Derivation of Identities Involving Bernoulli and Euler Numbers

Abstract

We derive some new and interesting identities involving Bernoulli and Euler numbers by using some polynomial identities and p-adic integrals on ℤ_𝑝.

1. Introduction and Preliminaries

Let 𝑝 be a fixed odd prime. Throughout this paper, ℤ_𝑝,ℚ_𝑝,ℂ_𝑝 will, respectively, denote the ring of 𝑝-adic integers, the field of 𝑝-adic rational numbers, and the completion of algebraic closure of ℚ_𝑝. The 𝑝-adic absolute value ||_𝑝 on ℂ_𝑝 is normalized so that |𝑝|_𝑝=1/𝑝. Let ℤ_>0 be the set of natural numbers and ℤ_≥0=ℤ_>0∪{0}.

As is well known, the Bernoulli polynomials 𝐵_𝑛(𝑥) are defined by the generating function as follows: 𝐹(𝑡,𝑥)=𝑡𝑒^𝑡−1𝑒^𝑥𝑡=𝑒^{𝐵(𝑥)𝑡}=^∞_𝑛=0𝐵_𝑛(𝑥)𝑡^𝑛𝑛!,(1.1) with the usual convention of replacing 𝐵(𝑥)^𝑛 by 𝐵_𝑛(𝑥).

In the special case, 𝑥=0, 𝐵_𝑛(0)=𝐵_𝑛 is referred to as the 𝑛th Bernoulli number. That is, the generating function of Bernoulli numbers is given by 𝐹(𝑡)=𝐹(𝑡,0)=𝑡𝑒^𝑡−1=^∞_𝑛=0𝐵_𝑛𝑡^𝑛𝑛!=𝑒^𝐵𝑡,(1.2) with the usual convention of replacing 𝐵^𝑛 by 𝐵_𝑛, (cf. [1–23]).

From (1.2), we see that the recurrence formula for the Bernoulli numbers is (𝐵+1)^𝑛−𝐵_𝑛=𝛿_1,𝑛,for𝑛∈ℤ_≥0,(1.3) where 𝛿_𝑘,𝑛 is the Kronecker symbol.

By (1.1) and (1.2), we easily get the following: 𝐵_𝑛(𝑥)=(𝐵+𝑥)^𝑛=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐵_𝑙𝑥^𝑛−𝑙=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐵_𝑛−𝑙𝑥^𝑙,for𝑛∈ℤ_≥0.(1.4) Let 𝑈𝐷(ℤ_𝑝) be the space of uniformly differentiable ℂ_𝑝-valued functions on ℤ_𝑝. For 𝑓∈𝑈𝐷(ℤ_𝑝), the bosonic 𝑝-adic integral on ℤ_𝑝 is defined by 𝐼(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇(𝑥)=lim_𝑁→∞1𝑝^{𝑁𝑝𝑁−1}_𝑥=0𝑓(𝑥),(1.5)(cf. [12]). Then it is easy to see that 𝐼𝑓₁=𝐼(𝑓)+𝑓^(0),(1.6) where 𝑓₁(𝑥)=𝑓(𝑥+1) and 𝑓′(0)=𝑑𝑓(𝑥)/𝑑𝑥|_𝑥=0.

By (1.6), we have the following: _ℤ𝑝𝑒^{(𝑥+𝑦)𝑡}𝑑𝜇(𝑦)=𝑡𝑒^𝑡−1𝑒^𝑥𝑡=^∞_𝑛=0𝐵_𝑛(𝑥)𝑡^𝑛𝑛!,(1.7)(cf. [12–14]). From (1.7), we can derive the Witt's formula for the 𝑛th Bernoulli polynomial as follows: _ℤ𝑝(𝑥+𝑦)^𝑛𝑑𝜇(𝑦)=𝐵_𝑛(𝑥),for𝑛∈ℤ_≥0.(1.8)

