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Abstract and Applied Analysis
VolumeΒ 2012, Article IDΒ 957350, 15 pages
http://dx.doi.org/10.1155/2012/957350
Research Article

Extended Laguerre Polynomials Associated with Hermite, Bernoulli, and Euler Numbers and Polynomials

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

Received 14 June 2012; Accepted 9 August 2012

Academic Editor: PekkaΒ Koskela

Copyright Β© 2012 Taekyun Kim 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

Let 𝐏𝑛={𝑝(π‘₯)βˆˆβ„[π‘₯]∣deg𝑝(π‘₯)≀𝑛} be an inner product space with the inner product βˆ«βŸ¨π‘(π‘₯),π‘ž(π‘₯)⟩=∞0π‘₯π›Όπ‘’βˆ’π‘₯𝑝(π‘₯)π‘ž(π‘₯)𝑑π‘₯, where 𝑝(π‘₯),π‘ž(π‘₯)βˆˆππ‘› and π›Όβˆˆβ„ with 𝛼>βˆ’1. In this paper we study the properties of the extended Laguerre polynomials which are an orthogonal basis for 𝐏𝑛. From those properties, we derive some interesting relations and identities of the extended Laguerre polynomials associated with Hermite, Bernoulli, and Euler numbers and polynomials.

1. Introduction/Preliminaries

For π›Όβˆˆβ„ with 𝛼>βˆ’1, the extended Laguerre polynomials are defined by the generating function as follows: exp(βˆ’π‘₯𝑑/(1βˆ’π‘‘))(1βˆ’π‘‘)𝛼+1=βˆžξ“π‘›=0𝐿𝛼𝑛(π‘₯)𝑑𝑛,(1.1) see [1–6].

From (1.1), we can derive the following: 𝐿𝛼𝑛(π‘₯)=π‘›ξ“π‘Ÿ=0(βˆ’1)π‘Ÿ(𝑛+π›Όπ‘›βˆ’π‘Ÿ)π‘₯π‘Ÿ!π‘Ÿ,(1.2) see [1–9].

As is well known, Rodrigues' formula for 𝐿𝛼𝑛(π‘₯) is given by 𝐿𝛼𝑛1(π‘₯)=π‘₯𝑛!βˆ’π›Όπ‘’π‘₯𝑑𝑛𝑑π‘₯π‘›π‘’βˆ’π‘₯π‘₯𝑛+𝛼,(1.3) see [1–6, 8, 9].

From (1.3), we note that ξ€œβˆž0π‘₯π›Όπ‘’βˆ’π‘₯πΏπ›Όπ‘š(π‘₯)𝐿𝛼𝑛1(π‘₯)𝑑π‘₯=𝑛!Ξ“(𝛼+𝑛+1)π›Ώπ‘š,𝑛,(𝛼>βˆ’1),(1.4) where π›Ώπ‘š,𝑛 is the Kronecker symbol.

From (1.1), (1.2), and (1.3), we can derive the following identities: (𝑛+1)𝐿𝛼𝑛+1(π‘₯)+(π‘₯βˆ’π›Όβˆ’2π‘›βˆ’1)𝐿𝛼𝑛(π‘₯)+(𝑛+𝛼)πΏπ›Όπ‘›βˆ’1(𝑑π‘₯)=0,(π‘›βˆˆβ„•),(1.5)𝐿𝑑π‘₯𝛼𝑛𝑑(π‘₯)βˆ’πΏπ‘‘π‘₯π›Όπ‘›βˆ’1(π‘₯)+πΏπ›Όπ‘›βˆ’1π‘₯𝑑(π‘₯)=0,for𝑛β‰₯1,(1.6)𝐿𝑑π‘₯𝛼𝑛(π‘₯)=𝑛𝐿𝛼𝑛(π‘₯)βˆ’(𝑛+𝛼)πΏπ›Όπ‘›βˆ’1(π‘₯)=0,(𝑛β‰₯1),(1.7) and 𝐿𝛼𝑛(π‘₯) is a solution of π‘₯π‘¦ξ…žξ…ž+(𝛼+1βˆ’π‘₯)π‘¦ξ…ž+π‘₯𝑦=0.

The derivatives of general Laguerre polynomials are given by 𝑑𝐿𝑑π‘₯𝛼𝑛(π‘₯)=βˆ’πΏπ›Ό+1π‘›βˆ’1𝑑(π‘₯),ξ€·π‘₯𝑑π‘₯𝛼𝐿𝛼𝑛(π‘₯)=(𝑛+𝛼)π‘₯π›Όβˆ’1πΏπ‘›π›Όβˆ’1𝑑(π‘₯),𝑒𝑑π‘₯βˆ’π‘₯𝐿𝛼𝑛(π‘₯)=βˆ’π‘’βˆ’π‘₯𝐿𝑛𝛼+1𝑑(π‘₯),ξ€·π‘₯𝑑π‘₯π›Όπ‘’βˆ’π‘₯𝐿𝛼𝑛(π‘₯)=(𝑛+1)π‘₯π›Όβˆ’1π‘’βˆ’π‘₯πΏπ›Όβˆ’1𝑛+1(π‘₯).(1.8) The 𝑛th Bernoulli polynomials, 𝐡𝑛(π‘₯), are defined by the generating function to be π‘‘π‘’π‘‘π‘’βˆ’1π‘₯𝑑=𝑒𝐡(π‘₯)𝑑=βˆžξ“π‘›=0𝐡𝑛(𝑑π‘₯)𝑛,𝑛!(1.9) see [10–17], with the usual convention about replacing 𝐡𝑛(π‘₯) by 𝐡𝑛(π‘₯). In the special case, π‘₯=0, 𝐡𝑛(0)=𝐡𝑛 are called the 𝑛th Bernoulli numbers.

It is well known that the 𝑛th Euler polynomials are also defined by the generating function to be 2𝑒𝑑𝑒+1π‘₯𝑑=𝑒𝐸(π‘₯)𝑑=βˆžξ“π‘›=0𝐸𝑛(𝑑π‘₯)𝑛,𝑛!(1.10) see [18–22], with the usual convention about replacing 𝐸𝑛(π‘₯) by 𝐸𝑛(π‘₯).

The Hermite polynomials are given by 𝐻𝑛(π‘₯)=(𝐻+2π‘₯)𝑛=𝑛𝑙=0βŽ›βŽœβŽœβŽπ‘›π‘™βŽžβŽŸβŽŸβŽ 2𝑙π‘₯π‘™π»π‘›βˆ’π‘™,(1.11) see [23, 24], with the usual convention about replacing 𝐻𝑛 by 𝐻𝑛. In the special case, π‘₯=0, 𝐻𝑛(0)=𝐻𝑛 are called the 𝑛th Hermite numbers.

From (1.11), we note that 𝑑𝐻𝑑π‘₯𝑛(π‘₯)=2𝑛(𝐻+2π‘₯)π‘›βˆ’1=2π‘›π»π‘›βˆ’1(π‘₯),(1.12) see [23, 24], and 𝐻𝑛(π‘₯) is a solution of Hermite differential equation which is given by π‘¦ξ…žξ…žβˆ’2π‘₯π‘¦ξ…ž+𝑛𝑦=0,(1.13) (see [1–6, 23–32]).

