𝑛−𝑘,𝑛}(1−𝑥)=𝐵_𝑘,𝑛(𝑥).

Let us take the fermionic 𝑝-adic integral on ℤ_𝑝 for the product of 𝑥^𝑚 and 𝐵_𝑘,𝑛(𝑥) as follows:_ℤ𝑝𝑥^𝑚𝐵_𝑘,𝑛(𝑥)𝑑𝜇₋₁(𝑥)=1𝑚+1^𝑚_𝑙=0𝑙𝑚+1_ℤ𝑝𝐵_𝑙(𝑥)𝐵_𝑘,𝑛(𝑥)𝑑𝜇₋₁(𝑥)=(^𝑛_𝑘)𝑚+1^𝑚𝑙_𝑙=0𝑗=0𝑙𝑙𝑗𝐵𝑚+1_{𝑙−𝑗ℤ𝑝}𝑥^𝑗+𝑘(1−𝑥)^𝑛−𝑘𝑑𝜇₋₁(𝑥)=(^𝑛_𝑘)𝑚+1^𝑚𝑙_𝑙=0𝑗=0^𝑛−𝑘_𝑖=0(−1)^𝑖𝐵_𝑙−𝑗𝑙𝑙𝑗𝑖𝑚+1𝑛−𝑘_ℤ𝑝𝑥^{𝑖+𝑗+𝑘}𝑑𝜇₋₁(𝑥)=(^𝑛_𝑘)𝑚+1^𝑚𝑙_𝑙=0𝑗=0^𝑛−𝑘_𝑖=0(−1)^𝑖𝑙𝑙𝑗𝑖𝐵𝑚+1𝑛−𝑘_𝑙−𝑗𝐸_{𝑖+𝑗+𝑘}. (2.19) From (2.18), we note that_ℤ𝑝𝑥^𝑚𝐵_𝑘,𝑛(𝑥)𝑑𝜇₋₁(𝑥)=𝑛𝑘^ℤ𝑝𝑥^𝑚+𝑘(1−𝑥)^𝑛−𝑘𝑑𝜇₋₁(𝑥)=𝑛𝑘^{𝑛−𝑘𝑗=0}𝑗()𝑛−𝑘−1^𝑗_ℤ𝑝𝑥^{𝑚+𝑘+𝑗}𝑑𝜇₋₁(𝑥)=𝑛𝑘^{𝑛−𝑘𝑗=0}𝑗()𝑛−𝑘−1^𝑗𝐸_{𝑚+𝑘+𝑗}. (2.20) Therefore, by (2.19) and (2.20), we obtain the following theorem.

Theorem 2.8. For 𝑚,𝑛,𝑘∈ℤ₊, one has ^{𝑛−𝑘𝑗=0}𝑗()𝑛−𝑘−1^𝑗𝐸_{𝑚+𝑘+𝑗}=1𝑚+1^𝑚𝑙_𝑙=0𝑗=0^𝑛−𝑘_𝑖=0(−1)^𝑖𝑙𝑙𝑗𝑖𝐵𝑚+1𝑛−𝑘_𝑙−𝑗𝐸_{𝑖+𝑗+𝑘}. (2.21) In particular, (𝑚+1)𝐸_𝑚+𝑛=^𝑚𝑙_𝑙=0𝑗=0𝑙𝑙𝑗𝐵𝑚+1_𝑙−𝑗𝐸_𝑗+𝑛. (2.22)

