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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012Β (2012), Article IDΒ 463659, 12 pages
http://dx.doi.org/10.1155/2012/463659
Research Article

Bernoulli Basis and the Product of Several Bernoulli Polynomials

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

Received 25 June 2012; Accepted 9 August 2012

Academic Editor: YilmazΒ Simsek

Copyright Β© 2012 Dae San Kim and Taekyun 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 develop methods for computing the product of several Bernoulli and Euler polynomials by using Bernoulli basis for the vector space of polynomials of degree less than or equal to 𝑛.

1. Introduction

It is well known that, the 𝑛th Bernoulli and Euler numbers are defined by 𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π΅π‘™βˆ’π΅π‘›=𝛿1,𝑛,𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ πΈπ‘™+𝐸𝑛=2𝛿0,𝑛,(1.1) where 𝐡0=𝐸0=1 and π›Ώπ‘˜,𝑛 is the Kronecker symbol (see [1–20]).

The Bernoulli and Euler polynomials are also defined by 𝐡𝑛(π‘₯)=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ π΅π‘›βˆ’π‘™π‘₯𝑙,𝐸𝑛(π‘₯)=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ πΈπ‘›βˆ’π‘™π‘₯𝑙.(1.2) Note that {𝐡0(π‘₯),𝐡1(1),…,𝐡𝑛(π‘₯)} forms a basis for the space ℙ𝑛={𝑝(π‘₯)βˆˆβ„š[π‘₯]∣deg𝑝(π‘₯)≀𝑛}.

So, for a given 𝑝(π‘₯)βˆˆβ„™π‘›, we can write 𝑝(π‘₯)=π‘›ξ“π‘˜=0π‘Žπ‘˜π΅π‘˜(π‘₯),(1.3) (see [8–12]) for uniquely determined π‘Žπ‘˜βˆˆβ„š.

Further, π‘Žπ‘˜=1ξ€½π‘π‘˜!(π‘˜βˆ’1)(1)βˆ’π‘(π‘˜βˆ’1)ξ€Ύ(0),where𝑝(π‘˜)𝑑(π‘₯)=π‘˜π‘(π‘₯)𝑑π‘₯π‘˜,π‘Ž0=ξ€œ10𝑝(𝑑)𝑑𝑑,whereπ‘˜=1,2,…,𝑛.(1.4) Probably, {1,π‘₯,…,π‘₯𝑛} is the most natural basis for the space ℙ𝑛. But {𝐡0(π‘₯),𝐡1(π‘₯),…,𝐡𝑛(π‘₯)} is also a good basis for the space ℙ𝑛, for our purpose of arithmetical and combinatorial applications.

What are common to 𝐡𝑛(π‘₯), 𝐸𝑛(π‘₯), π‘₯𝑛? A few proportion common to them are as follows: (i)they are all monic polynomials of degree 𝑛 with rational coefficients; (ii)(𝐡𝑛(π‘₯))β€²=π‘›π΅π‘›βˆ’1(π‘₯), (𝐸𝑛(π‘₯))β€²=π‘›πΈπ‘›βˆ’1(π‘₯), (π‘₯𝑛)β€²=𝑛π‘₯π‘›βˆ’1; (iii)βˆ«π΅π‘›(π‘₯)𝑑π‘₯=𝐡𝑛+1(π‘₯)/(𝑛+1)+𝑐, βˆ«πΈπ‘›(π‘₯)𝑑π‘₯=𝐸𝑛+1(π‘₯)/(𝑛+1)+𝑐, ∫π‘₯𝑛𝑑π‘₯=π‘₯𝑛+1/(𝑛+1)+𝑐.