Throughout this paper we assume that π›Όβˆˆβ„ with 𝛼>βˆ’1. Let 𝐏𝑛={𝑝(π‘₯)βˆˆβ„[π‘₯]|deg𝑝(π‘₯)≀𝑛}. Then 𝐏𝑛 is an inner product space with the inner product βˆ«βŸ¨π‘(π‘₯),π‘ž(π‘₯)⟩=∞0π‘₯π›Όπ‘’βˆ’π‘₯𝑝(π‘₯)π‘ž(π‘₯)𝑑π‘₯, where 𝑝(π‘₯),π‘ž(π‘₯)βˆˆππ‘›. By (1.4) the set of the extended Laguerre polynomials {𝐿𝛼0(π‘₯),𝐿𝛼1(π‘₯),…,𝐿𝛼𝑛(π‘₯)} is an orthogonal basis for 𝐏𝑛. In this paper we study the properties of the extended Laguerre polynomials which are an orthogonal basis for 𝐏𝑛. From those properties, we derive some new and interesting relations and identities of the extended Laguerre polynomials associated with Hermite, Bernoulli and Euler numbers and polynomials.

2. On the Extended Laguerre Polynomials Associated with Hermite, Bernoulli, and Euler Polynomials

For 𝑝(π‘₯)βˆˆππ‘›, 𝑝(π‘₯) is given by 𝑝(π‘₯)=π‘›ξ“π‘˜=0πΆπ‘˜πΏπ›Όπ‘˜(π‘₯),foruniquelydeterminedrealnumbersπΆπ‘˜.(2.1) From (1.3), (1.4), and (2.1), we note that 𝑝(π‘₯),πΏπ›Όπ‘˜ξ¬(π‘₯)=πΆπ‘˜ξ«πΏπ›Όπ‘˜(π‘₯),πΏπ›Όπ‘˜ξ¬(π‘₯)=πΆπ‘˜ξ€œβˆž0π‘₯π›Όπ‘’βˆ’π‘₯πΏπ›Όπ‘˜(π‘₯)πΏπ›Όπ‘˜(π‘₯)𝑑π‘₯=πΆπ‘˜Ξ“(𝛼+π‘˜+1).π‘˜!(2.2) Thus, by (2.2), we get πΆπ‘˜=π‘˜!Γ(𝛼+π‘˜+1)𝑝(π‘₯),πΏπ›Όπ‘˜ξ¬=(π‘₯)π‘˜!Ξ“1(𝛼+π‘˜+1)ξ€œπ‘˜!∞0ξ‚΅π‘‘π‘˜π‘‘π‘₯π‘˜π‘₯π‘˜+π›Όπ‘’βˆ’π‘₯ξ‚Ά=1𝑝(π‘₯)𝑑π‘₯ξ€œΞ“(𝛼+π‘˜+1)∞0ξ‚΅π‘‘π‘˜π‘‘π‘₯π‘˜π‘₯π‘˜+π›Όπ‘’βˆ’π‘₯𝑝(π‘₯)𝑑π‘₯.(2.3) Therefore, by (2.1) and (2.3), we obtain the following proposition.

Proposition 2.1. For 𝑝(π‘₯)βˆˆππ‘›, let 𝑝(π‘₯)=π‘›ξ“π‘˜=0πΆπ‘˜πΏπ›Όπ‘˜(π‘₯),(𝛼>βˆ’1).(2.4) Then one has the following: πΆπ‘˜=1ξ€œΞ“(𝛼+π‘˜+1)∞0ξ‚΅π‘‘π‘˜π‘‘π‘₯π‘˜π‘₯π‘˜+π›Όπ‘’βˆ’π‘₯𝑝(π‘₯)𝑑π‘₯.(2.5)

To derive inverse formula of (1.2), let take one 𝑝(π‘₯)=π‘₯π‘›βˆˆππ‘›. Then, by Proposition 2.1, one gets πΆπ‘˜=1ξ€œΞ“(𝛼+π‘˜+1)∞0ξ‚΅π‘‘π‘˜π‘‘π‘₯π‘˜π‘’βˆ’π‘₯π‘₯π‘˜+𝛼π‘₯𝑛=𝑑π‘₯(βˆ’1)π‘˜π‘›(π‘›βˆ’1)β‹―(π‘›βˆ’π‘˜+1)ξ€œΞ“(𝛼+π‘˜+1)∞0π‘’βˆ’π‘₯π‘₯𝛼+𝑛𝑑π‘₯=(βˆ’1)π‘˜π‘›(π‘›βˆ’1)β‹―(π‘›βˆ’π‘˜+1)=Ξ“(𝛼+π‘˜+1)Ξ“(𝛼+𝑛+1)(βˆ’1)π‘˜π‘›!(𝛼+𝑛)⋯𝛼Γ(𝛼)(𝛼+π‘˜)⋯𝛼Γ(𝛼)(π‘›βˆ’π‘˜)!=(βˆ’1)π‘˜π‘›!(𝛼+𝑛)β‹―(𝛼+π‘˜+1)(π‘›βˆ’π‘˜)!=(βˆ’1)π‘˜βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ .𝑛!𝛼+π‘›π‘›βˆ’π‘˜(2.6) Therefore, by (2.6), we obtain the following corollary.

Corollary 2.2 (Inverse formula of 𝐿𝛼𝑛(π‘₯)). For π‘›βˆˆZ+, one has π‘₯𝑛=𝑛!π‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ π›Ό+π‘›π‘›βˆ’π‘˜(βˆ’1)π‘˜πΏπ›Όπ‘˜(π‘₯).(2.7)

Let one takes Bernoulli polynomials of degree 𝑛 with 𝑝(π‘₯)=𝐡𝑛(π‘₯)βˆˆππ‘›. Then 𝐡𝑛(π‘₯) can be written as 𝐡𝑛(π‘₯)=π‘›ξ“π‘˜=0πΆπ‘˜πΏπ›Όπ‘˜(π‘₯),(π›Όβˆˆβ„with𝛼>βˆ’1).(2.8) From Proposition 2.1, one has πΆπ‘˜=1ξ€œΞ“(𝛼+π‘˜+1)∞0ξ‚΅π‘‘π‘˜π‘‘π‘₯π‘˜π‘’βˆ’π‘₯π‘₯π‘˜+𝛼𝐡𝑛=(π‘₯)𝑑π‘₯(βˆ’1)π‘˜π‘›(π‘›βˆ’1)β‹―(π‘›βˆ’π‘˜+1)ξ€œΞ“(𝛼+π‘˜+1)∞0π‘’βˆ’π‘₯π‘₯π‘˜+π›Όπ΅π‘›βˆ’π‘˜=(π‘₯)𝑑π‘₯(βˆ’1)π‘˜π‘›(π‘›βˆ’1)β‹―(π‘›βˆ’π‘˜+1)Ξ“(𝛼+π‘˜+1)π‘›βˆ’π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π΅π‘›βˆ’π‘˜π‘›βˆ’π‘˜βˆ’π‘™ξ€œβˆž0π‘’βˆ’π‘₯π‘₯π‘˜+𝛼+𝑙=𝑑π‘₯(βˆ’1)π‘˜π‘›(π‘›βˆ’1)β‹―(π‘›βˆ’π‘˜+1)Ξ“(𝛼+π‘˜+1)π‘›βˆ’π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π΅π‘›βˆ’π‘˜π‘›βˆ’π‘˜βˆ’π‘™Ξ“(𝛼+π‘˜+𝑙+1).(2.9) By the fundamental property of gamma function, one gets Ξ“(𝛼+π‘˜+𝑙+1)βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ =(Ξ“(𝛼+π‘˜+1)(π‘›βˆ’π‘˜)!π‘›βˆ’π‘˜π›Ό+𝑙+π‘˜)β‹―(𝛼+π‘˜+1)Ξ“(𝛼+π‘˜+1)(π‘›βˆ’π‘˜)!=ξ€·Ξ“(𝛼+π‘˜+1)(π‘›βˆ’π‘˜)!(π‘›βˆ’π‘˜βˆ’π‘™)!𝑙!𝑙𝛼+π‘˜+𝑙(π‘›βˆ’π‘˜βˆ’π‘™)!.(2.10) Therefore, by (2.8), (2.9), and (2.10), we obtain the following theorem.

