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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012Β (2012), Article IDΒ 343981, 15 pages
http://dx.doi.org/10.1155/2012/343981
Research Article

Euler Basis, Identities, and Their Applications

D. S. Kim1Β and T. Kim2

1Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
2Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 11 June 2012; Accepted 9 August 2012

Academic Editor: YilmazΒ Simsek

Copyright Β© 2012 D. S. Kim and T. 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

Let 𝑉𝑛={𝑝(π‘₯)βˆˆβ„š[π‘₯]|deg𝑝(π‘₯)≀𝑛} be the (𝑛+1)-dimensional vector space over β„š. We show that {𝐸0(π‘₯),𝐸1(π‘₯),…,𝐸𝑛(π‘₯)} is a good basis for the space 𝑉𝑛, for our purpose of arithmetical and combinatorial applications. Thus, if 𝑝(π‘₯)βˆˆβ„š[π‘₯] is of degree 𝑛, then βˆ‘π‘(π‘₯)=𝑛𝑙=0𝑏𝑙𝐸𝑙(π‘₯) for some uniquely determined π‘π‘™βˆˆβ„š. In this paper we develop method for computing 𝑏𝑙 from the information of 𝑝(π‘₯).

1. Introduction

The Euler polynomials, 𝐸𝑛(π‘₯), are given by 2𝑒𝑑𝑒+1π‘₯𝑑=𝑒𝐸(π‘₯)𝑑=βˆžξ“π‘›=0𝐸𝑛(𝑑π‘₯)𝑛,𝑛!(1.1)(see [1–20]) with the usual convention about replacing 𝐸𝑛(π‘₯) by 𝐸𝑛(π‘₯). In the special case, π‘₯=0,𝐸𝑛(0)=𝐸𝑛 are called the 𝑛th Euler numbers. The Bernoulli numbers are also defined by π‘‘π‘’π‘‘βˆ’1=𝑒𝐡𝑑=βˆžξ“π‘›=0𝐡𝑛𝑑𝑛,𝑛!(1.2) (see [1–20]) with the usual convention about replacing 𝐡𝑛 by 𝐡𝑛. As is well known, the Bernoulli polynomials are given by 𝐡𝑛(π‘₯)=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,(1.3) (see [9–15]) From (1.1), (1.2), and (1.3), we note that 𝐡𝑛(1)βˆ’π΅π‘›=𝛿1,𝑛,𝐸𝑛(1)+𝐸𝑛=2𝛿0,𝑛,(1.4) where π›Ώπ‘˜,𝑛 is the kronecker symbol.

Let π‘š,π‘›βˆˆβ„€+ with π‘š+𝑛β‰₯2. The formula π΅π‘š(π‘₯)𝐡𝑛(π‘₯)=π‘ŸβŽ§βŽͺ⎨βŽͺβŽ©βŽ›βŽœβŽœβŽπ‘šβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π‘šβŽ«βŽͺ⎬βŽͺ⎭𝐡2π‘Ÿπ‘›+2π‘Ÿ2π‘Ÿπ΅π‘š+π‘›βˆ’2π‘Ÿ(π‘₯)π‘š+π‘›βˆ’2π‘Ÿ+(βˆ’1)π‘š+1π‘š!𝑛!π΅π‘š+𝑛(π‘š+𝑛)!,(1.5) is proved in [4–6]. Let 𝑉𝑛={𝑝(π‘₯)βˆˆβ„š[π‘₯]∣deg𝑝(π‘₯)≀𝑛} be the (𝑛+1)-dimensional vector space over β„š. Probably, {1,π‘₯,…,π‘₯𝑛} is the most natural basis for this space. But {𝐸0(π‘₯),𝐸1(π‘₯),…,𝐸𝑛(π‘₯)} is also a good basis for the space 𝑉𝑛, for our purpose of arithmetical and combinatorial applications. Thus, if 𝑝(π‘₯)βˆˆβ„š[π‘₯] is of degree 𝑛, then 𝑝(π‘₯)=𝑛𝑙=0𝑏𝑙𝐸𝑙(π‘₯),(1.6) for some uniquely determined π‘π‘™βˆˆβ„š. Further, π‘π‘˜=1𝑝2π‘˜!(π‘˜)(1)+𝑝(π‘˜)ξ€Ύ(0)(π‘˜=0,1,2,…,𝑛),(1.7) where 𝑝(π‘˜)(π‘₯)=π‘‘π‘˜π‘(π‘₯)/𝑑π‘₯π‘˜. In this paper we develop methods for computing 𝑏𝑙 from the information of 𝑝(π‘₯). Apply these results to arithmetically and combinatorially interesting identities involving 𝐸0(π‘₯),𝐸1(π‘₯),…,𝐸𝑛(π‘₯),𝐡0(π‘₯),…,𝐡𝑛(π‘₯). Finally, we give some applications of those obtained identities.