In [5, 6], Carlitz introduced the identities of the product of several Bernoulli polynomials: π΅π‘š(π‘₯)𝐡𝑛(π‘₯)=βˆžξ“π‘Ÿ=0⎧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+1Γ—π‘š!𝑛!𝐡(π‘š+𝑛)!π‘š+π‘›ξ€œ(π‘š+𝑛β‰₯2),10π΅π‘š(π‘₯)𝐡𝑛(π‘₯)𝐡𝑝(π‘₯)π΅π‘ž(π‘₯)𝑑π‘₯=(βˆ’1)βˆžπ‘š+𝑛+𝑝+π‘žξ“π‘Ÿ,𝑠=0⎧βŽͺ⎨βŽͺβŽ©βŽ›βŽœβŽœβŽπ‘šβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π‘šβŽ«βŽͺ⎬βŽͺ⎭⎧βŽͺ⎨βŽͺβŽ©βŽ›βŽœβŽœβŽπ‘βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘žβŽžβŽŸβŽŸβŽ π‘βŽ«βŽͺ⎬βŽͺβŽ­Γ—2π‘Ÿπ‘›+2π‘Ÿ2π‘ π‘ž+2𝑠(π‘š+π‘›βˆ’2π‘Ÿβˆ’1)!(𝑝+π‘žβˆ’2π‘ βˆ’1)!𝐡(π‘š+𝑛+𝑝+π‘žβˆ’2π‘Ÿβˆ’2𝑠)!π‘Ÿπ΅π‘ π΅π‘š+𝑛+𝑝+π‘žβˆ’2π‘Ÿβˆ’2+(βˆ’1)π‘š+π‘π‘š!𝑛!(π‘š+𝑛)!𝑝!π‘ž!(𝐡𝑝+π‘ž)!π‘š+𝑛𝐡𝑝+π‘ž.(1.5) In this paper, we will use (1.4) to derive the identities of the product of several Bernoulli and Euler polynomials.

2. The Product of Several Bernoulli and Euler Polynomials

Let us consider the following polynomials of degree 𝑛: 𝑝(π‘₯)=𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=𝑛𝐡𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯),(2.1) where the sum runs over all nonnegative integers 𝑖1,…,π‘–π‘Ÿ,  𝑗1,…𝑗𝑠 satisfying 𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=𝑛,β€‰β€‰π‘Ÿ+𝑠=1, π‘Ÿ,𝑠β‰₯0.

Thus, from (2.1), we have 𝑝(π‘˜)Γ—(π‘₯)=(𝑛+π‘Ÿ+π‘ βˆ’1)(𝑛+π‘Ÿ+π‘ βˆ’2)β‹―(𝑛+π‘Ÿ+π‘ βˆ’π‘˜)βˆžξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜π΅π‘–1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯).(2.2) For π‘˜=1,2,…,𝑛, by (1.4), we get π‘Žπ‘˜=1ξ€½π‘π‘˜!(π‘˜βˆ’1)(1)βˆ’π‘(π‘˜βˆ’1)ξ€Ύ=ξ€·(0)π‘˜π‘›+π‘Ÿ+𝑠𝑛+π‘Ÿ+𝑠𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜+1𝐡𝑖1(1)β‹―π΅π‘–π‘Ÿ(1)𝐸𝑗1(1)⋯𝐸𝑗𝑠(1)βˆ’π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠=ξ€·π‘˜π‘›+π‘Ÿ+π‘ ξ€ΈβŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ“π‘›+π‘Ÿ+𝑠0β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ π‘˜+π‘Ÿβˆ’π‘›βˆ’1β‰€π‘Žβ‰€π‘ŸβŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘Γ—βˆžξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=𝑛+π‘Ž+1βˆ’π‘˜βˆ’π‘Ÿπ΅π‘–1β‹―π΅π‘–π‘ŽπΈπ‘—1β‹―πΈπ‘—π‘βˆ’ξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜+1𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1β‹―πΈπ‘—π‘ βŽ«βŽͺβŽͺ⎬βŽͺβŽͺ⎭.(2.3) From (2.3), we note that π‘Žπ‘›=(𝑛𝑛+π‘Ÿ+𝑠)𝑛+π‘Ÿ+𝑠𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=1𝐡𝑖1(1)β‹―π΅π‘–π‘Ÿ(1)𝐸𝑗1(1)⋯𝐸𝑗𝑠(1)βˆ’π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠=(𝑛𝑛+π‘Ÿ+𝑠)βˆ’1𝑛+π‘Ÿ+𝑠2ξ‚ξ‚€βˆ’1+1π‘Ÿβˆ’2ξ‚ξ‚€βˆ’1π‘ βˆ’2=((π‘Ÿ+𝑠)𝑛𝑛+π‘Ÿ+𝑠)βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ ,π‘Žπ‘›+π‘Ÿ+𝑠(π‘Ÿ+𝑠)=𝑛+π‘Ÿ+π‘ βˆ’1π‘›βˆ’1=1βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ Γ—ξ“π‘›+π‘Ÿ+𝑠𝑛+π‘Ÿ+π‘ π‘›βˆ’1𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=2𝐡𝑖1(1)β‹―π΅π‘–π‘Ÿ(1)𝐸𝑗1(1)⋯𝐸𝑗𝑠(1)βˆ’π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠=1βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ§βŽͺ⎨βŽͺ⎩1𝑛+π‘Ÿ+𝑠𝑛+π‘Ÿ+π‘ π‘›βˆ’161π‘Ÿ+212βŽ›βŽœβŽœβŽ2βŽžβŽŸβŽŸβŽ βˆ’1π‘Ÿ+𝑠6ξ‚€βˆ’1π‘Ÿβˆ’2βˆ’12ξ‚βŽ›βŽœβŽœβŽ2⎞⎟⎟⎠⎫βŽͺ⎬βŽͺβŽ­π‘Žπ‘Ÿ+𝑠=0,0=ξ€œ10=𝑝(𝑑)π‘‘π‘‘βˆžξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠𝑖=𝑛1𝑙1=0β‹―π‘–π‘Ÿξ“π‘™π‘Ÿπ‘—=01𝑝1=0⋯𝑗𝑠𝑝𝑠=0βŽ›βŽœβŽœβŽπ‘–1𝑙1βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘–π‘Ÿπ‘™π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘—1𝑝1βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘—π‘ π‘π‘ βŽžβŽŸβŽŸβŽ Γ—π΅π‘–1βˆ’π‘™1β‹―π΅π‘–π‘Ÿβˆ’π‘™π‘ŸπΈπ‘—1βˆ’π‘1πΈπ‘—π‘ βˆ’π‘π‘ π‘™1+β‹―+π‘™π‘Ÿ+𝑝1+β‹―+𝑝𝑠.+1(2.4) Therefore, by (1.3), (2.1), (2.3), and (2.4), we obtain the following theorem.

