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

Abstract

Let 𝐏_𝑛={𝑝(𝑥)∈ℝ[𝑥]∣deg𝑝(𝑥)≤𝑛} be an inner product space with the inner product ⟨𝑝(𝑥),𝑞(𝑥)⟩=∫^∞0𝑥^𝛼𝑒^−𝑥𝑝(𝑥)𝑞(𝑥)𝑑𝑥, where 𝑝(𝑥),𝑞(𝑥)∈𝐏_𝑛 and 𝛼∈ℝ with 𝛼>−1. In this paper we study the properties of the extended Laguerre polynomials which are an orthogonal basis for 𝐏_𝑛. 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 𝛼∈ℝ with 𝛼>−1, the extended Laguerre polynomials are defined by the generating function as follows: exp(−𝑥𝑡/(1−𝑡))(1−𝑡)^𝛼+1=^∞_𝑛=0𝐿^𝛼𝑛(𝑥)𝑡^𝑛,(1.1) see [1–6].

From (1.1), we can derive the following: 𝐿^𝛼𝑛(𝑥)=^𝑛_𝑟=0(−1)^𝑟(^{𝑛+𝛼𝑛−𝑟})𝑟!𝑥^𝑟,(1.2) see [1–9].

As is well known, Rodrigues' formula for 𝐿^𝛼𝑛(𝑥) is given by 𝐿^𝛼𝑛(𝑥)=1𝑛!𝑥^−𝛼𝑒^𝑥𝑑𝑑𝑥^𝑛𝑛𝑒^−𝑥𝑥^𝑛+𝛼,(1.3) see [1–6, 8, 9].

From (1.3), we note that ^∞0𝑥^𝛼𝑒^−𝑥𝐿^𝛼𝑚(𝑥)𝐿^𝛼𝑛(𝑥)𝑑𝑥=1𝑛!Γ(𝛼+𝑛+1)𝛿^𝑚,𝑛,(𝛼>−1),(1.4) where 𝛿_𝑚,𝑛 is the Kronecker symbol.

From (1.1), (1.2), and (1.3), we can derive the following identities: (𝑛+1)𝐿^𝛼𝑛+1(𝑑𝑥)+(𝑥−𝛼−2𝑛−1)𝐿^𝛼𝑛(𝑥)+(𝑛+𝛼)𝐿^𝛼𝑛−1(𝑥)=0,(𝑛∈ℕ),(1.5)𝑑𝑥𝐿^𝛼𝑛(𝑥)−𝑑𝑑𝑥𝐿^𝛼𝑛−1𝑥𝑑(𝑥)+𝐿^𝛼𝑛−1(𝑥)=0,for𝑛≥1,(1.6)𝑑𝑥𝐿^𝛼𝑛(𝑥)=𝑛𝐿^𝛼𝑛(𝑥)−(𝑛+𝛼)𝐿^𝛼𝑛−1(𝑥)=0,(𝑛≥1),(1.7) and 𝐿^𝛼𝑛(𝑥) is a solution of 𝑥𝑦^+(𝛼+1−𝑥)𝑦^+𝑥𝑦=0.

The derivatives of general Laguerre polynomials are given by 𝑑𝑑𝑥𝐿^𝛼𝑛(𝑥)=−𝐿^{𝛼+1𝑛−1}(𝑥),𝑑𝑑𝑥𝑥^𝛼𝑑𝐿^𝛼𝑛(𝑥)=(𝑛+𝛼)𝑥^𝛼−1𝐿^𝑛𝛼−1(𝑥),𝑑𝑥𝑒^−𝑥𝐿^𝛼𝑛(𝑥)=−𝑒^−𝑥𝐿^𝑛𝛼+1(𝑥),𝑑𝑑𝑥𝑥^𝛼𝑒^−𝑥𝐿^𝛼𝑛(𝑥)=(𝑛+1)𝑥^𝛼−1𝑒^−𝑥𝐿^{𝛼−1𝑛+1}(𝑥).(1.8) The 𝑛th Bernoulli polynomials, 𝐵_𝑛(𝑥), are defined by the generating function to be 𝑡𝑒^𝑡−1𝑒^𝑥𝑡=𝑒^{𝐵(𝑥)𝑡}=^∞_𝑛=0𝐵_𝑛(𝑥)𝑡^𝑛𝑛!,(1.9) see [10–17], with the usual convention about replacing 𝐵^𝑛(𝑥) by 𝐵_𝑛(𝑥). In the special case, 𝑥=0, 𝐵_𝑛(0)=𝐵_𝑛 are called the 𝑛th Bernoulli numbers.

It is well known that the 𝑛th Euler polynomials are also defined by the generating function to be 2𝑒^𝑡+1𝑒^𝑥𝑡=𝑒^{𝐸(𝑥)𝑡}=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛𝑛!,(1.10) see [18–22], with the usual convention about replacing 𝐸^𝑛(𝑥) by 𝐸_𝑛(𝑥).