Theorem 2.3. For π‘›βˆˆβ„€+, π›Όβˆˆβ„ with 𝛼>βˆ’1, one has 𝐡𝑛(π‘₯)=𝑛!π‘›ξ“π‘˜=0π‘›βˆ’π‘˜ξ“π‘™=0(βˆ’1)π‘˜βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π΅π›Ό+π‘˜+π‘™π‘›βˆ’π‘˜βˆ’π‘™πΏ(π‘›βˆ’π‘˜βˆ’π‘™)!π›Όπ‘˜(π‘₯).(2.11)

As is known, relationships between Hermite and Laguerre polynomials are given by 𝐻2π‘š(π‘₯)=(βˆ’1)π‘š22π‘šπ‘š!πΏπ‘šβˆ’1/2ξ€·π‘₯2ξ€Έ,𝐻(2.12)2π‘š+1(π‘₯)=(βˆ’1)π‘š22π‘š+1π‘š!πΏπ‘šβˆ’1/2ξ€·π‘₯2ξ€Έ,(2.13) see [1–6]. In the special case 𝛼=βˆ’1/2, by (2.12) and (2.13), we obtain the following corollary.

Corollary 2.4. For π‘›βˆˆβ„€+, one has 𝐡𝑛π‘₯2ξ€Έ=𝑛!π‘›ξ“π‘˜=0π‘›βˆ’π‘˜ξ“π‘™=0𝐻2π‘˜(π‘₯)22π‘˜βŽ›βŽœβŽœβŽβˆ’1π‘˜!2π‘™βŽžβŽŸβŽŸβŽ π΅+π‘˜+π‘™π‘›βˆ’π‘˜βˆ’π‘™(π‘›βˆ’π‘˜βˆ’π‘™)!.(2.14)

By the same method as Theorem 2.3, one gets 𝐸𝑛(π‘₯)=𝑛!π‘›ξ“π‘˜=0π‘›βˆ’π‘˜ξ“π‘™=0(βˆ’1)π‘˜βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ πΈπ›Ό+π‘˜+π‘™π‘›βˆ’π‘˜βˆ’π‘™πΏ(π‘›βˆ’π‘˜βˆ’π‘™)!π›Όπ‘˜(π‘₯),(2.15) where 𝐸𝑛(π‘₯) are the 𝑛th Euler polynomials. In the special case, π‘₯=0, 𝐸𝑛(0)=𝐸𝑛 are called the 𝑛th Euler numbers.

Let one considers the 𝑛th Hermite polynomials with 𝑝(π‘₯)=𝐻𝑛(π‘₯)βˆˆππ‘›. Then 𝐻𝑛(π‘₯) can be written as 𝐻𝑛(π‘₯)=π‘›ξ“π‘˜=0πΆπ‘˜πΏπ›Όπ‘˜(π‘₯),(π›Όβˆˆβ„with𝛼>βˆ’1).(2.16) From Proposition 2.1, one notes that πΆπ‘˜=1ξ€œΞ“(𝛼+π‘˜+1)∞0ξ‚΅π‘‘π‘˜π‘‘π‘₯π‘˜π‘’βˆ’π‘₯π‘₯π‘˜+𝛼𝐻𝑛=(π‘₯)𝑑π‘₯(βˆ’2𝑛)ξ€œΞ“(𝛼+π‘˜+1)∞0ξ‚΅π‘‘π‘˜βˆ’1𝑑π‘₯π‘˜βˆ’1π‘’βˆ’π‘₯π‘₯π‘˜+π›Όξ‚Άπ»π‘›βˆ’1=(π‘₯)𝑑π‘₯=β‹―(βˆ’2𝑛)(βˆ’2(π‘›βˆ’1))β‹―(βˆ’2(π‘›βˆ’π‘˜+1))ξ€œΞ“(𝛼+π‘˜+1)∞0π‘’βˆ’π‘₯π‘₯π‘˜+π›Όπ»π‘›βˆ’π‘˜=(π‘₯)𝑑π‘₯(βˆ’1)π‘˜2π‘˜π‘›!Ξ“(𝛼+π‘˜+1)(π‘›βˆ’π‘˜)!π‘›βˆ’π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π»π‘›βˆ’π‘˜π‘›βˆ’π‘˜βˆ’π‘™2π‘™ξ€œβˆž0π‘’βˆ’π‘₯π‘₯π‘˜+𝛼+𝑙=𝑑π‘₯(βˆ’1)π‘˜2π‘˜π‘›!Ξ“(𝛼+π‘˜+1)(π‘›βˆ’π‘˜)!π‘›βˆ’π‘˜ξ“π‘™=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π»π‘›βˆ’π‘˜π‘›βˆ’π‘˜βˆ’π‘™2𝑙Γ(𝛼+π‘˜+𝑙+1).(2.17) It is not difficult to show that ξ€·π‘™π‘›βˆ’π‘˜ξ€ΈΞ“(𝛼+π‘˜+𝑙+1)=Ξ“(𝛼+π‘˜+1)(π‘›βˆ’π‘˜)!(π‘›βˆ’π‘˜)!(𝛼+π‘˜+𝑙)β‹―(𝛼+π‘˜+1)Ξ“(𝛼+π‘˜+1)=ξ€·(π‘›βˆ’π‘˜βˆ’π‘™)!𝑙!Ξ“(𝛼+π‘˜+1)(π‘›βˆ’π‘˜)!𝑙𝛼+π‘˜+𝑙.(π‘›βˆ’π‘˜βˆ’π‘™)!(2.18) Therefore, by (2.16), (2.17), and (2.18), we obtain the following theorem.