By (1.1), we have the following: 𝐵_𝑛(1−𝑥)=(−1)^𝑛𝐵_𝑛(𝑥),for𝑛∈ℤ_≥0.(1.9) Thus, from (1.3), (1.4), and (1.9), we have the following: 𝐵_𝑛(1)=𝐵_𝑛+𝛿_1,𝑛=(−1)^𝑛𝐵_𝑛,for𝑛∈ℤ_≥0.(1.10) By (1.4), we have the following: 𝐵_𝑛(𝑥+𝑦)=^𝑛_𝑘=0⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝐵_𝑘(𝑥)𝑦^𝑛−𝑘,for𝑛∈ℤ_≥0.(1.11) Especially, for 𝑥=1 and 𝑦=1, 𝐵_𝑛(2)=^𝑛_𝑘=0⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝐵_𝑘(1)=^𝑛_𝑘=0⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝐵_𝑘+𝛿_1,𝑘,for𝑛∈ℤ_≥0.(1.12) Therefore, from (1.9), (1.10), and (1.12), we can derive the following relation. For 𝑛∈ℤ_≥0, (−1)^𝑛𝐵_𝑛(−1)=𝐵_𝑛(2)=𝑛+𝐵_𝑛(1)=𝑛+𝐵_𝑛+𝛿_1,𝑛=𝑛+(−1)^𝑛𝐵_𝑛.(1.13) Let 𝑓(𝑦)=(𝑥+𝑦)^𝑛+1. By (1.6), we have the following: _ℤ𝑝(𝑥+𝑦+1)^𝑛+1𝑑𝜇(𝑦)−_ℤ𝑝(𝑥+𝑦)^𝑛+1𝑑𝜇(𝑦)=(𝑛+1)𝑥^𝑛,for𝑛∈ℤ_≥0.(1.14) By (1.8) and (1.14), we have the following: 𝐵_𝑛+1(𝑥+1)−𝐵_𝑛+1(𝑥)=(𝑛+1)𝑥^𝑛,for𝑛∈ℤ_≥0.(1.15) Thus, by (1.11) and (1.15), we have the following identity. 𝑥^𝑛=1𝑛+1^𝑛_𝑙=0⎛⎜⎜⎝𝑛+𝑙1⎞⎟⎟⎠𝐵_𝑙(𝑥),for𝑛∈ℤ_≥0.(1.16)

As is well known, the Euler polynomials 𝐸_𝑛(𝑥) are defined by the generating function as follows: 𝐺(𝑡,𝑥)=2𝑒^𝑡+1𝑒^𝑥𝑡=𝑒^{𝐸(𝑥)𝑡}=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛𝑛!,(1.17) with the usual convention of replacing 𝐸(𝑥)^𝑛 by 𝐸_𝑛(𝑥).

In the special case, 𝑥=0, 𝐸_𝑛(0)=𝐸_𝑛 is referred to as the 𝑛th Euler number. That is, the generating function of Euler numbers is given by 𝐺(𝑡)=𝐺(𝑡,0)=2𝑒^𝑡+1=^∞_𝑛=0𝐸_𝑛𝑡^𝑛𝑛!=𝑒^𝐸𝑡,(1.18) with the usual convention of replacing 𝐸^𝑛 by 𝐸_𝑛, (cf. [1–23]).

From (1.18), we see that the recurrence formula for the Euler numbers is (𝐸+1)^𝑛+𝐸_𝑛=2𝛿_0,𝑛,for𝑛∈ℤ_≥0.(1.19) By (1.17) and (1.18), we easily get the following: 𝐸_𝑛(𝑥)=(𝐸+𝑥)^𝑛=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐸_𝑙𝑥^𝑛−𝑙=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐸_𝑛−𝑙𝑥^𝑙,for𝑛∈ℤ_≥0.(1.20) Let 𝐶(ℤ_𝑝) be the space of continuous ℂ_𝑝-valued functions on ℤ_𝑝. For 𝑓∈𝐶(ℤ_𝑝), the fermionic 𝑝-adic integral on ℤ_𝑝 is defined by Kim as follows: 𝐼₋₁(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥)=lim_{𝑝𝑁→∞}^𝑁−1_𝑥=0𝑓(𝑥)(−1)^𝑥,(1.21)(cf. [9]). Then it is easy to see that 𝐼₋₁𝑓₁+𝐼₋₁(𝑓)=2𝑓(0),(1.22) where 𝑓₁(𝑥)=𝑓(𝑥+1).