2. Euler Basis, Identities, and Their Applications

Let us take 𝑝(π‘₯) the polynomial of degree 𝑛 as follows: 𝑝(π‘₯)=π‘›ξ“π‘˜=0π΅π‘˜(π‘₯)π΅π‘›βˆ’π‘˜(π‘₯).(2.1) From (2.1), we have 𝑝(π‘˜)(π‘₯)=(𝑛+1)!(π‘›βˆ’π‘˜+1)!𝑛𝑙=π‘˜π΅π‘™βˆ’π‘˜(π‘₯)π΅π‘›βˆ’π‘™(π‘₯).(2.2) By (1.7) and (2.2), we get π‘π‘˜=1𝑝2π‘˜!(π‘˜)(1)+𝑝(π‘˜)ξ€Ύ=1(0)2βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘›+1𝑛𝑙=π‘˜π΅ξ€½ξ€·π‘™βˆ’π‘˜+𝛿1,π‘™βˆ’π‘˜π΅ξ€Έξ€·π‘›βˆ’π‘™+𝛿1,π‘›βˆ’π‘™ξ€Έ+π΅π‘™βˆ’π‘˜π΅π‘›βˆ’π‘™ξ€Ύ,(2.3) Thus, we have π‘π‘˜=βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξƒ©π‘›+1𝑛𝑙=π‘˜π΅π‘™βˆ’π‘˜π΅π‘›βˆ’π‘™+π΅π‘›βˆ’π‘˜βˆ’1ξƒͺ𝑏,(0β‰€π‘˜β‰€π‘›βˆ’3),(2.4)π‘›βˆ’2=7𝑛𝑛722ξ€Έβˆ’1,𝑏𝑛=𝑛+1,π‘π‘›βˆ’1=0.(2.5) By (1.6), (2.1), (2.3), and (2.4), we get π‘›ξ“π‘˜=0π΅π‘˜(π‘₯)π΅π‘›βˆ’π‘˜=(π‘₯)π‘›βˆ’3ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξƒ©π‘›+1𝑛𝑙=π‘˜π΅π‘™βˆ’π‘˜π΅π‘›βˆ’π‘™+π΅π‘›βˆ’π‘˜βˆ’1ξƒͺπΈπ‘˜7(π‘₯)+𝑛𝑛722ξ€ΈπΈβˆ’1π‘›βˆ’2(π‘₯)+(𝑛+1)𝐸𝑛(π‘₯).(2.6) Let us consider the following triple identities: 𝑝(π‘₯)=π‘Ÿ+𝑠+𝑑=π‘›π΅π‘Ÿ(π‘₯)𝐡𝑠(π‘₯)𝐡𝑑(π‘₯)=π‘›ξ“π‘˜=0π‘π‘˜πΈπ‘˜(π‘₯),(2.7) where the sum runs over all π‘Ÿ,𝑠,π‘‘βˆˆβ„€+ with π‘Ÿ+𝑠+𝑑=𝑛. Thus, by (2.7), we get 𝑝(π‘˜)(π‘₯)=(𝑛+2)(𝑛+1)𝑛(π‘›βˆ’1)β‹―(π‘›βˆ’π‘˜+3)π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿ(π‘₯)𝐡𝑠(π‘₯)𝐡𝑑(π‘₯).(2.8) From (1.7) and (2.8), we have π‘π‘˜=1𝑝2π‘˜!(π‘˜)(1)+𝑝(π‘˜)ξ€Ύ=ξ€·(0)π‘˜π‘›+2ξ€Έ2ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜ξ€½π΅π‘Ÿ(1)𝐡𝑠(1)𝐡𝑑(1)+π΅π‘Ÿπ΅π‘ π΅π‘‘ξ€Ύ=ξ€·π‘˜π‘›+2ξ€Έ2ξƒ―2ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿπ΅π‘ π΅π‘‘+ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π›Ώ1,π‘Ÿπ΅π‘ π΅π‘‘+ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿπ›Ώ1,𝑠𝐡𝑑+ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿπ΅π‘ π›Ώ1,𝑑+ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π›Ώ1,π‘Ÿπ›Ώ1,𝑠𝐡𝑑+ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π›Ώ1,π‘Ÿπ΅π‘ π›Ώ1,𝑑+ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿπ›Ώ1,𝑠𝛿1,𝑑+ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π›Ώ1,π‘Ÿπ›Ώ1,𝑠𝛿1,𝑑.(2.9) Therefore, by (2.7) and (2.9), we obtain the following theorem.