Theorem 2.5. For π‘›βˆˆβ„€+, π›Όβˆˆβ„ with 𝛼>βˆ’1, one has 𝐻𝑛(π‘₯)=𝑛!π‘›ξ“π‘˜=0π‘›βˆ’π‘˜ξ“π‘™=0(βˆ’1)π‘˜2π‘˜+π‘™βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π»π›Ό+π‘˜+π‘™π‘›βˆ’π‘˜βˆ’π‘™πΏ(π‘›βˆ’π‘˜βˆ’π‘™)!π›Όπ‘˜(π‘₯).(2.19)

In the special case, 𝛼=βˆ’1/2, we obtain the following corollary.

Corollary 2.6. For π‘›βˆˆβ„€+, one has 𝐻𝑛π‘₯2ξ€Έ=𝑛!π‘›ξ“π‘˜=0π‘›βˆ’π‘˜ξ“π‘™=02π‘™βˆ’π‘˜π»2π‘˜(π‘₯)βŽ›βŽœβŽœβŽβˆ’1π‘˜!2π‘™βŽžβŽŸβŽŸβŽ π»+π‘˜+π‘™π‘›βˆ’π‘˜βˆ’π‘™(π‘›βˆ’π‘˜βˆ’π‘™)!.(2.20)

For π›½βˆˆβ„ with 𝛽>βˆ’1, let one takes 𝑝(π‘₯)=𝐿𝛽𝑛(π‘₯)βˆˆππ‘›.(2.21) Then 𝐿𝛽𝑛(π‘₯) is also written as 𝐿𝛽𝑛(π‘₯)=π‘›ξ“π‘˜=0πΆπ‘˜πΏπ›Όπ‘˜(π‘₯).(2.22) From Proposition 2.1, one can determine the coefficients of (2.22) as follows: πΆπ‘˜=1ξ€œΞ“(𝛼+π‘˜+1)∞0ξ‚΅π‘‘π‘˜π‘‘π‘₯π‘˜π‘’βˆ’π‘₯π‘₯π‘˜+𝛼𝐿𝛽𝑛=1(π‘₯)𝑑π‘₯ξ€œΞ“(𝛼+π‘˜+1)∞0ξ‚΅π‘‘π‘˜βˆ’1𝑑π‘₯π‘˜βˆ’1π‘’βˆ’π‘₯π‘₯π‘˜+𝛼𝐿𝛽+1π‘›βˆ’1=1(π‘₯)𝑑π‘₯=β‹―ξ€œΞ“(𝛼+π‘˜+1)∞0π‘’βˆ’π‘₯π‘₯π‘˜+𝛼𝐿𝛽+π‘˜π‘›βˆ’π‘˜=1(π‘₯)𝑑π‘₯Ξ“(𝛼+π‘˜+1)π‘›βˆ’π‘˜ξ“π‘Ÿ=0(βˆ’1)π‘Ÿξ€·π‘›+π›½π‘›βˆ’π‘˜βˆ’π‘Ÿξ€Έξ€œπ‘Ÿ!∞0π‘’βˆ’π‘₯π‘₯π‘˜+𝛼+π‘Ÿ=1𝑑π‘₯Ξ“(𝛼+π‘˜+1)π‘›βˆ’π‘˜ξ“π‘Ÿ=0(βˆ’1)π‘Ÿξ€·π‘›+π›½π‘›βˆ’π‘˜βˆ’π‘Ÿξ€Έπ‘Ÿ!Ξ“(π‘˜+𝛼+π‘Ÿ+1).(2.23) By the fundamental property of gamma function, one gets Ξ“(π‘˜+𝛼+π‘Ÿ+1)=(π‘Ÿ!Ξ“(𝛼+π‘˜+1)π‘˜+𝛼+π‘Ÿ)β‹―(𝛼+π‘˜+1)Ξ“(𝛼+π‘˜+1)=βŽ›βŽœβŽœβŽπ‘ŸβŽžβŽŸβŽŸβŽ π‘Ÿ!Ξ“(𝛼+π‘˜+1)π‘˜+𝛼+π‘Ÿ.(2.24) Therefore, by (2.22), (2.23), and (2.24), we obtain the following theorem.

Theorem 2.7. For π›½βˆˆβ„ with 𝛽>βˆ’1, and π‘›βˆˆβ„€+, one has 𝐿𝛽𝑛(π‘₯)=π‘›ξ“π‘˜=0π‘›βˆ’π‘˜ξ“π‘Ÿ=0(βˆ’1)π‘ŸβŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ŸβŽžβŽŸβŽŸβŽ πΏπ‘›+π›½π‘›βˆ’π‘˜βˆ’π‘Ÿπ›Ό+π‘˜+π‘Ÿπ›Όπ‘˜(π‘₯).(2.25)

In the special case, 𝛼=𝛽, one has π‘›βˆ’1ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›βˆ’π‘˜ξ“π‘Ÿ=0(βˆ’1)π‘ŸβŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ŸβŽžβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽ πΏπ‘›+π›Όπ‘›βˆ’π‘˜βˆ’π‘Ÿπ›Ό+π‘˜+π‘Ÿπ›Όπ‘˜(π‘₯)=0.(2.26) Thus, by (2.26), we obtain the following corollary.

Corollary 2.8. For 0β‰€π‘˜β‰€π‘›βˆ’1, π›Όβˆˆβ„ with 𝛼>βˆ’1, one has π‘›βˆ’π‘˜ξ“π‘Ÿ=0(βˆ’1)π‘ŸβŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ŸβŽžβŽŸβŽŸβŽ π‘›+π›Όπ‘›βˆ’π‘˜βˆ’π‘Ÿπ›Ό+π‘˜+π‘Ÿ=0.(2.27)

Let one assumes that 𝑝(π‘₯)=𝑛𝑙=0𝐡𝑙(π‘₯)π΅π‘›βˆ’π‘™(π‘₯)βˆˆππ‘›.(2.28) Then 𝑝(π‘₯) can be rewritten as a linear combination of 𝐿𝛼0(π‘₯),𝐿𝛼1(π‘₯),…,𝐿𝛼𝑛(π‘₯) as follows: 𝑝(π‘₯)=𝑛𝑙=0𝐡𝑙(π‘₯)π΅π‘›βˆ’π‘™(π‘₯)=π‘›ξ“π‘˜=0πΆπ‘˜πΏπ›Όπ‘˜(π‘₯).(2.29) By Proposition 2.1, one can determine the coefficients of (2.29) as follows: πΆπ‘˜=1Ξ“(𝛼+π‘˜+1)𝑛𝑙=0ξ€œβˆž0ξ‚΅π‘‘π‘˜π‘‘π‘₯π‘˜π‘’βˆ’π‘₯π‘₯π‘˜+𝛼𝐡𝑙(π‘₯)π΅π‘›βˆ’π‘™(π‘₯)𝑑π‘₯.(2.30) It is known that 𝑛𝑙=0𝐡𝑙(π‘₯)π΅π‘›βˆ’π‘™2(π‘₯)=𝑛+2π‘›βˆ’2𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π΅π‘›+2π‘›βˆ’π‘™π΅π‘™(π‘₯)+(𝑛+1)𝐡𝑛(π‘₯),(2.31) see [25].