Theorem 2.1. For π‘›βˆˆβ„• with 𝑛β‰₯2, we have 𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=𝑛𝐡𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(=1π‘₯)𝑛+π‘Ÿ+π‘ π‘›βˆ’2ξ“π‘˜=1βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ Γ—βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ“π‘›+π‘Ÿ+𝑠0β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ π‘˜+π‘Ÿβˆ’π‘›βˆ’1β‰€π‘Žβ‰€π‘ŸβŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘ξ“π‘–1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=𝑛+π‘Ž+1βˆ’π‘˜βˆ’π‘Ÿπ΅π‘–1β‹―π΅π‘–π‘ŽπΈπ‘—1β‹―πΈπ‘—π‘βˆ’βˆžξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜+1𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘˜1β‹―πΈπ‘—π‘ βŽ«βŽͺβŽͺ⎬βŽͺβŽͺβŽ­π΅π‘˜βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π΅(π‘₯)+𝑛+π‘Ÿ+π‘ βˆ’1𝑛+(π‘₯)𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠𝑖=𝑛1𝑙1=0β€¦π‘–π‘Ÿξ“π‘™π‘Ÿπ‘—=01𝑝1=0⋯𝑗𝑠𝑝𝑠=0βŽ›βŽœβŽœβŽπ‘–1𝑙1βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘–π‘Ÿπ‘™π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘—1𝑝1βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘—π‘ π‘π‘ βŽžβŽŸβŽŸβŽ Γ—π΅π‘–1βˆ’π‘™1β‹―π΅π‘–π‘Ÿβˆ’π‘™π‘ŸπΈπ‘—1βˆ’π‘1πΈπ‘—π‘ βˆ’π‘π‘ π‘™1+β‹―+π‘™π‘Ÿ+𝑝1+⋯𝑝𝑠.+1(2.5)