Theorem 2.1. For π‘Ÿ,𝑠,π‘‘βˆˆβ„€+, and π‘›βˆˆβ„• with 𝑛β‰₯3, one has ξ“π‘Ÿ+𝑠+𝑑=π‘›π΅π‘Ÿ(π‘₯)𝐡𝑠(π‘₯)𝐡𝑑(=1π‘₯)2π‘›βˆ’2ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξƒ―2𝑛+2π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿπ΅π‘ π΅π‘‘ξ“+3π‘Ÿ+𝑠=π‘›βˆ’π‘˜βˆ’1π΅π‘Ÿπ΅π‘ +3π΅π‘›βˆ’π‘˜βˆ’2+π›Ώπ‘˜,π‘›βˆ’3ξƒ°πΈπ‘˜+βŽ›βŽœβŽœβŽ2⎞⎟⎟⎠𝐸(π‘₯)𝑛+2𝑛(π‘₯).(2.10)

Let us take the polynomial 𝑝(π‘₯) as follows: 𝑝(π‘₯)=π‘Ÿ+𝑠+𝑑=π‘›π΅π‘Ÿ(π‘₯)𝐡𝑠(π‘₯)𝐸𝑑(π‘₯).(2.11) Then, by (2.11), we get 𝑝(π‘˜)(π‘₯)=(𝑛+2)(𝑛+1)𝑛(π‘›βˆ’1)β‹―(π‘›βˆ’π‘˜+3)π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿ(π‘₯)𝐡𝑠(π‘₯)𝐸𝑑(π‘₯).(2.12)

From (1.6), (1.7), and (2.12), we have π‘π‘˜=1𝑝2π‘˜!(π‘˜)(1)+𝑝(π‘˜)ξ€Ύ=ξ€·(0)π‘˜π‘›+2ξ€Έ2ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜ξ€½π΅π‘Ÿ(1)𝐡𝑠(1)𝐸𝑑(1)+π΅π‘Ÿπ΅π‘ πΈπ‘‘ξ€Ύ=ξ€·π‘˜π‘›+2ξ€Έ2ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅ξ€½ξ€·π‘Ÿ+𝛿1,π‘Ÿπ΅ξ€Έξ€·π‘ +𝛿1,π‘ ξ€Έξ€·βˆ’πΈπ‘‘+2𝛿0,𝑑+π΅π‘Ÿπ΅π‘ πΈπ‘‘ξ€Ύ=ξ€·π‘˜π‘›+2ξ€Έ2ξƒ―βˆ’ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π›Ώ1,π‘Ÿπ΅π‘ πΈπ‘‘βˆ’ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿπ›Ώ1,𝑠𝐸𝑑+2π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿπ΅π‘ π›Ώ0,π‘‘βˆ’ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π›Ώ1,π‘Ÿπ›Ώ1,𝑠𝐸𝑑+2π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π›Ώ1,π‘Ÿπ΅π‘ π›Ώ0,𝑑+2π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿπ›Ώ1,𝑠𝛿0,𝑑+2π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π›Ώ1,π‘Ÿπ›Ώ1,𝑠𝛿0,𝑑.(2.13) Note that π‘π‘›βˆ’1=βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ ξƒ―βˆ’ξ“π‘›+2π‘›βˆ’1𝑠+𝑑=0π΅π‘ πΈπ‘‘βˆ’ξ“π‘Ÿ+𝑑=0π΅π‘ŸπΈπ‘‘ξ“+2π‘Ÿ+𝑠=1π΅π‘Ÿπ΅π‘ βˆ’0+2𝐡0+2𝐡0ξƒ°=1+2β‹…02βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ ξ€½ξ€·π΅π‘›+2π‘›βˆ’1βˆ’1βˆ’1+21+𝐡1ξ€Έξ€Ύ+2+2=0.(2.14) Therefore, we obtain the following theorem.

Theorem 2.2. For π‘›βˆˆβ„• with 𝑛β‰₯2, one has ξ“π‘Ÿ+𝑠+𝑑=π‘›π΅π‘Ÿ(π‘₯)𝐡𝑠(π‘₯)𝐸𝑑(=1π‘₯)2π‘›βˆ’2ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξƒ―2𝑛+2π‘Ÿ+𝑠=π‘›βˆ’π‘˜π΅π‘Ÿπ΅π‘ ξ“βˆ’2π‘Ÿ+𝑑=π‘›βˆ’π‘˜βˆ’1π΅π‘ŸπΈπ‘‘βˆ’πΈπ‘›βˆ’π‘˜βˆ’2+4π΅π‘›βˆ’π‘˜βˆ’1+2π›Ώπ‘˜,π‘›βˆ’2ξƒ°πΈπ‘˜+βŽ›βŽœβŽœβŽ2⎞⎟⎟⎠𝐸(π‘₯)𝑛+2𝑛(π‘₯).(2.15)