By (1.22), we have the following: _ℤ𝑝𝑒^{(𝑥+𝑦)𝑡}𝑑𝜇₋₁(𝑦)=2𝑒^𝑡+1𝑒^𝑥𝑡=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛𝑛!.(1.23) From (1.23), we can derive the Witt's formula for the 𝑛-th Euler polynomial as follows: _ℤ𝑝(𝑥+𝑦)^𝑛𝑑𝜇₋₁(𝑦)=𝐸_𝑛(𝑥),for𝑛∈ℤ_≥0.(1.24) By (1.17), we have the following: 𝐸_𝑛(1−𝑥)=(−1)^𝑛𝐸_𝑛(𝑥),for𝑛∈ℤ_≥0.(1.25) Thus, from (1.19), (1.20), and (1.25), we have the following: 𝐸_𝑛(1)=−𝐸_𝑛+2𝛿_0,𝑛=(−1)^𝑛𝐸_𝑛,for𝑛∈ℤ_≥0.(1.26) By (1.20), we have the following: 𝐸_𝑛(𝑥+𝑦)=^𝑛_𝑘=0⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝐸_𝑘(𝑥)𝑦^𝑛−𝑘,for𝑛∈ℤ_≥0.(1.27) Especially, for 𝑥=1 and 𝑦=1, 𝐸_𝑛(2)=^𝑛_𝑘=0⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝐸_𝑘(1)=^𝑛_𝑘=0⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠−𝐸_𝑛+2𝛿_0,𝑘,for𝑛∈ℤ_≥0.(1.28) Therefore, from (1.25), (1.26), and (1.28), we can derive the following relations. For 𝑛∈ℤ_≥0, (−1)^𝑛𝐸_𝑛(−1)=𝐸_𝑛(2)=2−𝐸_𝑛(1)=2+𝐸_𝑛−2𝛿_0,𝑛=2−(−1)^𝑛𝐸_𝑛.(1.29) Let 𝑓(𝑦)=(𝑥+𝑦)^𝑛. By (1.22), we have the following: _ℤ𝑝(𝑥+𝑦+1)^𝑛𝑑𝜇₋₁(𝑦)+_ℤ𝑝(𝑥+𝑦)^𝑛𝑑𝜇₋₁(𝑦)=2𝑥^𝑛,for𝑛∈ℤ_≥0.(1.30) By (1.24) and (1.30), we have the following: 𝐸_𝑛(𝑥+1)+𝐸_𝑛(𝑥)=2𝑥^𝑛,for𝑛∈ℤ_≥0.(1.31) Thus, by (1.27) and (1.31), we get the following identity. 𝑥^𝑛=12^𝑛−1_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐸_𝑙(𝑥)+𝐸_𝑛(𝑥),for𝑛∈ℤ_≥0.(1.32)

The Bernstein polynomials are defined by 𝐵_𝑘,𝑛(𝑥)=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝑥^𝑘(1−𝑥)^𝑛−𝑘,for𝑘,𝑛∈ℤ_≥0,(1.33) with 0≤𝑘≤𝑛 (cf. [14]).

By the definition of 𝐵_𝑘,𝑛(𝑥), we note that 𝐵_𝑘,𝑛(𝑥)=𝐵_{𝑛−𝑘,𝑛}(1−𝑥).(1.34)

In this paper, we derive some new and interesting identities involving Bernoulli and Euler numbers from well-known polynomial identities. Here, we note that our results are “complementary” to those in [6], in the sense that we take a fermionic 𝑝-adic integral where a bosonic 𝑝-adic integral is taken and vice versa, and we use the identity involving Euler polynomials in (1.32) where that involving Bernoulli polynomials in (1.16) is used and vice versa. Finally, we report that there have been a lot of research activities on this direction of research, namely, on derivation of identities involving Bernoulli and Euler numbers and polynomials by exploiting bosonic and fermionic 𝑝-adic integrals (cf. [6–8]).

