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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012Β (2012), Article IDΒ 598543, 14 pages
http://dx.doi.org/10.1155/2012/598543
Research Article

Derivation of Identities Involving Bernoulli and Euler Numbers

Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

Received 14 June 2012; Accepted 3 August 2012

Academic Editor: CheonΒ Ryoo

Copyright Β© 2012 Imju Lee and Dae San Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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

1. Introduction and Preliminaries

Let 𝑝 be a fixed odd prime. Throughout this paper, ℀𝑝,β„šπ‘,ℂ𝑝 will, respectively, denote the ring of 𝑝-adic integers, the field of 𝑝-adic rational numbers, and the completion of algebraic closure of β„šπ‘. The 𝑝-adic absolute value ||𝑝 on ℂ𝑝 is normalized so that |𝑝|𝑝=1/𝑝. Let β„€>0 be the set of natural numbers and β„€β‰₯0=β„€>0βˆͺ{0}.

As is well known, the Bernoulli polynomials 𝐡𝑛(π‘₯) are defined by the generating function as follows: 𝑑𝐹(𝑑,π‘₯)=π‘’π‘‘π‘’βˆ’1π‘₯𝑑=𝑒𝐡(π‘₯)𝑑=βˆžξ“π‘›=0𝐡𝑛(𝑑π‘₯)𝑛,𝑛!(1.1) with the usual convention of replacing 𝐡(π‘₯)𝑛 by 𝐡𝑛(π‘₯).

In the special case, π‘₯=0, 𝐡𝑛(0)=𝐡𝑛 is referred to as the 𝑛th Bernoulli number. That is, the generating function of Bernoulli numbers is given by 𝑑𝐹(𝑑)=𝐹(𝑑,0)=𝑒𝑑=βˆ’1βˆžξ“π‘›=0𝐡𝑛𝑑𝑛𝑛!=𝑒𝐡𝑑,(1.2) with the usual convention of replacing 𝐡𝑛 by 𝐡𝑛, (cf. [1–23]).

From (1.2), we see that the recurrence formula for the Bernoulli numbers is (𝐡+1)π‘›βˆ’π΅π‘›=𝛿1,𝑛,forπ‘›βˆˆβ„€β‰₯0,(1.3) where π›Ώπ‘˜,𝑛 is the Kronecker symbol.

By (1.1) and (1.2), we easily get the following: 𝐡𝑛(π‘₯)=(𝐡+π‘₯)𝑛=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π΅π‘™π‘₯π‘›βˆ’π‘™=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π΅π‘›βˆ’π‘™π‘₯𝑙,forπ‘›βˆˆβ„€β‰₯0.(1.4) Let π‘ˆπ·(℀𝑝) be the space of uniformly differentiable ℂ𝑝-valued functions on ℀𝑝. For π‘“βˆˆπ‘ˆπ·(℀𝑝), the bosonic 𝑝-adic integral on ℀𝑝 is defined by ξ€œπΌ(𝑓)=℀𝑝𝑓(π‘₯)π‘‘πœ‡(π‘₯)=limπ‘β†’βˆž1π‘π‘π‘π‘βˆ’1π‘₯=0𝑓(π‘₯),(1.5)(cf. [12]). Then it is easy to see that 𝐼𝑓1ξ€Έ=𝐼(𝑓)+π‘“ξ…ž(0),(1.6) where 𝑓1(π‘₯)=𝑓(π‘₯+1) and 𝑓′(0)=𝑑𝑓(π‘₯)/𝑑π‘₯|π‘₯=0.