Remark 2.3. By the same method, we obtain the following identities.
(I) ξ“π‘Ÿ+𝑠+𝑑=π‘›π΅π‘Ÿ(π‘₯)𝐸𝑠(π‘₯)𝐸𝑑(=1π‘₯)2π‘›βˆ’2ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξƒ―2𝑛+2π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘ŸπΈπ‘ πΈπ‘‘+𝑠+𝑑=π‘›βˆ’π‘˜βˆ’1πΈπ‘ πΈπ‘‘ξ“βˆ’4π‘Ÿ+𝑠=π‘›βˆ’π‘˜π΅π‘ŸπΈπ‘ +4π΅π‘›βˆ’π‘˜βˆ’4πΈπ‘›βˆ’π‘˜βˆ’1ξƒ°πΈπ‘˜+βŽ›βŽœβŽœβŽ2⎞⎟⎟⎠𝐸(π‘₯)𝑛+2𝑛(π‘₯).(2.16)
(II) ξ“π‘Ÿ+𝑠+𝑑=π‘›πΈπ‘Ÿ(π‘₯)𝐸𝑠(π‘₯)𝐸𝑑(π‘₯)=3π‘›βˆ’2ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξƒ―ξ“π‘›+2π‘Ÿ+𝑠=nβˆ’π‘˜πΈπ‘ŸπΈπ‘ βˆ’2πΈπ‘›βˆ’π‘˜ξƒ°πΈπ‘˜βŽ›βŽœβŽœβŽ2⎞⎟⎟⎠𝐸(π‘₯)+𝑛+2𝑛(π‘₯).(2.17) Let us consider the polynomial 𝑝(π‘₯) as follows: 𝑝(π‘₯)=π‘Ÿ+𝑠+𝑑=π‘›π΅π‘Ÿ(π‘₯)𝐡𝑠(π‘₯)π‘₯𝑑.(2.18) Thus, by (2.18), we get 𝑝(π‘˜)(π‘₯)=(𝑛+2)(𝑛+1)𝑛(π‘›βˆ’1)β‹―(π‘›βˆ’π‘˜+3)π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿ(π‘₯)𝐡𝑠(π‘₯)π‘₯𝑑.(2.19) From (1.6), (1.7), (2.18), and (2.19), we have π‘π‘˜=1𝑝2π‘˜!(π‘˜)(1)+𝑝(π‘˜)ξ€Ύ=ξ€·(0)π‘˜π‘›+2ξ€Έ2ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜ξ€½π΅π‘Ÿ(1)𝐡𝑠(1)+π΅π‘Ÿπ΅π‘ 0𝑑=ξ€·π‘˜π‘›+2ξ€Έ2ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅ξ€½ξ€·π‘Ÿ+𝛿1,π‘Ÿπ΅ξ€Έξ€·π‘ +𝛿1,𝑠+π΅π‘Ÿπ΅π‘ 0𝑑=ξ€·π‘˜π‘›+2ξ€Έ2ξƒ―ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿπ΅π‘ +ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿπ›Ώ1,𝑠+ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π›Ώ1,π‘Ÿπ΅π‘ +ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π›Ώ1,π‘Ÿπ›Ώ1,𝑠+ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿπ΅π‘ 0𝑑.(2.20) Here we note that ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿπ΅π‘ =π‘›βˆ’π‘˜ξ“π‘‘=0ξ“π‘Ÿ+𝑠=π‘›βˆ’π‘˜βˆ’π‘‘π΅π‘Ÿπ΅π‘ =π‘›βˆ’π‘˜ξ“π‘‘=0ξ“π‘Ÿ+𝑠=π‘‘π΅π‘Ÿπ΅π‘ ξ“π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿπ›Ώ1,𝑠=⎧βŽͺ⎨βŽͺβŽ©π‘›βˆ’π‘˜βˆ’1βˆ‘π‘Ÿ=0π΅π‘Ÿξ“,ifπ‘˜β‰€π‘›βˆ’1,0,ifπ‘˜=𝑛,π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘ π›Ώ1,π‘Ÿ=⎧βŽͺ⎨βŽͺβŽ©π‘›βˆ’π‘˜βˆ’1βˆ‘π‘Ÿ=0π΅π‘Ÿξ“,ifπ‘˜β‰€π‘›βˆ’1,0,ifπ‘˜=𝑛,π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π›Ώ1,π‘Ÿπ›Ώ1,𝑠=1,ifπ‘˜β‰€π‘›βˆ’2,0,ifπ‘˜=π‘›βˆ’1or𝑛,π‘Ÿ+𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘Ÿπ΅π‘ 0𝑑=ξ“π‘Ÿ+𝑠=π‘›βˆ’π‘˜π΅π‘Ÿπ΅π‘ ,βˆ€π‘˜.(2.21) It is easy to show that π‘π‘›βˆ’1=12βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ ξƒ―ξ“π‘›+2π‘›βˆ’1π‘Ÿ+𝑠=0π΅π‘Ÿπ΅π‘ ξ“+2π‘Ÿ+𝑠=1π΅π‘Ÿπ΅π‘ +2𝐡0ξƒ°=12βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ ξ€½ξ€·π΅π‘›+2π‘›βˆ’11+21+𝐡2ξ€Έξ€Ύ=1+22βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ .𝑛+2π‘›βˆ’1(2.22) Therefore, by (1.6), (2.18), (2.20), and (2.22), we obtain the following theorem.