2. Identities Involving Bernoulli Numbers

Taking the bosonic 𝑝-adic integral on both sides of (1.16), we have the following: _ℤ𝑝𝑥^𝑚𝑑𝜇(𝑥)=_ℤ𝑝1𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠𝐵_𝑘=1(𝑥)𝑑𝜇(𝑥)𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠_ℤ𝑝=1𝐵_𝑘(𝑥)𝑑𝜇(𝑥)𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠^𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝐵_𝑘−𝑗_ℤ𝑝=1𝑥^𝑗𝑑𝜇(𝑥)𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠^𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝐵_𝑘−𝑗𝐵_𝑗.(2.1) Therefore, we obtain the following theorem.

Theorem 2.1. Let 𝑚∈ℤ_≥0. Then on has the following: 𝐵_𝑚=1𝑚+1^𝑚_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝐵_𝑘−𝑗𝐵_𝑗.(2.2)

Let us apply (1.9) to the bosonic 𝑝-adic integral of (1.16). _ℤ𝑝𝑥^𝑚𝑑𝜇(𝑥)=1𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠_ℤ𝑝=1𝐵_𝑘(𝑥)𝑑𝜇(𝑥)𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠(−1)_𝑘_ℤ𝑝=1𝐵_𝑘(1−𝑥)𝑑𝜇(𝑥)𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠(−1)_𝑘𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝐵_𝑘−𝑗_ℤ𝑝=1(1−𝑥)^𝑗𝑑𝜇(𝑥)𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠(−1)_𝑘𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝐵_𝑘−𝑗(−1)^𝑗𝐵_𝑗(−1).(2.3) Then, we can express (2.3) in three different ways.

By (1.13), (2.3) can be written as _ℤ𝑝𝑥^𝑚𝑑𝜇(𝑥)=1𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠(−1)_𝑘𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝐵_𝑘−𝑗=1𝑗+𝐵_𝑗+𝛿_1,𝑗𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠(−1)_𝑘⎛⎜⎜⎝𝑘𝐵_𝑘−1(1)+𝑘𝐵_𝑘−1+^𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝐵_𝑘−𝑗𝐵_𝑗=−_𝑚−1⎞⎟⎟⎠_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝐵_𝑘+(−1)^𝑘𝐵_𝑘+1𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠(−1)_𝑘𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝐵_𝑘−𝑗𝐵_𝑗=−^𝑚−1_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝐵_𝑘+𝐵_𝑘+𝛿_1,𝑘+1𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠(−1)_𝑘𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝐵_𝑘−𝑗𝐵_𝑗=−2𝐵_𝑚(1)−𝐵_𝑚−𝑚−𝛿_1,𝑚+1𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠(−1)_𝑘𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝐵_𝑘−𝑗𝐵_𝑗=−𝛿_1,𝑚−𝑚+1𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠(−1)_𝑘𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝐵_𝑘−𝑗𝐵_𝑗.(2.4) Thus, we have the following theorem.

Theorem 2.2. Let 𝑚∈ℤ_≥0. Then one has the following: 𝐵_𝑚=−𝛿_1,𝑚−𝑚+1𝑚+1^𝑚_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘𝐵_𝑘−𝑗𝐵_𝑗.(2.5)

Corollary 2.3. Let 𝑚 be an integer ≥2. Then one has the following: 𝐵_𝑚+𝑚=1𝑚+1^𝑚_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘𝐵_𝑘−𝑗𝐵_𝑗.(2.6)

Especially, for an odd integer 𝑚 with 𝑚≥3, we obtain the following corollary.