From (2.30) and (2.31), one notes that 𝐢𝑛=𝑛+1ξ€œΞ“(𝛼+𝑛+1)∞0𝑑𝑛𝑑π‘₯π‘›π‘’βˆ’π‘₯π‘₯𝑛+𝛼𝐡𝑛=(π‘₯)𝑑π‘₯𝑛+1Ξ“(𝛼+𝑛+1)(βˆ’1)π‘›ξ€œπ‘›!∞0π‘’βˆ’π‘₯π‘₯𝑛+𝛼𝑑π‘₯=(𝑛+1)!(βˆ’1)𝑛ΓΓ(𝛼+𝑛+1)(𝑛+𝛼+1)=(𝑛+1)!(βˆ’1)𝑛,πΆπ‘›βˆ’1=𝑛+1ξ€œΞ“(𝛼+𝑛)∞0ξ‚΅π‘‘π‘›βˆ’1𝑑π‘₯π‘›βˆ’1π‘’βˆ’π‘₯π‘₯π‘›βˆ’1+𝛼𝐡𝑛=(π‘₯)𝑑π‘₯𝑛+1(Ξ“(𝛼+𝑛)βˆ’1)π‘›βˆ’1ξ€œπ‘›!∞0π‘’βˆ’π‘₯π‘₯π‘›βˆ’1+𝛼𝐡1(=π‘₯)𝑑π‘₯𝑛+1(Ξ“(𝛼+𝑛)βˆ’1)π‘›βˆ’11𝑛!Ξ“(𝛼+𝑛+1)βˆ’2Γ(𝛼+𝑛)=(𝑛+1)!(βˆ’1)π‘›βˆ’1ξ‚€1𝑛+π›Όβˆ’2.(2.32) For 0β‰€π‘˜β‰€π‘›βˆ’2, one has πΆπ‘˜=1⎧βŽͺ⎨βŽͺ⎩2Ξ“(𝛼+π‘˜+1)𝑛+2π‘›βˆ’2𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π΅π‘›+2π‘›βˆ’π‘™ξ€œβˆž0ξ‚΅π‘‘π‘˜π‘‘π‘₯π‘˜π‘’βˆ’π‘₯π‘₯π‘˜+π›Όξ‚Άπ΅π‘™ξ€œ(π‘₯)𝑑π‘₯+(𝑛+1)∞0ξ‚΅π‘‘π‘˜π‘‘π‘₯π‘˜π‘’βˆ’π‘₯π‘₯π‘˜+𝛼𝐡𝑛(⎫βŽͺ⎬βŽͺ⎭=1π‘₯)𝑑π‘₯⎧βŽͺ⎨βŽͺ⎩2Ξ“(𝛼+π‘˜+1)𝑛+2π‘›βˆ’2𝑙=π‘˜βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π΅π‘›+2π‘›βˆ’π‘™(βˆ’1)π‘˜ξ€œπ‘™(π‘™βˆ’1)β‹―(π‘™βˆ’π‘˜+1)∞0π‘’βˆ’π‘₯π‘₯π‘˜+π›Όπ΅π‘™βˆ’π‘˜(π‘₯)𝑑π‘₯+(𝑛+1)(βˆ’1)π‘˜ξ€œπ‘›(π‘›βˆ’1)β‹―(π‘›βˆ’π‘˜+1)∞0π‘’βˆ’π‘₯π‘₯π‘˜+π›Όπ΅π‘›βˆ’π‘˜βŽ«βŽͺ⎬βŽͺ⎭=1(π‘₯)𝑑π‘₯⎧βŽͺ⎨βŽͺ⎩2Ξ“(𝛼+π‘˜+1)𝑛+2π‘›βˆ’2𝑙=π‘˜βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π΅π‘›+2π‘›βˆ’π‘™(βˆ’1)π‘˜π‘™!(π‘™βˆ’π‘˜)!π‘™βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ π΅π‘™βˆ’π‘˜π‘™βˆ’π‘˜βˆ’π‘—ξ€œβˆž0π‘’βˆ’π‘₯π‘₯π‘˜+𝛼+𝑗+𝑑π‘₯(𝑛+1)(βˆ’1)π‘˜π‘›!(π‘›βˆ’π‘˜)!π‘›βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ π΅π‘›βˆ’π‘˜π‘›βˆ’π‘˜βˆ’π‘—ξ€œβˆž0π‘’βˆ’π‘₯π‘₯π‘˜+𝛼+π‘—βŽ«βŽͺ⎬βŽͺ⎭=1𝑑π‘₯⎧βŽͺ⎨βŽͺ⎩2Ξ“(𝛼+π‘˜+1)𝑛+2π‘›βˆ’2𝑙=π‘˜π‘™βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π΅π‘›+2π‘›βˆ’π‘™(βˆ’1)π‘˜π‘™!βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ π΅(π‘™βˆ’π‘˜)!π‘™βˆ’π‘˜π‘™βˆ’π‘˜βˆ’π‘—Ξ“(𝛼+π‘˜+𝑗+1)+(𝑛+1)(βˆ’1)π‘˜π‘›!(π‘›βˆ’π‘˜)!π‘›βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ π΅π‘›βˆ’π‘˜π‘›βˆ’π‘˜βˆ’π‘—βŽ«βŽͺ⎬βŽͺ⎭=2Ξ“(𝛼+π‘˜+𝑗+1)𝑛+2π‘›βˆ’2𝑙=π‘˜π‘™βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π΅π‘›+2π‘›βˆ’π‘™(βˆ’1)π‘˜π‘™!(𝛼+π‘˜+𝑗)β‹―(𝛼+π‘˜+1)π΅π‘™βˆ’π‘˜βˆ’π‘—π‘—!(π‘™βˆ’π‘˜βˆ’π‘—)!+(𝑛+1)(βˆ’1)π‘˜π‘›!π‘›βˆ’π‘˜ξ“π‘—=0(𝛼+π‘˜+𝑗)β‹―(𝛼+π‘˜+1)𝐡𝑗!(π‘›βˆ’π‘˜βˆ’π‘—)!π‘›βˆ’π‘˜βˆ’π‘—=2𝑛+2π‘›βˆ’2𝑙=π‘˜π‘™βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π΅π‘›+2π‘›βˆ’π‘™(βˆ’1)π‘˜βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ π΅π‘™!𝛼+π‘˜+π‘—π‘™βˆ’π‘˜βˆ’π‘—(π‘™βˆ’π‘˜βˆ’π‘—)!+(𝑛+1)(βˆ’1)π‘˜π‘›!π‘›βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ π΅π›Ό+π‘˜+π‘—π‘›βˆ’π‘˜βˆ’π‘—.(π‘›βˆ’π‘˜βˆ’π‘—)!(2.33) Therefore, by (2.29) and (2.32), we obtain the following theorem.