Let us take the polynomial 𝑝(π‘₯) of degree 𝑛 as follows: 𝑝(π‘₯)=𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=𝑛𝐡𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯)π‘₯𝑑,(2.6) Then, from (2.6), 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,(2.7) By (1.4) and (2.7), we get, for π‘˜=1,2,…,𝑛, π‘Žπ‘˜=1ξ€½π‘π‘˜!(π‘˜βˆ’1)(1)βˆ’π‘(π‘˜βˆ’1)ξ€Ύ=1(0)βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ Γ—ξ“π‘›+π‘Ÿ+𝑠+1𝑛+π‘Ÿ+𝑠+1𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=π‘›βˆ’π‘˜+1𝐡𝑖1(1)β‹―π΅π‘–π‘Ÿ(1)𝐸𝑗1(1)⋯𝐸𝑗𝑠(1)βˆ’π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠0𝑑=ξ€·π‘˜π‘›+π‘Ÿ+𝑠+1ξ€ΈβŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ“π‘›+π‘Ÿ+𝑠+10β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ π‘˜+π‘Ÿβˆ’π‘›βˆ’1β‰€π‘Žβ‰€π‘ŸβŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘Γ—π‘›+π‘Ž+1βˆ’π‘˜βˆ’π‘Ÿξ“π‘‘=0𝑖1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=𝑑𝐡𝑖1β‹―π΅π‘–π‘ŽπΈπ‘—1β‹―πΈπ‘—π‘βˆ’ξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜+1𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1β‹―πΈπ‘—π‘ βŽ«βŽͺβŽͺ⎬βŽͺβŽͺ⎭,(2.8) Now, we look at π‘Žπ‘› and π‘Žπ‘›βˆ’1. π‘Žπ‘›=𝑛𝑛+π‘Ÿ+𝑠+1𝑛+π‘Ÿ+𝑠+1𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=1𝐡𝑖1(1)β‹―π΅π‘–π‘Ÿ(1)𝐸𝑗1(1)⋯𝐸𝑗𝑠(1)βˆ’π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠0𝑑=𝑛𝑛+π‘Ÿ+𝑠+11𝑛+π‘Ÿ+𝑠+12ξ‚€βˆ’1(π‘Ÿ+𝑠)+1βˆ’2=(π‘Ÿ+𝑠)π‘Ÿ+𝑠+1βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ =βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ ,π‘Žπ‘›+π‘Ÿ+𝑠+1𝑛+π‘Ÿ+𝑠+1𝑛+π‘Ÿ+π‘ π‘›βˆ’1=𝑛+π‘Ÿ+𝑠+1π‘›βˆ’1𝑛+π‘Ÿ+𝑠+1𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=2𝐡𝑖1(1)β‹―π΅π‘–π‘Ÿ(1)𝐸𝑗1(1)⋯𝐸𝑗𝑠(1)βˆ’π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠0𝑑=𝑛+π‘Ÿ+𝑠+1π‘›βˆ’1ξ€ΈβŽ§βŽͺ⎨βŽͺ⎩1𝑛+π‘Ÿ+𝑠+161π‘Ÿ+1+212βŽ›βŽœβŽœβŽ2⎞⎟⎟⎠+1π‘Ÿ+𝑠21(π‘Ÿ+𝑠)βˆ’6ξ‚€βˆ’1π‘Ÿβˆ’2βˆ’12ξ‚βŽ›βŽœβŽœβŽ2⎞⎟⎟⎠⎫βŽͺ⎬βŽͺ⎭=1π‘Ÿ+π‘ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ π‘›+π‘Ÿ+𝑠+1𝑛+π‘Ÿ+𝑠+1π‘›βˆ’1π‘Ÿ+𝑠+22=12βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ ,𝑛+π‘Ÿ+π‘ π‘›βˆ’1(2.9) From (2.6), we note that π‘Ž0=ξ€œ10=𝑝(𝑑)𝑑𝑑𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠𝑖+𝑑=𝑛1𝑙1=0β‹―π‘–π‘Ÿξ“π‘™π‘Ÿπ‘—=01𝑝1=0⋯𝑗𝑠𝑝𝑠=0βŽ›βŽœβŽœβŽπ‘–1𝑙1βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘–π‘Ÿπ‘™π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘—1𝑝1βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘—π‘ π‘π‘ βŽžβŽŸβŽŸβŽ Γ—π΅π‘–1βˆ’π‘™1β‹―π΅π‘–π‘Ÿβˆ’π‘™π‘ŸπΈπ‘—1βˆ’π‘1πΈπ‘—π‘ βˆ’π‘π‘ 1𝑙1+β‹―+π‘™π‘Ÿ+𝑝1+⋯𝑝𝑠.+𝑑+1(2.10) Therefore, by (1.3), (2.6), (2.8), (2.9), and (2.10), we obtain the following theorem.