Theorem 2.4. For π‘›βˆˆβ„• with 𝑛β‰₯2, one has ξ“π‘Ÿ+𝑠+𝑑=π‘›π΅π‘Ÿ(π‘₯)𝐡𝑠(π‘₯)π‘₯𝑑=12π‘›βˆ’2ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξƒ―π‘›+2π‘›βˆ’π‘˜βˆ’1𝑑=0ξ“π‘Ÿ+𝑠=π‘‘π΅π‘Ÿπ΅π‘ ξ“+2π‘Ÿ+𝑠=π‘›βˆ’π‘˜π΅π‘Ÿπ΅π‘ +2π‘›βˆ’π‘˜βˆ’1ξ“π‘Ÿ=0π΅π‘Ÿξƒ°πΈ+1π‘˜+1(π‘₯)2βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πΈπ‘›+2π‘›βˆ’1π‘›βˆ’1βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ πΈ(π‘₯)+𝑛+2𝑛(π‘₯).(2.23)

Remark 2.5. By the same method, we can derive the following identities.
(I) ξ“π‘Ÿ+𝑠+𝑑=π‘›π΅π‘Ÿ(π‘₯)𝐸𝑠(π‘₯)π‘₯𝑑=12π‘›βˆ’2ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξƒ―βˆ’π‘›+2π‘›βˆ’π‘˜βˆ’1𝑑=0ξ“π‘Ÿ+𝑠=π‘‘π΅π‘ŸπΈπ‘ βˆ’π‘›βˆ’π‘˜βˆ’1𝑠=0𝐸𝑠+2π‘›βˆ’π‘˜ξ“π‘Ÿ=0π΅π‘Ÿξƒ°πΈ+2π‘˜+1(π‘₯)2βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πΈπ‘›+2π‘›βˆ’1π‘›βˆ’1βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ πΈ(π‘₯)+𝑛+2𝑛(π‘₯).(2.24)
(II) ξ“π‘Ÿ+𝑠+𝑑=π‘›πΈπ‘Ÿ(π‘₯)𝐸𝑠(π‘₯)π‘₯𝑑=12π‘›βˆ’2ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξƒ―π‘›+2π‘›βˆ’π‘˜βˆ’1𝑑=0ξ“π‘Ÿ+𝑠=π‘‘πΈπ‘ŸπΈπ‘ ξ“+2π‘Ÿ+𝑠=π‘›βˆ’π‘˜πΈπ‘ŸπΈπ‘ βˆ’4π‘›βˆ’π‘˜ξ“π‘Ÿ=0πΈπ‘Ÿξƒ°πΈ+4π‘˜+1(π‘₯)2βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πΈπ‘›+2π‘›βˆ’1π‘›βˆ’1βŽ›βŽœβŽœβŽ2⎞⎟⎟⎠𝐸(π‘₯)+𝑛+2𝑛(π‘₯).(2.25)

Now we generalize the above consideration to the completely arbitrary case. Let 𝑝(π‘₯)=𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=𝑛𝐡𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯),(2.26) where the sum runs over all nonnegative integers 𝑖1,𝑖2,…,π‘–π‘Ÿ,𝑗1,…,𝑗𝑠 satisfying 𝑖1+𝑖2+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=𝑛. From (2.26), we note that 𝑝(π‘˜)(π‘₯)=(𝑛+π‘Ÿ+π‘ βˆ’1)β‹―(𝑛+π‘Ÿ+π‘ βˆ’π‘˜)𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜π΅π‘–1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)×𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯).(2.27)