Corollary 2.4. Let 𝑚 be an odd integer with 𝑚≥3. Then one has the following: 𝑚(𝑚+1)=^𝑚_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘𝐵_𝑘−𝑗𝐵_𝑗.(2.7)

By (1.13), (2.3) can be written as _ℤ𝑝𝑥^𝑚𝑑𝜇(𝑥)=1𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠(−1)_𝑘𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝐵_𝑘−𝑗=1𝑗+(−1)^𝑗𝐵_𝑗𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠(−1)_𝑘⎛⎜⎜⎝𝑘𝐵_𝑘−1(1)+^𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑗𝐵_𝑘−𝑗𝐵_𝑗=−_𝑚−1⎞⎟⎟⎠_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝐵_𝑘+1𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠(−1)_𝑘𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑗𝐵_𝑘−𝑗𝐵_𝑗=−𝐵_𝑚(1)+𝐵_𝑚+1𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠(−1)_𝑘𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑗𝐵_𝑘−𝑗𝐵_𝑗.(2.8) By (1.10), (2.8) can be written as _ℤ𝑝𝑥^𝑚𝑑𝜇(𝑥)=(−1)^𝑚+1𝐵_𝑚+𝐵_𝑚+1𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠(−1)_𝑘𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑗𝐵_𝑘−𝑗𝐵_𝑗.(2.9) So, we get the following theorem.

Theorem 2.5. Let 𝑚∈ℤ_≥0. Then one has the following: 𝐵_𝑚=1𝑚+1^𝑚_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_{𝑚+𝑘+𝑗}𝐵_𝑘−𝑗𝐵_𝑗.(2.10)

By (1.10), (2.8) can also be written as _ℤ𝑝𝑥^𝑚𝑑𝜇(𝑥)=−𝛿_1,𝑚+1𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠(−1)_𝑘𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑗𝐵_𝑘−𝑗𝐵_𝑗.(2.11) Thus, we have the following theorem.

Theorem 2.6. Let 𝑚∈ℤ_≥0. Then one has the following: 𝐵_𝑚=−𝛿_1,𝑚+1𝑚+1^𝑚_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘+𝑗𝐵_𝑘−𝑗𝐵_𝑗.(2.12)

3. Identities Involving Euler Numbers

Taking the fermionic 𝑝-adic integral on both sides of (1.32), we have the following: _ℤ𝑝𝑥^𝑚𝑑𝜇₋₁(𝑥)=_ℤ𝑝⎛⎜⎜⎝𝐸_𝑚(𝑥)+12^𝑚−1_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝐸_𝑘=(𝑥)⎞⎟⎟⎠𝑑𝜇₋₁(𝑥)_𝑚_𝑙=0⎛⎜⎜⎝𝑚𝑙⎞⎟⎟⎠𝐸_𝑚−𝑙_ℤ𝑝𝑥^𝑙𝑑𝜇₋₁(𝑥)+12^𝑚−1_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠^𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝐸_𝑘−𝑗_ℤ𝑝=𝑥^𝑗𝑑𝜇₋₁(𝑥)_𝑚_𝑙=0⎛⎜⎜⎝𝑚𝑙⎞⎟⎟⎠𝐸_𝑚−𝑙𝐸_𝑙+12^𝑚−1_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠^𝑘_𝑗=0⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝐸_𝑘−𝑗𝐸_𝑗.(3.1) So, we obtain the following theorem.

Theorem 3.1. Let 𝑚∈ℤ_≥0. Then one has the following: 𝐸_𝑚=^𝑚_𝑙=0⎛⎜⎜⎝𝑚𝑙⎞⎟⎟⎠𝐸_𝑚−𝑙𝐸_𝑙+12^𝑚−1_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠𝐸_𝑘−𝑗𝐸_𝑗.(3.2)

Let us apply (1.25) to the fermionic 𝑝-adic integral of (1.32). _ℤ𝑝𝑥^𝑚𝑑𝜇₋₁(𝑥)=(−1)^𝑚_ℤ𝑝𝐸_𝑚(1−𝑥)𝑑𝜇₋₁(𝑥)+12^𝑚−1_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠(−1)_𝑘_ℤ𝑝=(−1)_𝑚𝑚𝐸_𝑘(1−𝑥)𝑑𝜇₋₁(𝑥)_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝐸_𝑚−𝑘+1(−1)^𝑘𝐸_𝑘(−1)2^𝑚−1_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘𝐸_𝑘−𝑗(−1)^𝑗𝐸_𝑗(−1).(3.3) Then, we can express (3.3) in two different ways.