By (1.6), we have the following: ξ€œβ„€π‘π‘’(π‘₯+𝑦)π‘‘π‘‘π‘‘πœ‡(𝑦)=π‘’π‘‘π‘’βˆ’1π‘₯𝑑=βˆžξ“π‘›=0𝐡𝑛(𝑑π‘₯)𝑛,𝑛!(1.7)(cf. [12–14]). From (1.7), we can derive the Witt's formula for the 𝑛th Bernoulli polynomial as follows: ξ€œβ„€π‘(π‘₯+𝑦)π‘›π‘‘πœ‡(𝑦)=𝐡𝑛(π‘₯),forπ‘›βˆˆβ„€β‰₯0.(1.8)

By (1.1), we have the following: 𝐡𝑛(1βˆ’π‘₯)=(βˆ’1)𝑛𝐡𝑛(π‘₯),forπ‘›βˆˆβ„€β‰₯0.(1.9) Thus, from (1.3), (1.4), and (1.9), we have the following: 𝐡𝑛(1)=𝐡𝑛+𝛿1,𝑛=(βˆ’1)𝑛𝐡𝑛,forπ‘›βˆˆβ„€β‰₯0.(1.10) By (1.4), we have the following: 𝐡𝑛(π‘₯+𝑦)=π‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π΅π‘˜(π‘₯)π‘¦π‘›βˆ’π‘˜,forπ‘›βˆˆβ„€β‰₯0.(1.11) Especially, for π‘₯=1 and 𝑦=1, 𝐡𝑛(2)=π‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π΅π‘˜(1)=π‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ€·π΅π‘˜+𝛿1,π‘˜ξ€Έ,forπ‘›βˆˆβ„€β‰₯0.(1.12) Therefore, from (1.9), (1.10), and (1.12), we can derive the following relation. For π‘›βˆˆβ„€β‰₯0, (βˆ’1)𝑛𝐡𝑛(βˆ’1)=𝐡𝑛(2)=𝑛+𝐡𝑛(1)=𝑛+𝐡𝑛+𝛿1,𝑛=𝑛+(βˆ’1)𝑛𝐡𝑛.(1.13) Let 𝑓(𝑦)=(π‘₯+𝑦)𝑛+1. By (1.6), we have the following: ξ€œβ„€π‘(π‘₯+𝑦+1)𝑛+1ξ€œπ‘‘πœ‡(𝑦)βˆ’β„€π‘(π‘₯+𝑦)𝑛+1π‘‘πœ‡(𝑦)=(𝑛+1)π‘₯𝑛,forπ‘›βˆˆβ„€β‰₯0.(1.14) By (1.8) and (1.14), we have the following: 𝐡𝑛+1(π‘₯+1)βˆ’π΅π‘›+1(π‘₯)=(𝑛+1)π‘₯𝑛,forπ‘›βˆˆβ„€β‰₯0.(1.15) Thus, by (1.11) and (1.15), we have the following identity. π‘₯𝑛=1𝑛+1𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π΅π‘›+1𝑙(π‘₯),forπ‘›βˆˆβ„€β‰₯0.(1.16)

As is well known, the Euler polynomials 𝐸𝑛(π‘₯) are defined by the generating function as follows: 2𝐺(𝑑,π‘₯)=𝑒𝑑𝑒+1π‘₯𝑑=𝑒𝐸(π‘₯)𝑑=βˆžξ“π‘›=0𝐸𝑛(𝑑π‘₯)𝑛,𝑛!(1.17) with the usual convention of replacing 𝐸(π‘₯)𝑛 by 𝐸𝑛(π‘₯).

In the special case, π‘₯=0, 𝐸𝑛(0)=𝐸𝑛 is referred to as the 𝑛th Euler number. That is, the generating function of Euler numbers is given by 2𝐺(𝑑)=𝐺(𝑑,0)=𝑒𝑑=+1βˆžξ“π‘›=0𝐸𝑛𝑑𝑛𝑛!=𝑒𝐸𝑑,(1.18) with the usual convention of replacing 𝐸𝑛 by 𝐸𝑛, (cf. [1–23]).