By (1.17) and the symmetric property of 𝐵_𝑘,𝑛(𝑥), we get_ℤ𝑝𝑥^𝑚𝐵_𝑘,𝑛(𝑥)𝑑𝜇₋₁(𝑥)=_ℤ𝑝𝑥^𝑚𝐵_{𝑛−𝑘,𝑛}(1−𝑥)𝑑𝜇₋₁(𝑥)=1𝑚+1^𝑚_𝑙=0(−1)^𝑙𝑙𝑚+1_ℤ𝑝𝐵_𝑙(1−𝑥)𝐵_{𝑛−𝑘,𝑛}(1−𝑥)𝑑𝜇₋₁(𝑥)=(^𝑛_𝑘)𝑚+1^𝑚𝑙_𝑙=0^𝑘_𝑗=0𝑖=0(−1)^𝑖+𝑙𝑙𝑙𝑗𝑘𝑖𝐵𝑚+1_{𝑙−𝑗ℤ𝑝}(1−𝑥)^{𝑖+𝑗+𝑛−𝑘}𝑑𝜇₋₁(𝑥). (2.23) From (1.4) and (2.2), we note that_ℤ𝑝(1−𝑥)^𝑛𝑑𝜇₋₁(𝑥)=(−1)^𝑛𝐸_𝑛(−1)=𝐸_𝑛(2)=2+𝐸_𝑛−2𝛿_0,𝑛. (2.24) By (2.23) and (2.24), we see that_ℤ𝑝𝑥^𝑚𝐵_𝑘,𝑛(𝑥)𝑑𝜇₋₁(𝑥)=(^𝑛_𝑘)𝑚+1^𝑚𝑙_𝑙=0^𝑘_𝑗=0𝑖=0(−1)^𝑖+𝑙𝑙𝑙𝑗𝑘𝑖𝐵𝑚+1_𝑙−𝑗2+𝐸_{𝑖+𝑗+𝑛−𝑘}−2𝛿_{0,𝑖+𝑗+𝑛−𝑘}. (2.25) From (2.20) and (2.25), we have^{𝑛−𝑘𝑗=0}𝑗()𝑛−𝑘−1^𝑗𝐸_{𝑚+𝑘+𝑗}=2𝛿_0,𝑘𝑚+1^𝑚𝑙_𝑙=0𝑗=0(−1)^𝑙𝑙𝑙𝑗𝐵𝑚+1_𝑙−𝑗−2𝑚+1^𝑚_𝑙=0(−1)^𝑙𝑙𝐵𝑚+1_𝑙𝛿_𝑘,𝑛+1𝑚+1^𝑚𝑙_𝑙=0^𝑘_𝑗=0𝑖=0(−1)^𝑖+𝑙𝑙𝑙𝑗𝑘𝑖𝐵𝑚+1_𝑙−𝑗𝐸_{𝑖+𝑗+𝑛−𝑘}=2𝛿_0,𝑘𝑚+1^𝑚𝑙_𝑙=0𝑗=0(−1)^𝑙𝑙𝑙𝑗𝐵𝑚+1_𝑙−𝑗−2𝐵𝑚+1^𝑚+1(2)+(−1)^𝑚𝐵_𝑚+1𝛿_𝑘,𝑛+1𝑚+1^𝑚𝑙_𝑙=0^𝑘_𝑗=0𝑖=0(−1)^𝑖+𝑙𝑙𝑙𝑗𝑘𝑖𝐵𝑚+1_𝑙−𝑗𝐸_{𝑖+𝑗+𝑛−𝑘}. (2.26) Therefore, by (1.19) and (2.26), we obtain the following theorem.

Theorem 2.9. For 𝑚,𝑛,𝑘∈ℕ with 𝑛≥𝑘, one has ^{𝑛−𝑘𝑗=0}𝑗()𝑛−𝑘−1^𝑗𝐸_{𝑚+𝑘+𝑗}=1𝑚+1^𝑚𝑙_𝑙=0^𝑘_𝑗=0𝑖=0(−1)^𝑖+𝑙𝑙𝑙𝑗𝑘𝑖𝐵𝑚+1_𝑙−𝑗𝐸_{𝑖+𝑗+𝑛−𝑘}−2𝐵𝑚+1^𝑚+1+𝑚+1+(−1)^𝑚𝐵_𝑚+1. (2.27) In particular, (2𝑚+2)𝐸_2𝑚+𝑛+1+2=^2𝑚+1𝑙_𝑙=0^𝑛_𝑗=0𝑖=0(−1)^𝑖+𝑙𝑙𝑙𝑗𝑛𝑖𝐵2𝑚+2_𝑙−𝑗𝐸_𝑖+𝑗. (2.28)

Acknowledgment

The first author was supported by National Research Foundation of Korea Grant funded by the Korean Government 2011-0002486.