Theorem 2.2. 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=1𝑛+π‘Ÿ+𝑠+1π‘›βˆ’2ξ“π‘˜=1βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ Γ—βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ“π‘›+π‘Ÿ+𝑠+10β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ π‘˜+π‘Ÿβˆ’π‘›βˆ’1β‰€π‘Žβ‰€π‘ŸβŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘π‘›+π‘Ž+1βˆ’π‘˜βˆ’π‘Ÿξ“βˆžπ‘‘=0𝑖1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=𝑑𝐡𝑖1β‹―π΅π‘–π‘ŽπΈπ‘—1β‹―πΈπ‘—π‘βˆ’ξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜+1𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1β‹―πΈπ‘—π‘ βŽ«βŽͺβŽͺ⎬βŽͺβŽͺβŽ­π΅π‘˜+1(π‘₯)2βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ π΅π‘›+π‘Ÿ+π‘ π‘›βˆ’1π‘›βˆ’1(βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π΅π‘₯)+𝑛+π‘Ÿ+𝑠𝑛(+π‘₯)𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠𝑖+𝑑=𝑛1𝑙1=0β‹―π‘–π‘Ÿξ“π‘™π‘Ÿπ‘—=01𝑝1=0⋯𝑗𝑠𝑝𝑠=0⎧βŽͺ⎨βŽͺβŽ©βŽ›βŽœβŽœβŽπ‘–1𝑙1βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘–π‘Ÿπ‘™π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘—1𝑝1βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘—π‘ π‘π‘ βŽžβŽŸβŽŸβŽ Γ—π΅π‘–1βˆ’π‘™1β‹―π΅π‘–π‘Ÿβˆ’π‘™π‘ŸπΈπ‘—1βˆ’π‘1β‹―πΈπ‘—π‘ βˆ’π‘π‘ Γ—1𝑙1+β‹―+π‘™π‘Ÿ+𝑝1+β‹―π‘π‘ βŽ«βŽͺ⎬βŽͺ⎭.+𝑑+1(2.11)

Consider the following polynomial of degree 𝑛: 𝑝(π‘₯)=βˆžξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=𝑛1𝑖1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!𝐡𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯).(2.12) Then, from (2.12), one has 𝑝(π‘˜)(π‘₯)=(π‘Ÿ+𝑠)π‘˜ξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜π΅π‘–1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯)𝑖1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!.(2.13) By (1.4) and (2.13), one gets, for π‘˜=1,2,…,𝑛, π‘Žπ‘˜=1ξ€½π‘π‘˜!(π‘˜βˆ’1)(1)βˆ’π‘(π‘˜βˆ’1)ξ€Ύ=(0)(π‘Ÿ+𝑠)π‘˜βˆ’1ξ“π‘˜!𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=π‘›βˆ’π‘˜+1𝐡𝑖1(1)β‹―π΅π‘–π‘Ÿ(1)𝐸𝑗1(1)⋯𝐸𝑗𝑠(1)βˆ’π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠𝑖1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!ξ‚Ό=(π‘Ÿ+𝑠)π‘˜βˆ’1⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ“π‘˜!0β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ π‘˜+π‘Ÿβˆ’π‘›βˆ’1β‰€π‘Žβ‰€π‘ŸβŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘ξ“π‘–1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=𝑛+π‘Ž+1βˆ’π‘˜βˆ’π‘Ÿπ΅π‘–1β‹―π΅π‘–π‘ŽπΈπ‘—1⋯𝐸𝑗𝑐𝑖1!𝑖2!β‹―π‘–π‘Ž!𝑗1!⋯𝑗𝑐!βˆ’ξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜+11𝑖1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1β‹―πΈπ‘—π‘ βŽ«βŽͺβŽͺ⎬βŽͺβŽͺ⎭.(2.14) Now look at π‘Žπ‘› and π‘Žπ‘›βˆ’1: π‘Žπ‘›=(π‘Ÿ+𝑠)π‘›βˆ’1𝑛!𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=1𝐡𝑖1(1)β‹―π΅π‘–π‘Ÿ(1)𝐸𝑗1(1)⋯𝐸𝑗𝑠(1)βˆ’π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠𝑖1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!ξ‚Ό=(π‘Ÿ+𝑠)π‘›βˆ’11𝑛!2ξ‚€βˆ’1(π‘Ÿ+𝑠)βˆ’2=(π‘Ÿ+𝑠)(π‘Ÿ+𝑠)𝑛,π‘Žπ‘›!π‘›βˆ’1=(π‘Ÿ+𝑠)π‘›βˆ’2(π‘›βˆ’1)!𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=2𝐡𝑖1(1)β‹―π΅π‘–π‘Ÿ(1)𝐸𝑗1(1)⋯𝐸𝑗𝑠(1)βˆ’π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠𝑖1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!ξ‚Ό=(π‘Ÿ+𝑠)π‘›βˆ’2⎧βŽͺ⎨βŽͺ⎩1(π‘›βˆ’1)!2161π‘Ÿ+212βŽ›βŽœβŽœβŽ2βŽžβŽŸβŽŸβŽ βˆ’1π‘Ÿ+𝑠216ξ‚€1π‘Ÿβˆ’2βˆ’12ξ‚βŽ›βŽœβŽœβŽ2⎞⎟⎟⎠⎫βŽͺ⎬βŽͺβŽ­π‘Ÿ+𝑠=0.(2.15)