From (1.18), we see that the recurrence formula for the Euler numbers is (𝐸+1)𝑛+𝐸𝑛=2𝛿0,𝑛,forπ‘›βˆˆβ„€β‰₯0.(1.19) By (1.17) and (1.18), we easily get the following: 𝐸𝑛(π‘₯)=(𝐸+π‘₯)𝑛=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ πΈπ‘™π‘₯π‘›βˆ’π‘™=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ πΈπ‘›βˆ’π‘™π‘₯𝑙,forπ‘›βˆˆβ„€β‰₯0.(1.20) Let 𝐢(℀𝑝) be the space of continuous ℂ𝑝-valued functions on ℀𝑝. For π‘“βˆˆπΆ(℀𝑝), the fermionic 𝑝-adic integral on ℀𝑝 is defined by Kim as follows: πΌβˆ’1(ξ€œπ‘“)=℀𝑝𝑓(π‘₯)π‘‘πœ‡βˆ’1(π‘₯)=limπ‘π‘β†’βˆžπ‘βˆ’1π‘₯=0𝑓(π‘₯)(βˆ’1)π‘₯,(1.21)(cf. [9]). Then it is easy to see that πΌβˆ’1𝑓1ξ€Έ+πΌβˆ’1(𝑓)=2𝑓(0),(1.22) where 𝑓1(π‘₯)=𝑓(π‘₯+1).

By (1.22), we have the following: ξ€œβ„€π‘π‘’(π‘₯+𝑦)π‘‘π‘‘πœ‡βˆ’1(2𝑦)=𝑒𝑑𝑒+1π‘₯𝑑=βˆžξ“π‘›=0𝐸𝑛(𝑑π‘₯)𝑛.𝑛!(1.23) From (1.23), we can derive the Witt's formula for the 𝑛-th Euler polynomial as follows: ξ€œβ„€π‘(π‘₯+𝑦)π‘›π‘‘πœ‡βˆ’1(𝑦)=𝐸𝑛(π‘₯),forπ‘›βˆˆβ„€β‰₯0.(1.24) By (1.17), we have the following: 𝐸𝑛(1βˆ’π‘₯)=(βˆ’1)𝑛𝐸𝑛(π‘₯),forπ‘›βˆˆβ„€β‰₯0.(1.25) Thus, from (1.19), (1.20), and (1.25), we have the following: 𝐸𝑛(1)=βˆ’πΈπ‘›+2𝛿0,𝑛=(βˆ’1)𝑛𝐸𝑛,forπ‘›βˆˆβ„€β‰₯0.(1.26) By (1.20), we have the following: 𝐸𝑛(π‘₯+𝑦)=π‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ πΈπ‘˜(π‘₯)π‘¦π‘›βˆ’π‘˜,forπ‘›βˆˆβ„€β‰₯0.(1.27) Especially, for π‘₯=1 and 𝑦=1, 𝐸𝑛(2)=π‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ πΈπ‘˜(1)=π‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ€·βˆ’πΈπ‘›+2𝛿0,π‘˜ξ€Έ,forπ‘›βˆˆβ„€β‰₯0.(1.28) Therefore, from (1.25), (1.26), and (1.28), we can derive the following relations. For π‘›βˆˆβ„€β‰₯0, (βˆ’1)𝑛𝐸𝑛(βˆ’1)=𝐸𝑛(2)=2βˆ’πΈπ‘›(1)=2+πΈπ‘›βˆ’2𝛿0,𝑛=2βˆ’(βˆ’1)𝑛𝐸𝑛.(1.29) Let 𝑓(𝑦)=(π‘₯+𝑦)𝑛. By (1.22), we have the following: ξ€œβ„€π‘(π‘₯+𝑦+1)π‘›π‘‘πœ‡βˆ’1(ξ€œπ‘¦)+℀𝑝(π‘₯+𝑦)π‘›π‘‘πœ‡βˆ’1(𝑦)=2π‘₯𝑛,forπ‘›βˆˆβ„€β‰₯0.(1.30) By (1.24) and (1.30), we have the following: 𝐸𝑛(π‘₯+1)+𝐸𝑛(π‘₯)=2π‘₯𝑛,forπ‘›βˆˆβ„€β‰₯0.(1.31) Thus, by (1.27) and (1.31), we get the following identity. π‘₯𝑛=12π‘›βˆ’1𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ πΈπ‘™(π‘₯)+𝐸𝑛(π‘₯),forπ‘›βˆˆβ„€β‰₯0.(1.32)

The Bernstein polynomials are defined by π΅π‘˜,π‘›βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘₯(π‘₯)=π‘˜(1βˆ’π‘₯)π‘›βˆ’π‘˜,forπ‘˜,π‘›βˆˆβ„€β‰₯0,(1.33) with 0β‰€π‘˜β‰€π‘› (cf. [14]).