By (1.29), (3.3) can be written as _ℤ𝑝𝑥^𝑚𝑑𝜇₋₁(𝑥)=(−1)^𝑚𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝐸_𝑚−𝑘+12+𝐸_𝑘−2𝛿_0,𝑘2^𝑚−1_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘=2𝐸_𝑚𝐸_𝑘−𝑗2+𝐸_𝑗−2𝛿_0,𝑗+(−1)^𝑚𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝐸_𝑚−𝑘𝐸_𝑘+2(−1)^𝑚+1𝐸_𝑚+^𝑚−1_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝐸_𝑘+12^𝑚−1_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘𝐸_𝑘−𝑗𝐸_𝑗+^𝑚−1_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠(−1)_𝑘+1𝐸_𝑘=2𝐸_𝑚+(−1)^𝑚𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝐸_𝑚−𝑘𝐸_𝑘+2(−1)^𝑚+1𝐸_𝑚+𝐸_𝑚(1)−𝐸_𝑚+12^𝑚−1_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘=−2+2𝛿_0,𝑚𝐸_𝑘−𝑗𝐸_𝑗+(−1)^𝑚+1𝐸_𝑚(−1)−𝐸_𝑚+(−1)^𝑚𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝐸_𝑚−𝑘𝐸_𝑘+12^𝑚−1_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘𝐸_𝑘−𝑗𝐸_𝑗.(3.4) Thus, we get the following theorem.

Theorem 3.2. Let 𝑚∈ℤ_≥0. Then one has the following: 𝐸_𝑚=−2+2𝛿_0,𝑚+(−1)^𝑚𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝐸_𝑚−𝑘𝐸_𝑘+12^𝑚−1_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘𝐸_𝑘−𝑗𝐸_𝑗.(3.5)

Corollary 3.3. Let 𝑚∈ℤ_>0. Then one has the following: 𝐸_𝑚+2=(−1)^𝑚𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝐸_𝑚−𝑘𝐸_𝑘+12^𝑚−1_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘𝐸_𝑘−𝑗𝐸_𝑗.(3.6)

By (1.29), (3.3) can be written as _ℤ𝑝𝑥^𝑚𝑑𝜇₋₁(𝑥)=(−1)^𝑚𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝐸_𝑚−𝑘+12−(−1)^𝑘𝐸_𝑘2^𝑚−1_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘=2𝐸_𝑚𝐸_𝑘−𝑗2−(−1)^𝑗𝐸_𝑗+(−1)^𝑚𝑚+1_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠(−1)_𝑘𝐸_𝑚−𝑘𝐸_𝑘+^𝑚−1_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝐸_𝑘−12^𝑚−1_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘(−1)^𝑗𝐸_𝑘−𝑗𝐸_𝑗=2𝐸_𝑚+(−1)^𝑚𝑚+1_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠(−1)_𝑘𝐸_𝑚−𝑘𝐸_𝑘+𝐸_𝑚(1)−𝐸_𝑚−12^𝑚−1_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘(−1)^𝑗𝐸_𝑘−𝑗𝐸_𝑗=2𝛿_0,𝑚+(−1)^𝑚𝑚+1_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠(−1)_𝑘𝐸_𝑚−𝑘𝐸_𝑘−12^𝑚−1_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘(−1)^𝑗𝐸_𝑘−𝑗𝐸_𝑗.(3.7) So, we have the following theorem.