It is easy to show that π‘Ž0=ξ€œ10𝑝(𝑑)𝑑𝑑=𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=𝑛1𝑖1!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!×𝑖1𝑙1=0β‹―π‘–π‘Ÿξ“π‘™π‘Ÿπ‘—=01𝑝1=0⋯𝑗𝑠𝑝𝑠=0⎧βŽͺ⎨βŽͺβŽ©π΅π‘–1βˆ’π‘™1β‹―π΅π‘–π‘Ÿβˆ’π‘™π‘ŸπΈπ‘—1βˆ’π‘1πΈπ‘—π‘ βˆ’π‘π‘ π‘™1+β‹―+π‘™π‘Ÿ+𝑝1+β‹―π‘π‘ βŽ›βŽœβŽœβŽπ‘–+11𝑙1βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘–π‘Ÿπ‘™π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘—1𝑝1βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘—π‘ π‘π‘ βŽžβŽŸβŽŸβŽ βŽ«βŽͺ⎬βŽͺ⎭.(2.16) Therefore, by (1.3), (2.14), and (2.15), one obtains the following theorem.

Theorem 2.3. For π‘›βˆˆβ„• with 𝑛β‰₯2, one has 𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=𝑛1𝑖1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!𝐡𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠=(π‘₯)π‘›βˆ’2ξ“π‘˜=1(π‘Ÿ+𝑠)π‘˜βˆ’1⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ“π‘˜!0β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ π‘˜+π‘Ÿβˆ’π‘›βˆ’1β‰€π‘Žβ‰€π‘ŸβŽ›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‘Ž+1βˆ’π‘˜βˆ’π‘Ÿπ΅π‘–1β‹―π΅π‘–π‘ŽπΈπ‘—1⋯𝐸𝑗𝑐𝑖1!𝑖2!β‹―π‘–π‘Ž!𝑗1!⋯𝑗𝑐!βˆ’ξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜+11𝑖1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!×𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1β‹―πΈπ‘—π‘ βŽ«βŽͺβŽͺ⎬βŽͺβŽͺβŽ­π΅π‘˜(π‘₯)+(π‘Ÿ+𝑠)𝑛𝐡𝑛!𝑛+(π‘₯)𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠𝑖=𝑛1𝑙1=0β‹―π‘–π‘Ÿξ“π‘™π‘Ÿπ‘—=01𝑝1=0⋯𝑗𝑠𝑝𝑠=0βŽ›βŽœβŽœβŽπ‘–1𝑙1βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘–π‘Ÿπ‘™π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘—1𝑝1βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘—π‘ π‘π‘ βŽžβŽŸβŽŸβŽ Γ—π΅π‘–1βˆ’π‘™1β‹―π΅π‘–π‘Ÿβˆ’π‘™π‘ŸπΈπ‘—1βˆ’π‘1πΈπ‘—π‘ βˆ’π‘π‘ π‘–1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!𝑙1+β‹―+π‘™π‘Ÿ+𝑝1+⋯𝑝𝑠.+1(2.17)