By the definition of π΅π‘˜,𝑛(π‘₯), we note that π΅π‘˜,𝑛(π‘₯)=π΅π‘›βˆ’π‘˜,𝑛(1βˆ’π‘₯).(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 β€œcomplementary” 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: ξ€œβ„€π‘π‘₯π‘šξ€œπ‘‘πœ‡(π‘₯)=℀𝑝1π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π΅π‘š+1π‘˜=1(π‘₯)π‘‘πœ‡(π‘₯)π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ€œπ‘š+1β„€π‘π΅π‘˜(=1π‘₯)π‘‘πœ‡(π‘₯)π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘š+1π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π΅π‘˜βˆ’π‘—ξ€œβ„€π‘π‘₯𝑗=1π‘‘πœ‡(π‘₯)π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘š+1π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π΅π‘˜βˆ’π‘—π΅π‘—.(2.1) Therefore, we obtain the following theorem.

Theorem 2.1. Let π‘šβˆˆβ„€β‰₯0. Then on has the following: π΅π‘š=1π‘š+1π‘šξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π΅π‘š+1π‘˜βˆ’π‘—π΅π‘—.(2.2)

Let us apply (1.9) to the bosonic 𝑝-adic integral of (1.16). ξ€œβ„€π‘π‘₯π‘š1π‘‘πœ‡(π‘₯)=π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ€œπ‘š+1β„€π‘π΅π‘˜=1(π‘₯)π‘‘πœ‡(π‘₯)π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ (π‘š+1βˆ’1)π‘˜ξ€œβ„€π‘π΅π‘˜(=11βˆ’π‘₯)π‘‘πœ‡(π‘₯)π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π΅π‘˜βˆ’π‘—ξ€œβ„€π‘(1βˆ’π‘₯)𝑗=1π‘‘πœ‡(π‘₯)π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π΅π‘˜βˆ’π‘—(βˆ’1)𝑗𝐡𝑗(βˆ’1).(2.3) Then, we can express (2.3) in three different ways.

By (1.13), (2.3) can be written as ξ€œβ„€π‘π‘₯π‘š1π‘‘πœ‡(π‘₯)=π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π΅π‘˜βˆ’π‘—ξ€·π‘—+𝐡𝑗+𝛿1,𝑗=1π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ (π‘š+1βˆ’1)π‘˜βŽ›βŽœβŽœβŽπ‘˜π΅π‘˜βˆ’1(1)+π‘˜π΅π‘˜βˆ’1+π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π΅π‘˜βˆ’π‘—π΅π‘—βŽžβŽŸβŽŸβŽ =βˆ’π‘šβˆ’1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ ξ€·π΅π‘˜+(βˆ’1)π‘˜π΅π‘˜ξ€Έ+1π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π΅π‘˜βˆ’π‘—π΅π‘—=βˆ’π‘šβˆ’1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ ξ€·π΅π‘˜+π΅π‘˜+𝛿1,π‘˜ξ€Έ+1π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π΅π‘˜βˆ’π‘—π΅π‘—ξ€·π΅=βˆ’2π‘š(1)βˆ’π΅π‘šξ€Έβˆ’ξ€·π‘šβˆ’π›Ώ1,π‘šξ€Έ+1π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π΅π‘˜βˆ’π‘—π΅π‘—=βˆ’π›Ώ1,π‘š1βˆ’π‘š+π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π΅π‘˜βˆ’π‘—π΅π‘—.(2.4) Thus, we have the following theorem.

Theorem 2.2. Let π‘šβˆˆβ„€β‰₯0. Then one has the following: π΅π‘š=βˆ’π›Ώ1,π‘š1βˆ’π‘š+π‘š+1π‘šξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜π΅π‘˜βˆ’π‘—π΅π‘—.(2.5)

Corollary 2.3. Let π‘š be an integer β‰₯2. Then one has the following: π΅π‘š1+π‘š=π‘š+1π‘šξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜π΅π‘˜βˆ’π‘—π΅π‘—.(2.6)

Especially, for an odd integer π‘š with π‘šβ‰₯3, we obtain the following corollary.