Theorem 2.9. For π‘›βˆˆβ„€+, π›Όβˆˆβ„ with 𝛼>βˆ’1, one has π‘›ξ“π‘˜=0π΅π‘˜(π‘₯)π΅π‘›βˆ’π‘˜=(π‘₯)π‘›βˆ’2ξ“π‘˜=0⎧βŽͺ⎨βŽͺ⎩2𝑛+2π‘›βˆ’2𝑙=π‘˜ξ“0β‰€π‘—β‰€π‘›βˆ’π‘˜(βˆ’1)π‘˜βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ π‘›+2𝑙!π΅π‘›βˆ’π‘™βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ π΅π›Ό+π‘˜+π‘—π‘™βˆ’π‘˜βˆ’π‘—ξ“(π‘™βˆ’π‘˜βˆ’π‘—)!+(𝑛+1)!0β‰€π‘—β‰€π‘›βˆ’π‘˜(βˆ’1)π‘˜βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ π΅π›Ό+π‘˜+π‘—π‘›βˆ’π‘˜βˆ’π‘—βŽ«βŽͺ⎬βŽͺ⎭𝐿(π‘›βˆ’π‘˜βˆ’π‘—)!π›Όπ‘˜(π‘₯)+(βˆ’1)𝑛𝐿(𝑛+1)!𝛼𝑛1(π‘₯)βˆ’π‘›+π›Όβˆ’2ξ‚πΏπ›Όπ‘›βˆ’1.(π‘₯)(2.34)

Let one takes the polynomial 𝑝(π‘₯) in 𝐏𝑛 as follows: 𝑝(π‘₯)=𝑖1+β‹―+π‘–π‘Ÿ=𝑛𝐡𝑖1(π‘₯)𝐡𝑖2(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)βˆˆππ‘›.(2.35) From the orthogonality of {𝐿𝛼0(π‘₯),…,𝐿𝛼𝑛(π‘₯)}, one notes that 𝑝(π‘₯)=𝑖1+β‹―+π‘–π‘Ÿ=𝑛𝐡𝑖1(π‘₯)𝐡𝑖2(π‘₯)β‹―π΅π‘–π‘Ÿ(π‘₯)=π‘›ξ“π‘˜=0πΆπ‘˜πΏπ›Όπ‘˜(π‘₯),(2.36) where πΆπ‘˜=1ξ€œΞ“(𝛼+π‘˜+1)∞0ξ‚΅π‘‘π‘˜π‘‘π‘₯π‘˜π‘’βˆ’π‘₯π‘₯π‘˜+𝛼𝑝(π‘₯)𝑑π‘₯.(2.37) It is known in [25] that 𝑖1+β‹―+π‘–π‘Ÿ=𝑛𝐡𝑖1(π‘₯)β‹―π΅π‘–π‘Ÿ(=1π‘₯)2π‘›βˆ’2ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ βŽ§βŽͺ⎨βŽͺβŽ©ξ“π‘›+π‘Ÿβˆ’1max{0,π‘˜+π‘Ÿβˆ’π‘›}β‰€π‘Žβ‰€π‘ŸβŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ ξ“π‘–1+β‹―+π‘–π‘Ž=𝑛+π‘Žβˆ’π‘˜βˆ’π‘Ÿπ΅π‘–1𝐡𝑖2β‹―π΅π‘–π‘Ž+𝑖1+β‹―+π‘–π‘Ÿ=π‘›βˆ’π‘˜π΅π‘–1β‹―π΅π‘–π‘ŸβŽ«βŽͺ⎬βŽͺβŽ­πΈπ‘˜βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ πΈ(π‘₯)+𝑛+π‘Ÿβˆ’1𝑛(π‘₯).(2.38) From (2.35), (2.37), and (2.38), one notes that 𝐢𝑛=𝑛𝑛+π‘Ÿβˆ’1ξ€Έξ€œΞ“(𝛼+𝑛+1)∞0𝑑𝑛𝑑π‘₯π‘›π‘’βˆ’π‘₯π‘₯𝑛+𝛼𝐸𝑛=ξ€·(π‘₯)𝑑π‘₯𝑛𝑛+π‘Ÿβˆ’1ξ€ΈΞ“(𝛼+𝑛+1)(βˆ’1)π‘›ξ€œπ‘›!∞0π‘₯𝑛+π›Όπ‘’βˆ’π‘₯=𝑑π‘₯𝑛𝑛+π‘Ÿβˆ’1ξ€ΈΞ“(𝛼+𝑛+1)(βˆ’1)π‘›βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π‘›!Ξ“(𝛼+𝑛+1)=𝑛+π‘Ÿβˆ’1(βˆ’1)𝑛𝐢𝑛!,π‘›βˆ’1=𝑛𝑛+π‘Ÿβˆ’1ξ€Έξ€œΞ“(𝛼+𝑛)∞0ξ‚΅π‘‘π‘›βˆ’1𝑑π‘₯π‘›βˆ’1π‘’βˆ’π‘₯π‘₯𝑛+π›Όβˆ’1𝐸𝑛=ξ€·(π‘₯)𝑑π‘₯𝑛𝑛+π‘Ÿβˆ’1ξ€ΈΞ“(𝛼+𝑛)(βˆ’1)π‘›βˆ’1ξ€œπ‘›!∞0π‘’βˆ’π‘₯π‘₯𝑛+π›Όβˆ’1𝐸1=ξ€·(π‘₯)𝑑π‘₯𝑛𝑛+π‘Ÿβˆ’1ξ€ΈΞ“(𝛼+𝑛)(βˆ’1)π‘›βˆ’11𝑛!Ξ“(𝛼+𝑛+1)βˆ’2=βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ (Ξ“(𝛼+𝑛)𝑛+π‘Ÿβˆ’1βˆ’1)π‘›βˆ’1ξ‚€1𝑛!𝑛+π›Όβˆ’2.(2.39) For 0β‰€π‘˜β‰€π‘›βˆ’2, by (2.37) and (2.38), one gets πΆπ‘˜=1⎧βŽͺ⎨βŽͺ⎩1Ξ“(𝛼+π‘˜+1)2π‘›βˆ’2𝑙=0βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽξ“π‘›+π‘Ÿβˆ’1max{0,𝑙+π‘Ÿβˆ’π‘›}β‰€π‘Žβ‰€π‘ŸβŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ ξ“π‘–1+β‹―+π‘–π‘Ž=𝑛+π‘Žβˆ’π‘™βˆ’π‘Ÿπ΅π‘–1𝐡𝑖2β‹―π΅π‘–π‘Ž+𝑖1+β‹―+π‘–π‘Ÿ=π‘›βˆ’π‘™π΅π‘–1β‹―π΅π‘–π‘ŸβŽžβŽŸβŽŸβŽ ξ€œβˆž0ξ‚΅π‘‘π‘˜π‘‘π‘₯π‘˜π‘’βˆ’π‘₯π‘₯𝑛+𝛼𝐸𝑙+βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ ξ€œ(π‘₯)𝑑π‘₯𝑛+π‘Ÿβˆ’1∞0ξ‚΅π‘‘π‘˜π‘‘π‘₯π‘˜π‘’βˆ’π‘₯π‘₯𝑛+π›Όξ‚ΆπΈπ‘›βŽ«βŽͺ⎬βŽͺ⎭=1(π‘₯)𝑑π‘₯⎧βŽͺ⎨βŽͺ⎩1Ξ“(𝛼+π‘˜+1)2π‘›βˆ’2𝑙=π‘˜βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽξ“π‘›+π‘Ÿβˆ’1max{0,𝑙+π‘Ÿβˆ’π‘›}β‰€π‘Žβ‰€π‘ŸβŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ ξ“π‘–1+β‹―+π‘–π‘Ž=𝑛+π‘Žβˆ’π‘™βˆ’π‘Ÿπ΅π‘–1𝐡𝑖2β‹―π΅π‘–π‘Ž+𝑖1+β‹―+π‘–π‘Ÿ=π‘›βˆ’π‘™π΅π‘–1β‹―π΅π‘–π‘ŸβŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜π‘™Γ—ξ€œ(π‘™βˆ’1)β‹―(π‘™βˆ’π‘˜+1)∞0π‘’βˆ’π‘₯π‘₯π‘˜+π›ΌπΈπ‘™βˆ’π‘˜+βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ (π‘₯)𝑑π‘₯𝑛+π‘Ÿβˆ’1(βˆ’1)π‘˜ξ€œπ‘›(π‘›βˆ’1)β‹―(π‘›βˆ’π‘˜+1)∞0π‘’βˆ’π‘₯π‘₯π‘˜+π›ΌπΈπ‘›βˆ’π‘˜βŽ«βŽͺ⎬βŽͺ⎭=1(π‘₯)𝑑π‘₯⎧βŽͺ⎨βŽͺ⎩1Ξ“(𝛼+π‘˜+1)2π‘›βˆ’2𝑙=π‘˜βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽξ“π‘›+π‘Ÿβˆ’1max{0,𝑙+π‘Ÿβˆ’π‘›}β‰€π‘Žβ‰€π‘Ÿξ“π‘–1+β‹―+π‘–π‘Ž=𝑛+π‘Žβˆ’π‘™βˆ’π‘Ÿπ΅π‘–1β‹―π΅π‘–π‘Ž+𝑖1+β‹―+π‘–π‘Ÿ=π‘›βˆ’π‘™π΅π‘–1β‹―π΅π‘–π‘ŸβŽžβŽŸβŽŸβŽ (βˆ’1)π‘˜π‘™!(π‘™βˆ’π‘˜)!π‘™βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ πΈπ‘™βˆ’π‘˜π‘™βˆ’π‘˜βˆ’π‘—Γ—ξ€œβˆž0π‘’βˆ’π‘₯π‘₯π‘˜+𝛼+𝑗+βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π‘‘π‘₯𝑛+π‘Ÿβˆ’1(βˆ’1)π‘˜π‘›!(π‘›βˆ’π‘˜)!π‘›βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ πΈπ‘›βˆ’π‘˜π‘›βˆ’π‘˜βˆ’π‘—ξ€œβˆž0π‘’βˆ’π‘₯π‘₯π‘˜+𝛼+π‘—βŽ«βŽͺ⎬βŽͺ⎭=1𝑑π‘₯2π‘›βˆ’2𝑙=π‘˜βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽξ“π‘›+π‘Ÿβˆ’1max{0,𝑙+π‘Ÿβˆ’π‘›}β‰€π‘Žβ‰€π‘Ÿξ“π‘–1+β‹―+π‘–π‘Ž=𝑛+π‘Žβˆ’π‘™βˆ’π‘Ÿπ΅π‘–1β‹―π΅π‘–π‘Ž+𝑖1+β‹―+π‘–π‘Ÿ=π‘›βˆ’π‘™π΅π‘–1β‹―π΅π‘–π‘ŸβŽžβŽŸβŽŸβŽ Γ—(βˆ’1)π‘˜π‘™!π‘™βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ πΈπ›Ό+π‘˜+π‘—π‘™βˆ’π‘˜βˆ’π‘—+βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ (π‘™βˆ’π‘˜βˆ’π‘—)!𝑛+π‘Ÿβˆ’1(βˆ’1)π‘˜π‘›!π‘›βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ πΈπ›Ό+π‘˜+π‘—π‘›βˆ’π‘˜βˆ’π‘—.(π‘›βˆ’π‘˜βˆ’π‘—)!(2.40) Therefore, by (2.36), (2.39), and (2.40), we obtain the following theorem.