Theorem 3.4. Let 𝑚∈ℤ_≥0. Then one has the following: 𝐸_𝑚=2𝛿_0,𝑚+(−1)^𝑚𝑚+1_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠(−1)_𝑘𝐸_𝑚−𝑘𝐸_𝑘−12^𝑚−1_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘+𝑗𝐸_𝑘−𝑗𝐸_𝑗.(3.8)

Corollary 3.5. Let 𝑚∈ℤ_>1. Then one has the following: 𝐸_𝑚=(−1)^𝑚𝑚+1_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠(−1)_𝑘𝐸_𝑚−𝑘𝐸_𝑘−12^𝑚−1_𝑘𝑘=0_𝑗=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠(−1)_𝑘+𝑗𝐸_𝑘−𝑗𝐸_𝑗.(3.9)

4. Identities Involving Bernoulli and Euler Numbers

By (1.16) and (1.32), we have the following: _ℤ𝑝𝑥^𝑚+𝑛𝑑𝜇(𝑥)=_ℤ𝑝⎛⎜⎜⎝1𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠𝐵_𝑘(𝑥)⎞⎟⎟⎠⎛⎜⎜⎝𝐸_𝑛(𝑥)+12^𝑛−1_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐸_𝑙=1(𝑥)⎞⎟⎟⎠𝑑𝜇(𝑥)𝑚+1^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠_ℤ𝑝+1𝐵_𝑘(𝑥)𝐸_𝑛(𝑥)𝑑𝜇(𝑥)2(𝑚+1)^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠^𝑛−1_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠_ℤ𝑝=1𝐵_𝑘(𝑥)𝐸_𝑙(𝑥)𝑑𝜇(𝑥)𝑚+1^𝑚_𝑘𝑘=0_𝑛𝑗=0_𝑙=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐵_𝑘−𝑗𝐸_𝑛−𝑙𝐵_𝑗+𝑙+12(𝑚+1)^𝑚_{𝑘=0𝑛−1𝑘𝑙=0}_𝑙𝑗=0_𝑖=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠⎛⎜⎜⎝𝑙𝑖⎞⎟⎟⎠𝐵_𝑘−𝑗𝐸_𝑙−𝑖𝐵_𝑗+𝑖.(4.1) Therefore, we get the following theorem.

Theorem 4.1. Let 𝑚,𝑛∈ℤ_≥0. Then one has the following: 𝐵_𝑚+𝑛=1𝑚+1^𝑚_𝑘𝑘=0_𝑛𝑗=0_𝑙=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐵_𝑘−𝑗𝐸_𝑛−𝑙𝐵_𝑗+𝑙+12(𝑚+1)^𝑚_{𝑘=0𝑛−1𝑘𝑙=0}_𝑙𝑗=0_𝑖=0⎛⎜⎜⎝𝑚𝑘+1⎞⎟⎟⎠⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑗⎞⎟⎟⎠⎛⎜⎜⎝𝑙𝑖⎞⎟⎟⎠𝐵_𝑘−𝑗𝐸_𝑙−𝑖𝐵_𝑗+𝑖.(4.2)