By (1.6), (1.7), (2.18), and (2.27), we get π‘π‘˜=1𝑝2π‘˜!(π‘˜)(1)+𝑝(π‘˜)ξ€Ύ=1(0)2βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ“π‘›+π‘Ÿ+π‘ βˆ’1𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜ξ€½π΅π‘–1(1)β‹―π΅π‘–π‘Ÿ(1)𝐸𝑗1(1)⋯𝐸𝑗𝑠(1)+𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠=12βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ Γ—ξ“π‘›+π‘Ÿ+π‘ βˆ’1𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜π΅ξ€½ξ€·π‘–1+𝛿1,𝑖1ξ€Έβ‹―ξ€·π΅π‘–π‘Ÿ+𝛿1,π‘–π‘Ÿξ€ΈΓ—ξ€·βˆ’πΈπ‘—1+2𝛿0,𝑗1ξ€Έβ‹―ξ€·βˆ’πΈj𝑠+2𝛿0,𝑗𝑠+𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠=12βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ“π‘›+π‘Ÿ+π‘ βˆ’10β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ π‘Žβ‰₯π‘˜+π‘Ÿβˆ’π‘›βŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘Γ—ξ“π‘–1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=𝑛+π‘Žβˆ’π‘˜βˆ’π‘Ÿπ΅π‘–1β‹―π΅π‘–π‘ŽπΈπ‘—1⋯𝐸𝑗𝑐+𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1β‹―πΈπ‘—π‘ βŽ«βŽͺβŽͺ⎬βŽͺβŽͺ⎭.(2.28) Note that 𝑏𝑛=12βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ βŽ§βŽͺ⎨βŽͺβŽ©ξ“π‘›+π‘Ÿ+π‘ βˆ’10β‰€π‘β‰€π‘ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘Γ—ξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑐=0𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑐+𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=0𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1β‹―πΈπ‘—π‘ βŽ«βŽͺ⎬βŽͺ⎭=12βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π‘›+π‘Ÿ+π‘ βˆ’1((2βˆ’1)π‘ βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ ,𝑏+1)=𝑛+π‘Ÿ+π‘ βˆ’1π‘›βˆ’1=12βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ§βŽͺ⎨βŽͺβŽ©ξ“π‘›+π‘Ÿ+π‘ βˆ’1π‘›βˆ’1π‘Ÿβˆ’1β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ βŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘Γ—ξ“π‘–1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=1+π‘Žβˆ’π‘Ÿπ΅π‘–1β‹―π΅π‘–π‘ŽπΈπ‘—1⋯𝐸𝑗𝑐+𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=1𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1β‹―πΈπ‘—π‘ βŽ«βŽͺ⎬βŽͺ⎭=12βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ§βŽͺ⎨βŽͺβŽ©π‘›+π‘Ÿ+π‘ βˆ’1π‘›βˆ’1π‘Ÿ(2βˆ’1)𝑠+0β‰€π‘β‰€π‘ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘ξ‚ƒβˆ’12ξ‚„βˆ’1(π‘Ÿ+𝑐)2⎫βŽͺ⎬βŽͺ⎭=1(π‘Ÿ+𝑠)2βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ ξ‚†1𝑛+π‘Ÿ+π‘ βˆ’1π‘›βˆ’1π‘Ÿβˆ’21π‘Ÿ+21π‘ βˆ’21π‘Ÿβˆ’2𝑠=0.(2.29) Therefore, by (1.6), (2.28), and (2.29), we obtain the following theorem.

Theorem 2.6. For π‘›βˆˆβ„• with 𝑛β‰₯2, one has 𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=𝑛𝐡𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(=1π‘₯)2π‘›βˆ’2ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ“π‘›+π‘Ÿ+π‘ βˆ’10β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ π‘Žβ‰₯π‘˜+π‘Ÿβˆ’π‘›βŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘Γ—ξ“π‘–1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=𝑛+π‘Žβˆ’π‘˜βˆ’π‘Ÿπ΅π‘–1β‹―π΅π‘–π‘ŽπΈπ‘—1⋯𝐸𝑗𝑐+𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1β‹―πΈπ‘—π‘ βŽ«βŽͺβŽͺ⎬βŽͺβŽͺβŽ­πΈπ‘˜+βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ πΈ(π‘₯)𝑛+π‘Ÿ+π‘ βˆ’1𝑛(π‘₯).(2.30)