Theorem 2.10. For π‘›βˆˆβ„€+, π‘Ÿβˆˆβ„•, and π›Όβˆˆβ„ with 𝛼>βˆ’1, one has 𝑖1+β‹―+π‘–π‘Ÿ=𝑛𝐡𝑖1(π‘₯)𝐡𝑖2(π‘₯)β‹―π΅π‘–π‘Ÿ(=π‘₯)π‘›βˆ’2ξ“π‘˜=0(βˆ’1)π‘˜2⎧βŽͺ⎨βŽͺβŽ©π‘›βˆ’2𝑙=π‘˜βŽ›βŽœβŽœβŽπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽξ“π‘™!𝑛+π‘Ÿβˆ’1max{0,𝑙+π‘Ÿβˆ’π‘›}β‰€π‘Žβ‰€π‘ŸβŽ›βŽœβŽœβŽπ‘Ÿπ‘ŽβŽžβŽŸβŽŸβŽ ξ“π‘–1+β‹―+π‘–π‘Ž=𝑛+π‘Žβˆ’π‘™βˆ’π‘Ÿπ΅π‘–1β‹―π΅π‘–π‘Ž+𝑖1+β‹―+π‘–π‘Ÿ=π‘›βˆ’π‘™π΅π‘–1β‹―π΅π‘–π‘ŸβŽžβŽŸβŽŸβŽ Γ—π‘™βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ πΈπ›Ό+π‘˜+π‘—π‘™βˆ’π‘˜βˆ’π‘—+βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ (π‘™βˆ’π‘˜βˆ’π‘—)!𝑛+π‘Ÿβˆ’1(βˆ’1)π‘˜Γ—π‘›!π‘›βˆ’π‘˜ξ“π‘—=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ πΈπ›Ό+π‘˜+π‘—π‘›βˆ’π‘˜βˆ’π‘—βŽ«βŽͺ⎬βŽͺ⎭𝐿(π‘›βˆ’π‘˜βˆ’π‘—)!π›Όπ‘˜βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ (π‘₯)+𝑛+π‘Ÿβˆ’1(βˆ’1)π‘›βˆ’1Γ—ξ‚€1𝑛!𝑛+π›Όβˆ’2ξ‚πΏπ›Όπ‘›βˆ’1βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ (π‘₯)+𝑛+π‘Ÿβˆ’1(βˆ’1)𝑛𝑛!𝐿𝛼𝑛(π‘₯).(2.41)

For π‘š,π‘ βˆˆβ„€+ with π‘š+𝑠=𝑛, let one assumes that 𝑝(π‘₯)=𝐿𝛼𝑠(π‘₯)πΏπ›Όπ‘š(π‘₯)βˆˆππ‘›.