By (1.16) and (1.33), we have the following: _ℤ𝑝𝑥^𝑚𝐵_𝑘,𝑛(𝑥)𝑑𝜇(𝑥)=_ℤ𝑝1𝑚+1^𝑚_𝑙=0⎛⎜⎜⎝𝑚+𝑙1⎞⎟⎟⎠𝐵_𝑙=1(𝑥)𝐵_𝑘,𝑛(𝑥)𝑑𝜇(𝑥)𝑚+1⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠^𝑚_𝑙𝑙=0_𝑖=0⎛⎜⎜⎝𝑚+𝑙1⎞⎟⎟⎠⎛⎜⎜⎝𝑙𝑖⎞⎟⎟⎠𝐵_𝑙−𝑖_ℤ𝑝=1𝑥^𝑖+𝑘(1−𝑥)^𝑛−𝑘𝑑𝜇(𝑥)𝑚+1⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠^𝑚_𝑙𝑙=0_{𝑖=0𝑛−𝑘𝑗=0}⎛⎜⎜⎝𝑚+𝑙1⎞⎟⎟⎠⎛⎜⎜⎝𝑙𝑖⎞⎟⎟⎠⎛⎜⎜⎝𝑛−𝑗𝑘⎞⎟⎟⎠(−1)_𝑗𝐵_𝑙−𝑖_ℤ𝑝=1𝑥^{𝑖+𝑘+𝑗}𝑑𝜇(𝑥)𝑚+1⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠^𝑚_𝑙𝑙=0_{𝑖=0𝑛−𝑘𝑗=0}⎛⎜⎜⎝𝑚+𝑙1⎞⎟⎟⎠⎛⎜⎜⎝𝑙𝑖⎞⎟⎟⎠⎛⎜⎜⎝𝑛−𝑗𝑘⎞⎟⎟⎠(−1)_𝑗𝐵_𝑙−𝑖𝐵_{𝑖+𝑘+𝑗}.(4.3) By (1.33), we have the following: _ℤ𝑝𝑥^𝑚𝐵_𝑘,𝑛(𝑥)𝑑𝜇(𝑥)==⎛⎜⎜⎝⎛⎜⎜⎝𝑘𝑘𝑛𝑛⎞⎟⎟⎠⎞⎟⎟⎠_ℤ𝑝𝑥^𝑚+𝑘(1−𝑥)^𝑛−𝑘𝑑𝜇(𝑥)_𝑛−𝑘_𝑗=0⎛⎜⎜⎝𝑛−𝑗𝑘⎞⎟⎟⎠(−1)_𝑗_ℤ𝑝=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝑥^{𝑚+𝑘+𝑗}𝑑𝜇(𝑥)_𝑛−𝑘_𝑗=0⎛⎜⎜⎝𝑛−𝑗𝑘⎞⎟⎟⎠(−1)_𝑗𝐵_{𝑚+𝑘+𝑗}.(4.4) By (4.3) and (4.4), we obtain the following theorem.

Theorem 4.2. Let 𝑚,𝑛,𝑘∈ℤ_≥0. Then one has the following: ^𝑛−𝑘_𝑗=0⎛⎜⎜⎝𝑛−𝑗𝑘⎞⎟⎟⎠(−1)_𝑗𝐵_{𝑚+𝑘+𝑗}=1𝑚+1^𝑚_𝑙𝑙=0_{𝑖=0𝑛−𝑘𝑗=0}⎛⎜⎜⎝𝑚+𝑙1⎞⎟⎟⎠⎛⎜⎜⎝𝑙𝑖⎞⎟⎟⎠⎛⎜⎜⎝𝑛−𝑗𝑘⎞⎟⎟⎠(−1)_𝑗𝐵_𝑙−𝑖𝐵_{𝑖+𝑘+𝑗}.(4.5) Especially, one has the following: (𝑚+1)𝐵_𝑚+𝑛=^𝑚_𝑙𝑙=0_𝑖=0⎛⎜⎜⎝𝑚+𝑙1⎞⎟⎟⎠⎛⎜⎜⎝𝑙𝑖⎞⎟⎟⎠𝐵_𝑙−𝑖𝐵_𝑖+𝑛.(4.6)

By (4.2) and (4.6), we have the following theorem. Note that (4.8) in the following was obtained in [6].

Theorem 4.3. Let 𝑚,𝑛∈ℤ_≥0. Then one has the following: 𝐵_𝑚+𝑛=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐸_𝑛−𝑙𝐵_𝑚+𝑙+12^𝑛−1_𝑙𝑙=0_𝑖=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑙𝑖⎞⎟⎟⎠𝐸_𝑙−𝑖𝐵_𝑚+𝑖.(4.7)

In particular, we have the following: 𝐵_𝑛=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐸_𝑛−1𝐵_𝑙+12^𝑛−1_𝑙𝑙=0_𝑖=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑙𝑖⎞⎟⎟⎠𝐸_𝑙−𝑖𝐵_𝑖.(4.8)