Let us consider the polynomial 𝑝(π‘₯) of degree 𝑛 as 𝑝(π‘₯)=𝑑+𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=𝑛𝐡𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯)π‘₯𝑑.(2.31) Then, from (2.31), we have 𝑝(π‘˜)×(π‘₯)=(𝑛+π‘Ÿ+𝑠)(𝑛+π‘Ÿ+π‘ βˆ’1)β‹―(𝑛+π‘Ÿ+π‘ βˆ’π‘˜+1)𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=π‘›βˆ’π‘˜π΅π‘–1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯)π‘₯𝑑.(2.32) By (1.7) and (2.32), we get π‘π‘˜=1𝑝2π‘˜!(π‘˜)(1)+𝑝(π‘˜)ξ€Ύ=1(0)2βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ ξ“π‘›+π‘Ÿ+𝑠𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=π‘›βˆ’π‘˜ξ€½π΅π‘–1(1)β‹―π΅π‘–π‘Ÿ(1)𝐸𝑗1(1)⋯𝐸𝑗𝑠(1)+𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠0𝑑=12βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ Γ—ξ“π‘›+π‘Ÿ+𝑠𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=π‘›βˆ’π‘˜π΅ξ€½ξ€·π‘–1+𝛿1,𝑖1ξ€Έβ‹―ξ€·π΅π‘–π‘Ÿ+𝛿1,π‘–π‘Ÿξ€ΈΓ—ξ€·βˆ’πΈπ‘—0+2𝛿0,𝑗1ξ€Έβ‹―ξ€·βˆ’πΈπ‘—π‘ +2𝛿1,𝑗𝑠+𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠0𝑑(2.33) From (2.33), we can derive the following equation: π‘π‘˜=12βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ“π‘›+π‘Ÿ+𝑠0β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ π‘Žβ‰₯π‘˜+π‘Ÿβˆ’π‘›βŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘Γ—π‘›+π‘Žβˆ’π‘˜βˆ’π‘Ÿξ“π‘‘=0𝑖1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=𝑑𝐡𝑖1β‹―π΅π‘–π‘ŽπΈπ‘—1⋯𝐸𝑗𝑐+𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1β‹―πΈπ‘—π‘ βŽ«βŽͺβŽͺ⎬βŽͺβŽͺ⎭.(2.34) Observe now that 𝑏𝑛=12βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ βŽ§βŽͺ⎨βŽͺβŽ©π‘›+π‘Ÿ+𝑠𝑠𝑐=0βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘Γ—ξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑐=0𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑐+𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=0𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1β‹―πΈπ‘—π‘ βŽ«βŽͺ⎬βŽͺ⎭=12βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ ξ€Ίπ‘›+π‘Ÿ+𝑠(2βˆ’1)𝑠=βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ ,𝑏+1𝑛+π‘Ÿ+𝑠(2.35)π‘›βˆ’1=12βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ§βŽͺ⎨βŽͺβŽ©ξ“π‘›+π‘Ÿ+π‘ π‘›βˆ’1π‘Ÿβˆ’1β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ βŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘Γ—1+π‘Žβˆ’π‘Ÿξ“π‘‘=0𝑖1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=𝑑𝐡𝑖1β‹―π΅π‘–π‘ŽπΈπ‘—1⋯𝐸𝑗𝑐+𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=1𝐡𝑖1⋯𝐡iπ‘ŸπΈπ‘—1β‹―πΈπ‘—π‘ βŽ«βŽͺ⎬βŽͺ⎭=12βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ ξ‚†1𝑛+π‘Ÿ+π‘ π‘›βˆ’1π‘Ÿ+1βˆ’21π‘Ÿ+21π‘ βˆ’21π‘Ÿβˆ’2𝑠=12βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ .𝑛+π‘Ÿ+π‘ π‘›βˆ’1(2.36) Therefore, by (1.6), (2.31), (2.34), (2.35), and (2.36), we obtain the following theorem.

Theorem 2.7. For π‘›βˆˆβ„• with 𝑛β‰₯2, one has 𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=𝑛𝐡𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯)π‘₯𝑑=12π‘›βˆ’2ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ“π‘›+π‘Ÿ+𝑠0β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ π‘Žβ‰₯π‘˜+π‘Ÿβˆ’π‘›βŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘Γ—π‘›+π‘Žβˆ’π‘˜βˆ’π‘Ÿξ“π‘‘=0𝑖1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=t𝐡𝑖1β‹―π΅π‘–π‘ŽπΈπ‘—1⋯𝐸𝑗𝑐+𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1β‹―πΈπ‘—π‘ βŽ«βŽͺβŽͺ⎬βŽͺβŽͺβŽ­πΈπ‘˜+1(π‘₯)2βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πΈπ‘›+π‘Ÿ+π‘ π‘›βˆ’1π‘›βˆ’1βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ πΈ(π‘₯)+𝑛+π‘Ÿ+𝑠𝑛(π‘₯).(2.37)