Corollary 2.4. Let π‘š be an odd integer with π‘šβ‰₯3. Then one has the following: π‘š(π‘š+1)=π‘šξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜π΅π‘˜βˆ’π‘—π΅π‘—.(2.7)

By (1.13), (2.3) can be written as ξ€œβ„€π‘π‘₯π‘š1π‘‘πœ‡(π‘₯)=π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π΅π‘˜βˆ’π‘—ξ€·π‘—+(βˆ’1)𝑗𝐡𝑗=1π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ (π‘š+1βˆ’1)π‘˜βŽ›βŽœβŽœβŽπ‘˜π΅π‘˜βˆ’1(1)+π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘—π΅π‘˜βˆ’π‘—π΅π‘—βŽžβŽŸβŽŸβŽ =βˆ’π‘šβˆ’1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ π΅π‘˜+1π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘—π΅π‘˜βˆ’π‘—π΅π‘—=βˆ’π΅π‘š(1)+π΅π‘š+1π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘—π΅π‘˜βˆ’π‘—π΅π‘—.(2.8) By (1.10), (2.8) can be written as ξ€œβ„€π‘π‘₯π‘šπ‘‘πœ‡(π‘₯)=(βˆ’1)π‘š+1π΅π‘š+π΅π‘š+1π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘—π΅π‘˜βˆ’π‘—π΅π‘—.(2.9) So, we get the following theorem.

Theorem 2.5. Let π‘šβˆˆβ„€β‰₯0. Then one has the following: π΅π‘š=1π‘š+1π‘šξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘š+π‘˜+π‘—π΅π‘˜βˆ’π‘—π΅π‘—.(2.10)

By (1.10), (2.8) can also be written as ξ€œβ„€π‘π‘₯π‘šπ‘‘πœ‡(π‘₯)=βˆ’π›Ώ1,π‘š+1π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘—π΅π‘˜βˆ’π‘—π΅π‘—.(2.11) Thus, we have the following theorem.

Theorem 2.6. Let π‘šβˆˆβ„€β‰₯0. Then one has the following: π΅π‘š=βˆ’π›Ώ1,π‘š+1π‘š+1π‘šξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ π‘š+1(βˆ’1)π‘˜+π‘—π΅π‘˜βˆ’π‘—π΅π‘—.(2.12)

3. Identities Involving Euler Numbers

Taking the fermionic 𝑝-adic integral on both sides of (1.32), we have the following: ξ€œβ„€π‘π‘₯π‘šπ‘‘πœ‡βˆ’1ξ€œ(π‘₯)=β„€π‘βŽ›βŽœβŽœβŽπΈπ‘š1(π‘₯)+2π‘šβˆ’1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ πΈπ‘˜βŽžβŽŸβŽŸβŽ (π‘₯)π‘‘πœ‡βˆ’1=(π‘₯)π‘šξ“π‘™=0βŽ›βŽœβŽœβŽπ‘šπ‘™βŽžβŽŸβŽŸβŽ πΈπ‘šβˆ’π‘™ξ€œβ„€π‘π‘₯π‘™π‘‘πœ‡βˆ’1(1π‘₯)+2π‘šβˆ’1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ πΈπ‘˜βˆ’π‘—ξ€œβ„€π‘π‘₯π‘—π‘‘πœ‡βˆ’1(=π‘₯)π‘šξ“π‘™=0βŽ›βŽœβŽœβŽπ‘šπ‘™βŽžβŽŸβŽŸβŽ πΈπ‘šβˆ’π‘™πΈπ‘™+12π‘šβˆ’1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ πΈπ‘˜βˆ’π‘—πΈπ‘—.(3.1) So, we obtain the following theorem.