Take the polynomial 𝑝(π‘₯) of degree 𝑛 as follows: 𝑝(π‘₯)=𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=𝑛1𝑖1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠𝐡!𝑑!𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯)π‘₯𝑑.(2.18) Then, from (2.18), one gets 𝑝(π‘˜)(π‘₯)=(π‘Ÿ+𝑠+1)π‘˜Γ—ξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=π‘›βˆ’π‘˜1𝑖1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠𝐡!𝑑!𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯)π‘₯𝑑.(2.19) By (1.4) and (2.19), one gets, for π‘˜=1,…,𝑛, π‘Žπ‘˜=1ξ€½π‘π‘˜!(π‘˜βˆ’1)(1)βˆ’π‘(π‘˜βˆ’1)ξ€Ύ=(0)(π‘Ÿ+𝑠+1)π‘˜βˆ’1ξ“π‘˜!𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=π‘›βˆ’π‘˜+11𝑖1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠×𝐡!𝑑!𝑖1(1)β‹―π΅π‘–π‘Ÿ(1)𝐸𝑗1(1)⋯𝐸𝑗𝑠(1)βˆ’π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠0𝑑=(π‘Ÿ+𝑠+1)π‘˜βˆ’1⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ“π‘˜!0β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ π‘˜+π‘Ÿβˆ’π‘›βˆ’1β‰€π‘Žβ‰€π‘ŸβŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘π‘›+π‘Ž+1βˆ’π‘˜βˆ’π‘Ÿξ“π‘‘=01×(𝑛+π‘Ž+1βˆ’π‘˜βˆ’π‘Ÿβˆ’π‘‘)!𝑖1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=𝑑1𝑖1!𝑖2!β‹―π‘–π‘Ž!𝑗1!⋯𝑗𝑐!𝐡𝑖1β‹―π΅π‘–π‘ŽπΈπ‘—1β‹―πΈπ‘—π‘βˆ’ξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜+1𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1β‹―πΈπ‘—π‘ βŽ«βŽͺβŽͺ⎬βŽͺβŽͺ⎭.(2.20)

Now look at π‘Žπ‘› and π‘Žπ‘›βˆ’1: π‘Žπ‘›=(π‘Ÿ+𝑠+1)π‘›βˆ’1𝑛!𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=11𝑖1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠×𝐡!𝑑!𝑖1(1)β‹―π΅π‘–π‘Ÿ(1)𝐸𝑗1(1)⋯𝐸𝑗𝑠(1)βˆ’π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠0𝑑=(π‘Ÿ+𝑠+1)π‘›βˆ’11𝑛!2(ξ‚€βˆ’1π‘Ÿ+𝑠)+1βˆ’2(=π‘Ÿ+𝑠)(π‘Ÿ+𝑠+1)π‘›βˆ’1𝑛!(π‘Ÿ+𝑠+1)=(π‘Ÿ+𝑠+1)𝑛,π‘Žπ‘›!π‘›βˆ’1=(π‘Ÿ+𝑠+1)π‘›βˆ’2(π‘›βˆ’1)!𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=21𝑖1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠×𝐡!𝑑!𝑖1(1)β‹―π΅π‘–π‘Ÿ(1)𝐸𝑗1(1)⋯𝐸𝑗𝑠(1)βˆ’π΅π‘–1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠0𝑑=(π‘Ÿ+𝑠+1)π‘›βˆ’2⎧βŽͺ⎨βŽͺ⎩1(π‘›βˆ’1)!2161π‘Ÿ+2+1212βŽ›βŽœβŽœβŽ2⎞⎟⎟⎠+1π‘Ÿ+𝑠21(π‘Ÿ+𝑠)βˆ’216ξ‚€βˆ’1π‘Ÿβˆ’2βˆ’12ξ‚βŽ›βŽœβŽœβŽ2⎞⎟⎟⎠⎫βŽͺ⎬βŽͺ⎭=π‘Ÿ+𝑠(π‘Ÿ+𝑠+1)π‘›βˆ’2(π‘›βˆ’1)!π‘Ÿ+𝑠+12=(π‘Ÿ+𝑠+1)π‘›βˆ’1.2(π‘›βˆ’1)!(2.21) From (2.18), one can derive the following identity: π‘Ž0=ξ€œ10=𝑝(𝑑)𝑑𝑑𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=𝑛1𝑖1!β‹―π‘–π‘Ÿ!𝑗1!β‹―π‘—π‘ ξ€œ!𝑑!10𝐡𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯)π‘₯𝑑=𝑑𝑑𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=𝑛1𝑖1!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!𝑑!𝑖1𝑙1=0β‹―π‘–π‘Ÿξ“π‘™π‘Ÿπ‘—=01𝑝1=0⋯×𝑗𝑠𝑝𝑠=0βŽ›βŽœβŽœβŽπ‘–1𝑙1βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘–π‘Ÿπ‘™π‘ŸβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘—1𝑝1βŽžβŽŸβŽŸβŽ β‹―βŽ›βŽœβŽœβŽπ‘—π‘ π‘π‘ βŽžβŽŸβŽŸβŽ π΅π‘–1βˆ’π‘™1β‹―π΅π‘–π‘Ÿβˆ’π‘™π‘ŸπΈπ‘—1βˆ’π‘1πΈπ‘—π‘ βˆ’π‘π‘ 1𝑙1+β‹―+π‘™π‘Ÿ+𝑝1+⋯𝑝𝑠.+𝑑+1(2.22) Therefore, by (1.3), (2.20), (2.21), and (2.22), one obtains the following theorem.