By Proposition 2.1, one sees that 𝑝(π‘₯) can be written as 𝑝(π‘₯)=𝐿𝛼𝑠(π‘₯)πΏπ›Όπ‘š(π‘₯)=π‘›ξ“π‘˜=0πΆπ‘˜πΏπ›Όπ‘˜(π‘₯),π›Όβˆˆβ„with𝛼>βˆ’1.(2.42) From the orthogonality of {𝐿𝛼0(π‘₯),𝐿𝛼1(π‘₯),…,𝐿𝛼𝑛(π‘₯)}, one has πΆπ‘˜=1ξ€œΞ“(𝛼+π‘˜+1)∞0ξ‚΅π‘‘π‘˜π‘‘π‘₯π‘˜π‘’βˆ’π‘₯π‘₯π‘˜+𝛼𝑝(π‘₯)𝑑π‘₯.(2.43) By (1.2), (1.3), and (1.8), one gets 𝐿𝛼𝑠(π‘₯)πΏπ›Όπ‘šβŽ›βŽœβŽœβŽ(π‘₯)=π‘ ξ“π‘Ÿ1=0(βˆ’1)π‘Ÿ1π‘Ÿ1!βŽ›βŽœβŽœβŽπ‘ +π›Όπ‘ βˆ’π‘Ÿ1⎞⎟⎟⎠π‘₯π‘Ÿ1βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘šξ“π‘Ÿ2=0(βˆ’1)π‘Ÿ2π‘Ÿ2!βŽ›βŽœβŽœβŽπ‘š+π›Όπ‘šβˆ’π‘Ÿ2⎞⎟⎟⎠π‘₯π‘Ÿ2⎞⎟⎟⎠=π‘›ξ“π‘Ÿ=0βŽ›βŽœβŽœβŽπ‘Ÿξ“π‘Ÿ1=0(βˆ’1)π‘ŸβŽ›βŽœβŽœβŽπ‘Ÿπ‘Ÿ1βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ +π›Όπ‘ βˆ’π‘Ÿ1βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›βˆ’π‘ +𝛼𝛼+π‘Ÿβˆ’π‘Ÿ1⎞⎟⎟⎠⎞⎟⎟⎠π‘₯π‘Ÿ.π‘Ÿ!(2.44) Thus, from (2.44), one has 𝐿𝛼𝑠(π‘₯)πΏπ›Όπ‘š(π‘₯)=π‘›ξ“π‘Ÿ=0βŽ›βŽœβŽœβŽπ‘Ÿξ“π‘™=0(βˆ’1)π‘ŸβŽ›βŽœβŽœβŽπ‘Ÿπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽ π‘₯𝑠+𝛼𝛼+π‘™π‘›βˆ’π‘ +𝛼𝛼+π‘Ÿβˆ’π‘™π‘Ÿπ‘Ÿ!.(2.45) By (2.44) and (2.45), one gets πΆπ‘˜=1Ξ“(𝛼+π‘˜+1)π‘›ξ“π‘Ÿ=0(βˆ’1)π‘ŸβŽ§βŽͺ⎨βŽͺβŽ©π‘Ÿξ“π‘™=0βŽ›βŽœβŽœβŽπ‘Ÿπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ«βŽͺ⎬βŽͺ⎭1𝑠+𝛼𝛼+π‘™π‘›βˆ’π‘ +𝛼𝛼+π‘Ÿβˆ’π‘™Γ—ξ€œπ‘Ÿ!∞0ξ‚΅π‘‘π‘˜π‘‘π‘₯π‘˜π‘’βˆ’π‘₯π‘₯π‘˜+𝛼π‘₯π‘Ÿ=1𝑑π‘₯Ξ“(𝛼+π‘˜+1)π‘›ξ“π‘Ÿ=π‘˜(βˆ’1)π‘ŸβŽ§βŽͺ⎨βŽͺβŽ©π‘Ÿξ“π‘™=0βŽ›βŽœβŽœβŽπ‘Ÿπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ 1𝑠+𝛼𝛼+π‘™π‘›βˆ’π‘ +𝛼𝛼+π‘Ÿβˆ’π‘™βŽ«βŽͺ⎬βŽͺβŽ­π‘Ÿ!Γ—(βˆ’1)π‘˜ξ€œπ‘Ÿ(π‘Ÿβˆ’1)β‹―(π‘Ÿβˆ’π‘˜+1)∞0π‘’βˆ’π‘₯π‘₯π‘Ÿ+𝛼=1𝑑π‘₯Ξ“(𝛼+π‘˜+1)π‘›ξ“π‘Ÿ=π‘˜(βˆ’1)π‘ŸβŽ§βŽͺ⎨βŽͺβŽ©π‘Ÿξ“π‘™=0βŽ›βŽœβŽœβŽπ‘Ÿπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ 1𝑠+𝛼𝛼+π‘™π‘›βˆ’π‘ +𝛼𝛼+π‘Ÿβˆ’π‘™βŽ«βŽͺ⎬βŽͺβŽ­Γ—(π‘Ÿ!βˆ’1)π‘˜π‘Ÿ!=(π‘Ÿβˆ’π‘˜)!Ξ“(𝛼+π‘Ÿ+1)π‘›ξ“π‘Ÿ=π‘˜(βˆ’1)π‘Ÿ+π‘˜βŽ›βŽœβŽœβŽπ‘Ÿξ“π‘™=0βŽ›βŽœβŽœβŽπ‘Ÿπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ π‘ +𝛼𝛼+π‘™π‘›βˆ’π‘ +𝛼𝛼+π‘Ÿβˆ’π‘™(𝛼+π‘Ÿ)(𝛼+π‘Ÿβˆ’1)β‹―(𝛼+π‘˜+1)(⎞⎟⎟⎠=π‘Ÿβˆ’π‘˜)!π‘›ξ“π‘Ÿ=π‘˜(βˆ’1)π‘Ÿ+π‘˜βŽ›βŽœβŽœβŽπ‘Ÿξ“π‘™=0βŽ›βŽœβŽœβŽπ‘Ÿπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽ .𝑠+𝛼𝛼+π‘™π‘›βˆ’π‘ +𝛼𝛼+π‘Ÿβˆ’π‘™π›Ό+π‘Ÿπ‘Ÿβˆ’π‘˜(2.46) Therefore, by (2.42) and (2.46), we obtain the following theorem.

Theorem 2.11. For 𝑠,π‘šβˆˆβ„€+ with 𝑠+π‘š=𝑛, π›Όβˆˆβ„ with 𝛼>βˆ’1, one has 𝐿𝛼𝑠(π‘₯)πΏπ›Όπ‘š(π‘₯)=π‘›ξ“π‘›π‘˜=0ξ“π‘Ÿ=π‘˜(βˆ’1)π‘Ÿ+π‘˜βŽ›βŽœβŽœβŽπ‘Ÿξ“π‘™=0βŽ›βŽœβŽœβŽπ‘Ÿπ‘™βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽ πΏπ‘ +𝛼𝛼+π‘™π‘›βˆ’π‘ +𝛼𝛼+π‘Ÿβˆ’π‘™π›Ό+π‘Ÿπ‘Ÿβˆ’π‘˜π›Όπ‘˜(π‘₯).(2.47)

Acknowledgments

The authors would like to express their sincere gratitude to referee for his/her valuable comments and information. 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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