Theorem 3.1. Let π‘šβˆˆβ„€β‰₯0. Then one has the following: πΈπ‘š=π‘šξ“π‘™=0βŽ›βŽœβŽœβŽπ‘šπ‘™βŽžβŽŸβŽŸβŽ πΈπ‘šβˆ’π‘™πΈπ‘™+12π‘šβˆ’1ξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ πΈπ‘˜βˆ’π‘—πΈπ‘—.(3.2)

Let us apply (1.25) to the fermionic 𝑝-adic integral of (1.32). ξ€œβ„€π‘π‘₯π‘šπ‘‘πœ‡βˆ’1(π‘₯)=(βˆ’1)π‘šξ€œβ„€π‘πΈπ‘š(1βˆ’π‘₯)π‘‘πœ‡βˆ’11(π‘₯)+2π‘šβˆ’1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜ξ€œβ„€π‘πΈπ‘˜(1βˆ’π‘₯)π‘‘πœ‡βˆ’1(π‘₯)=(βˆ’1)π‘šπ‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ πΈπ‘šβˆ’π‘˜(βˆ’1)π‘˜πΈπ‘˜(+1βˆ’1)2π‘šβˆ’1ξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜πΈπ‘˜βˆ’π‘—(βˆ’1)𝑗𝐸𝑗(βˆ’1).(3.3) Then, we can express (3.3) in two different ways.

By (1.29), (3.3) can be written as ξ€œβ„€π‘π‘₯π‘šπ‘‘πœ‡βˆ’1(π‘₯)=(βˆ’1)π‘šπ‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ πΈπ‘šβˆ’π‘˜ξ€·2+πΈπ‘˜βˆ’2𝛿0,π‘˜ξ€Έ+12π‘šβˆ’1ξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜πΈπ‘˜βˆ’π‘—ξ€·2+πΈπ‘—βˆ’2𝛿0,𝑗=2πΈπ‘š+(βˆ’1)π‘šπ‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ πΈπ‘šβˆ’π‘˜πΈπ‘˜+2(βˆ’1)π‘š+1πΈπ‘š+π‘šβˆ’1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ πΈπ‘˜+12π‘šβˆ’1ξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜πΈπ‘˜βˆ’π‘—πΈπ‘—+π‘šβˆ’1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜+1πΈπ‘˜=2πΈπ‘š+(βˆ’1)π‘šπ‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ πΈπ‘šβˆ’π‘˜πΈπ‘˜+2(βˆ’1)π‘š+1πΈπ‘š+πΈπ‘š(1)βˆ’πΈπ‘š+12π‘šβˆ’1ξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜πΈπ‘˜βˆ’π‘—πΈπ‘—+(βˆ’1)π‘š+1ξ€·πΈπ‘š(βˆ’1)βˆ’πΈπ‘šξ€Έ=βˆ’2+2𝛿0,π‘š+(βˆ’1)π‘šπ‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ πΈπ‘šβˆ’π‘˜πΈπ‘˜+12π‘šβˆ’1ξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜πΈπ‘˜βˆ’π‘—πΈπ‘—.(3.4) Thus, we get the following theorem.

Theorem 3.2. Let π‘šβˆˆβ„€β‰₯0. Then one has the following: πΈπ‘š=βˆ’2+2𝛿0,π‘š+(βˆ’1)π‘šπ‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ πΈπ‘šβˆ’π‘˜πΈπ‘˜+12π‘šβˆ’1ξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜πΈπ‘˜βˆ’π‘—πΈπ‘—.(3.5)

Corollary 3.3. Let π‘šβˆˆβ„€>0. Then one has the following: πΈπ‘š+2=(βˆ’1)π‘šπ‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ πΈπ‘šβˆ’π‘˜πΈπ‘˜+12π‘šβˆ’1ξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜πΈπ‘˜βˆ’π‘—πΈπ‘—.(3.6)