Let us consider the following polynomial of degree 𝑛. 𝑝(π‘₯)=𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=𝑛1𝑖1!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!𝐡𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯).(2.38) Thus, by (2.38), we get 𝑝(π‘˜)(π‘₯)=(π‘Ÿ+𝑠)π‘˜ξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜1𝑖1!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!×𝐡𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯).(2.39) From (1.7), we have π‘π‘˜=1𝑝2π‘˜!(π‘˜)(1)+𝑝(π‘˜)ξ€Ύ=(0)(π‘Ÿ+𝑠)π‘˜ξ“2π‘˜!𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜1𝑖1!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!×𝐡𝑖1(1)β‹―π΅π‘–π‘Ÿ(1)×𝐸𝑗1(1)⋯𝐸𝑗𝑠(1)+𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠=(π‘Ÿ+𝑠)π‘˜ξ“2π‘˜!𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜1𝑖1!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!×𝐡𝑖1+𝛿1,𝑖1ξ€Έβ‹―ξ€·π΅π‘–π‘Ÿ+𝛿1,π‘–π‘Ÿξ€ΈΓ—ξ€·βˆ’πΈπ‘—1+2𝛿0,𝑗1ξ€Έβ‹―ξ€·βˆ’πΈπ‘—π‘ +2𝛿0,𝑗𝑠+𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠.(2.40) Thus, by (2.40), we get π‘π‘˜=(π‘Ÿ+𝑠)π‘˜βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ“2π‘˜!0β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ π‘Žβ‰₯π‘˜+π‘Ÿβˆ’π‘›βŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2sβˆ’π‘Γ—ξ“π‘–1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=𝑛+π‘Žβˆ’π‘˜βˆ’π‘Ÿπ΅π‘–1β‹―π΅π‘–π‘ŽπΈπ‘—1⋯𝐸𝑗𝑐𝑖1!β‹―π‘–π‘Ž!𝑗1!⋯𝑗𝑐!+𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠𝑖1!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!⎫βŽͺβŽͺ⎬βŽͺβŽͺ⎭.(2.41) Now, we note that 𝑏𝑛=(π‘Ÿ+𝑠)π‘›βŽ§βŽͺ⎨βŽͺ⎩2𝑛!𝑠𝑐=0βŽ›βŽœβŽœβŽsπ‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘Γ—ξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑐=0𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑐𝑖1!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑐!+𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=0𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠𝑖1!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!⎫βŽͺ⎬βŽͺ⎭=(π‘Ÿ+𝑠)𝑛2𝑛!(2βˆ’1)𝑠=+1(π‘Ÿ+𝑠)𝑛,𝑏𝑛!π‘›βˆ’1=(π‘Ÿ+𝑠)π‘›βˆ’1⎧βŽͺ⎨βŽͺβŽ©ξ“2(π‘›βˆ’1)!π‘Ÿβˆ’1β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ βŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘Γ—ξ“π‘–1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=1+π‘Žβˆ’π‘Ÿπ΅π‘–1β‹―π΅π‘–π‘ŽπΈπ‘—1⋯𝐸𝑗𝑐𝑖1!β‹―π‘–π‘Ž!𝑗1!⋯𝑗𝑐!+𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=1𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠𝑖1!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!⎫βŽͺ⎬βŽͺ⎭=(π‘Ÿ+𝑠)π‘›βˆ’1⎧βŽͺ⎨βŽͺ⎩2(π‘›βˆ’1)!π‘Ÿ(2βˆ’1)𝑠+𝑠𝑐=0βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘ξ‚ƒβˆ’12ξ‚„βˆ’1(π‘Ÿ+𝑐)2⎫βŽͺ⎬βŽͺ⎭(π‘Ÿ+𝑠)=0.(2.42) Therefore, by (1.6), (2.38), (2.41), and (2.42), we obtain the following theorem.

Theorem 2.8. For π‘›βˆˆβ„• with 𝑛β‰₯2, one has 𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=𝑛𝐡𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯)𝑖1!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!=12π‘›βˆ’2ξ“π‘˜=0(π‘Ÿ+𝑠)π‘˜βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ“π‘˜!0β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ π‘Žβ‰₯π‘˜+π‘Ÿβˆ’π‘›βŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘Γ—ξ“π‘–1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=𝑛+π‘Žβˆ’π‘˜βˆ’π‘Ÿπ΅π‘–1β‹―π΅π‘–π‘ŽπΈπ‘—1⋯𝐸𝑗𝑐𝑖1!β‹―π‘–π‘Ž!𝑗1!⋯𝑗𝑐!+𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠𝑖1!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!⎫βŽͺβŽͺ⎬βŽͺβŽͺβŽ­πΈπ‘˜+(π‘₯)(π‘Ÿ+𝑠)𝑛𝐸𝑛!𝑛(π‘₯).(2.43)

By the same method, we can obtain the following identity: 𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=𝑛𝐡𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯)π‘₯𝑑𝑖1!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠=1!𝑑!2π‘›βˆ’2ξ“π‘˜=0(π‘Ÿ+𝑠+1)π‘˜βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ“π‘˜!0β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ π‘Žβ‰₯π‘˜+π‘Ÿβˆ’π‘›βŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘Γ—π‘›+π‘Žβˆ’π‘˜βˆ’π‘Ÿξ“π‘‘=01(𝑛+π‘Žβˆ’π‘˜βˆ’π‘Ÿβˆ’π‘‘)!𝑖1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=𝑑𝐡𝑖1β‹―π΅π‘–π‘ŽπΈπ‘—1⋯𝐸𝑗𝑐𝑖1!β‹―π‘–π‘Ž!𝑗1!⋯𝑗𝑐!+𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠𝑖1!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!⎫βŽͺβŽͺ⎬βŽͺβŽͺβŽ­πΈπ‘˜(+π‘₯)(π‘Ÿ+𝑠+1)π‘›βˆ’1𝐸2(π‘›βˆ’1)!π‘›βˆ’1(π‘₯)+(π‘Ÿ+𝑠+1)𝑛𝐸𝑛!𝑛(π‘₯).(2.44)

Acknowledgments

This paper was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology 2012R1A1A2003786.

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