Theorem 2.4. For π‘›βˆˆβ„• with 𝑛β‰₯2, one has 𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠+𝑑=𝑛1𝑖1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠𝐡!𝑑!𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)𝐸𝑗1(π‘₯)⋯𝐸𝑗𝑠(π‘₯)π‘₯𝑑=π‘›βˆ’2ξ“π‘˜=1(π‘Ÿ+𝑠+1)π‘˜βˆ’1⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ“π‘˜!0β‰€π‘Žβ‰€π‘Ÿ0β‰€π‘β‰€π‘ π‘˜+π‘Ÿβˆ’π‘›βˆ’1β‰€π‘Žβ‰€π‘ŸβŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ π‘βŽžβŽŸβŽŸβŽ (βˆ’1)𝑐2π‘ βˆ’π‘π‘›+π‘Ž+1βˆ’π‘˜βˆ’π‘Ÿξ“π‘‘=01(×𝑛+π‘Ž+1βˆ’π‘˜βˆ’π‘Ÿβˆ’π‘‘)!𝑖1+β‹―+π‘–π‘Ž+𝑗1+β‹―+𝑗𝑐=𝑑1𝑖1!𝑖2!β‹―π‘–π‘Ž!𝑗1!⋯𝑗𝑐!𝐡𝑖1β‹―π΅π‘–π‘ŽπΈπ‘—1β‹―πΈπ‘—π‘βˆ’ξ“π‘–1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠=π‘›βˆ’π‘˜+1𝐡𝑖1β‹―π΅π‘–π‘ŸπΈπ‘—1⋯𝐸𝑗𝑠𝑖1!𝑖2!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠!⎫βŽͺβŽͺ⎬βŽͺβŽͺβŽ­π΅π‘˜+(π‘₯)(π‘Ÿ+𝑠+1)π‘›βˆ’1𝐡2(π‘›βˆ’1)!π‘›βˆ’1(π‘₯)+(π‘Ÿ+𝑠+1)𝑛𝑛!𝑖1+β‹―+π‘–π‘Ÿ+𝑗1+β‹―+𝑗𝑠𝑖+𝑑=𝑛1𝑙1=0β‹―π‘–π‘Ÿξ“π‘™π‘Ÿπ‘—=01𝑝1=0⋯𝑗𝑠𝑝𝑠=0×𝑖1𝑙1ξ‚β‹―ξ‚€π‘–π‘Ÿπ‘™π‘Ÿξ‚ξ€·π‘—1𝑝1⋯𝑗𝑠𝑝𝑠𝑖1!β‹―π‘–π‘Ÿ!𝑗1!⋯𝑗𝑠𝐡!𝑑!𝑖1βˆ’π‘™1β‹―π΅π‘–π‘Ÿβˆ’π‘™π‘ŸπΈπ‘—1βˆ’π‘1πΈπ‘—π‘ βˆ’π‘π‘ 1𝑙1+β‹―+π‘™π‘Ÿ+𝑝1+⋯𝑝𝑠.+𝑑+1(2.23)

Acknowledgments

This research 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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