By (1.29), (3.3) can be written as ξ€œβ„€π‘π‘₯π‘šπ‘‘πœ‡βˆ’1(π‘₯)=(βˆ’1)π‘šπ‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ πΈπ‘šβˆ’π‘˜ξ€·2βˆ’(βˆ’1)π‘˜πΈπ‘˜ξ€Έ+12π‘šβˆ’1ξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜πΈπ‘˜βˆ’π‘—ξ€·2βˆ’(βˆ’1)𝑗𝐸𝑗=2πΈπ‘š+(βˆ’1)π‘šπ‘š+1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜πΈπ‘šβˆ’π‘˜πΈπ‘˜+π‘šβˆ’1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ πΈπ‘˜βˆ’12π‘šβˆ’1ξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜(βˆ’1)π‘—πΈπ‘˜βˆ’π‘—πΈπ‘—=2πΈπ‘š+(βˆ’1)π‘šπ‘š+1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜πΈπ‘šβˆ’π‘˜πΈπ‘˜+πΈπ‘š(1)βˆ’πΈπ‘šβˆ’12π‘šβˆ’1ξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜(βˆ’1)π‘—πΈπ‘˜βˆ’π‘—πΈπ‘—=2𝛿0,π‘š+(βˆ’1)π‘šπ‘š+1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜πΈπ‘šβˆ’π‘˜πΈπ‘˜βˆ’12π‘šβˆ’1ξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜(βˆ’1)π‘—πΈπ‘˜βˆ’π‘—πΈπ‘—.(3.7) So, we have the following theorem.

Theorem 3.4. Let π‘šβˆˆβ„€β‰₯0. Then one has the following: πΈπ‘š=2𝛿0,π‘š+(βˆ’1)π‘šπ‘š+1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜πΈπ‘šβˆ’π‘˜πΈπ‘˜βˆ’12π‘šβˆ’1ξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜+π‘—πΈπ‘˜βˆ’π‘—πΈπ‘—.(3.8)

Corollary 3.5. Let π‘šβˆˆβ„€>1. Then one has the following: πΈπ‘š=(βˆ’1)π‘šπ‘š+1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜πΈπ‘šβˆ’π‘˜πΈπ‘˜βˆ’12π‘šβˆ’1ξ“π‘˜π‘˜=0𝑗=0βŽ›βŽœβŽœβŽπ‘šπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜+π‘—πΈπ‘˜βˆ’π‘—πΈπ‘—.(3.9)

4. Identities Involving Bernoulli and Euler Numbers

By (1.16) and (1.32), we have the following: ξ€œβ„€π‘π‘₯π‘š+π‘›ξ€œπ‘‘πœ‡(π‘₯)=β„€π‘βŽ›βŽœβŽœβŽ1π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π΅π‘š+1π‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπΈ(π‘₯)𝑛1(π‘₯)+2π‘›βˆ’1𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ πΈπ‘™βŽžβŽŸβŽŸβŽ =1(π‘₯)π‘‘πœ‡(π‘₯)π‘š+1π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ€œπ‘š+1β„€π‘π΅π‘˜(π‘₯)𝐸𝑛(+1π‘₯)π‘‘πœ‡(π‘₯)2(π‘š+1)π‘šξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘š+1π‘›βˆ’1𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ ξ€œβ„€π‘π΅π‘˜(π‘₯)𝐸𝑙=1(π‘₯)π‘‘πœ‡(π‘₯)π‘š+1π‘šξ“π‘˜π‘˜=0𝑛𝑗=0𝑙=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π΅π‘š+1π‘˜βˆ’π‘—πΈπ‘›βˆ’π‘™π΅π‘—+𝑙+12(π‘š+1)π‘šξ“π‘˜=0π‘›βˆ’1ξ“π‘˜π‘™=0𝑙𝑗=0𝑖=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘™π‘–βŽžβŽŸβŽŸβŽ π΅π‘š+1π‘˜βˆ’π‘—πΈπ‘™βˆ’π‘–π΅π‘—+𝑖.(4.1) Therefore, we get the following theorem.

Theorem 4.1. Let π‘š,π‘›βˆˆβ„€β‰₯0. Then one has the following: π΅π‘š+𝑛=1π‘š+1π‘šξ“π‘˜π‘˜=0𝑛𝑗=0𝑙=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π΅π‘š+1π‘˜βˆ’π‘—πΈπ‘›βˆ’π‘™π΅π‘—+𝑙+12(π‘š+1)π‘šξ“π‘˜=0π‘›βˆ’1ξ“π‘˜π‘™=0𝑙𝑗=0𝑖=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘˜π‘—βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘™π‘–βŽžβŽŸβŽŸβŽ π΅π‘š+1π‘˜βˆ’π‘—πΈπ‘™βˆ’π‘–π΅π‘—+𝑖.(4.2)

By (1.16) and (1.33), we have the following: ξ€œβ„€π‘π‘₯π‘šπ΅π‘˜,π‘›ξ€œ(π‘₯)π‘‘πœ‡(π‘₯)=℀𝑝1π‘š+1π‘šξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π΅π‘š+1𝑙(π‘₯)π΅π‘˜,𝑛=1(π‘₯)π‘‘πœ‡(π‘₯)βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘š+1π‘šξ“π‘™π‘™=0𝑖=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘™π‘–βŽžβŽŸβŽŸβŽ π΅π‘š+1π‘™βˆ’π‘–ξ€œβ„€π‘π‘₯𝑖+π‘˜(1βˆ’π‘₯)π‘›βˆ’π‘˜=1π‘‘πœ‡(π‘₯)βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘š+1π‘šξ“π‘™π‘™=0𝑖=0π‘›βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘™π‘–βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ π‘š+1π‘›βˆ’π‘˜(βˆ’1)π‘—π΅π‘™βˆ’π‘–ξ€œβ„€π‘π‘₯𝑖+π‘˜+𝑗=1π‘‘πœ‡(π‘₯)βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘š+1π‘šξ“π‘™π‘™=0𝑖=0π‘›βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘™π‘–βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ π‘š+1π‘›βˆ’π‘˜(βˆ’1)π‘—π΅π‘™βˆ’π‘–π΅π‘–+π‘˜+𝑗.(4.3) By (1.33), we have the following: ξ€œβ„€π‘π‘₯π‘šπ΅π‘˜,π‘›βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ€œ(π‘₯)π‘‘πœ‡(π‘₯)=℀𝑝π‘₯π‘š+π‘˜(1βˆ’π‘₯)π‘›βˆ’π‘˜=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘‘πœ‡(π‘₯)π‘›βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ (π‘›βˆ’π‘˜βˆ’1)π‘—ξ€œβ„€π‘π‘₯π‘š+π‘˜+𝑗=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘‘πœ‡(π‘₯)π‘›βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ π‘›βˆ’π‘˜(βˆ’1)π‘—π΅π‘š+π‘˜+𝑗.(4.4) By (4.3) and (4.4), we obtain the following theorem.

Theorem 4.2. Let π‘š,𝑛,π‘˜βˆˆβ„€β‰₯0. Then one has the following: π‘›βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ π‘›βˆ’π‘˜(βˆ’1)π‘—π΅π‘š+π‘˜+𝑗=1π‘š+1π‘šξ“π‘™π‘™=0𝑖=0π‘›βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘™π‘–βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ π‘š+1π‘›βˆ’π‘˜(βˆ’1)π‘—π΅π‘™βˆ’π‘–π΅π‘–+π‘˜+𝑗.(4.5) Especially, one has the following: (π‘š+1)π΅π‘š+𝑛=π‘šξ“π‘™π‘™=0𝑖=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘™π‘–βŽžβŽŸβŽŸβŽ π΅π‘š+1π‘™βˆ’π‘–π΅π‘–+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 π‘š,π‘›βˆˆβ„€β‰₯0. Then one has the following: π΅π‘š+𝑛=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ πΈπ‘›βˆ’π‘™π΅π‘š+𝑙+12π‘›βˆ’1𝑙𝑙=0𝑖=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘™π‘–βŽžβŽŸβŽŸβŽ πΈπ‘™βˆ’π‘–π΅π‘š+𝑖.(4.7)

In particular, we have the following: 𝐡𝑛=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ πΈπ‘›βˆ’1𝐡𝑙+12π‘›βˆ’1𝑙𝑙=0𝑖=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘™π‘–βŽžβŽŸβŽŸβŽ πΈπ‘™βˆ’π‘–π΅π‘–.(4.8)

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