The Hermite polynomials are given by 𝐻_𝑛(𝑥)=(𝐻+2𝑥)^𝑛=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠2_𝑙𝑥^𝑙𝐻_𝑛−𝑙,(1.11) see [23, 24], with the usual convention about replacing 𝐻^𝑛 by 𝐻_𝑛. In the special case, 𝑥=0, 𝐻_𝑛(0)=𝐻_𝑛 are called the 𝑛th Hermite numbers.

From (1.11), we note that 𝑑𝑑𝑥𝐻^𝑛(𝑥)=2𝑛(𝐻+2𝑥)^𝑛−1=2𝑛𝐻_𝑛−1(𝑥),(1.12) see [23, 24], and 𝐻_𝑛(𝑥) is a solution of Hermite differential equation which is given by 𝑦^−2𝑥𝑦^+𝑛𝑦=0,(1.13) (see [1–6, 23–32]).

Throughout this paper we assume that 𝛼∈ℝ with 𝛼>−1. Let 𝐏_𝑛={𝑝(𝑥)∈ℝ[𝑥]|deg𝑝(𝑥)≤𝑛}. Then 𝐏_𝑛 is an inner product space with the inner product ⟨𝑝(𝑥),𝑞(𝑥)⟩=∫^∞0𝑥^𝛼𝑒^−𝑥𝑝(𝑥)𝑞(𝑥)𝑑𝑥, where 𝑝(𝑥),𝑞(𝑥)∈𝐏_𝑛. By (1.4) the set of the extended Laguerre polynomials {𝐿^𝛼0(𝑥),𝐿^𝛼1(𝑥),…,𝐿^𝛼𝑛(𝑥)} is an orthogonal basis for 𝐏_𝑛. In this paper we study the properties of the extended Laguerre polynomials which are an orthogonal basis for 𝐏_𝑛. 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(𝑥)∈𝐏_𝑛, 𝑝(𝑥) is given by 𝑝(𝑥)=^𝑛_𝑘=0𝐶_𝑘𝐿^𝛼𝑘(𝑥),foruniquelydeterminedrealnumbers𝐶_𝑘.(2.1) From (1.3), (1.4), and (2.1), we note that 𝑝(𝑥),𝐿^𝛼𝑘(𝑥)=𝐶_𝑘𝐿^𝛼𝑘(𝑥),𝐿^𝛼𝑘(𝑥)=𝐶_𝑘^∞0𝑥^𝛼𝑒^−𝑥𝐿^𝛼𝑘(𝑥)𝐿^𝛼𝑘(𝑥)𝑑𝑥=𝐶_𝑘Γ(𝛼+𝑘+1)𝑘!.(2.2) Thus, by (2.2), we get 𝐶_𝑘=𝑘!Γ(𝛼+𝑘+1)𝑝(𝑥),𝐿^𝛼𝑘=𝑘!(𝑥)Γ(𝛼+𝑘+1)𝑘1!^∞0𝑑𝑑𝑥^𝑘𝑘𝑥^𝑘+𝛼𝑒^−𝑥=1𝑝(𝑥)𝑑𝑥Γ(𝛼+𝑘+1)^∞0𝑑𝑑𝑥^𝑘𝑘𝑥^𝑘+𝛼𝑒^−𝑥𝑝(𝑥)𝑑𝑥.(2.3) Therefore, by (2.1) and (2.3), we obtain the following proposition.

Proposition 2.1. For 𝑝(𝑥)∈𝐏_𝑛, let 𝑝(𝑥)=^𝑛_𝑘=0𝐶_𝑘𝐿^𝛼𝑘(𝑥),(𝛼>−1).(2.4) Then one has the following: 𝐶_𝑘=1Γ(𝛼+𝑘+1)^∞0𝑑𝑑𝑥^𝑘𝑘𝑥^𝑘+𝛼𝑒^−𝑥𝑝(𝑥)𝑑𝑥.(2.5)

To derive inverse formula of (1.2), let take one 𝑝(𝑥)=𝑥^𝑛∈𝐏_𝑛. Then, by Proposition 2.1, one gets 𝐶_𝑘=1Γ(𝛼+𝑘+1)^∞0𝑑𝑑𝑥^𝑘𝑘𝑒^−𝑥𝑥^𝑘+𝛼𝑥^𝑛=(−1)_𝑘𝑑𝑥𝑛(𝑛−1)⋯(𝑛−𝑘+1)Γ(𝛼+𝑘+1)^∞0=(−1)_𝑘𝑒^−𝑥𝑥^𝛼+𝑛𝑑𝑥𝑛(𝑛−1)⋯(𝑛−𝑘+1)=(−1)_𝑘Γ(𝛼+𝑘+1)Γ(𝛼+𝑛+1)𝑛!(𝛼+𝑛)⋯𝛼Γ(𝛼)=((−𝛼1+)_𝑘𝑘)⋯𝛼Γ(𝛼)(𝑛−𝑘)!𝑛!(𝛼+𝑛)⋯(𝛼+𝑘+1)(𝑛−𝑘)!=(−1)^𝑘𝑛!⎛⎜⎜⎝𝛼𝑛−+𝑘𝑛⎞⎟⎟⎠.(2.6) Therefore, by (2.6), we obtain the following corollary.

Corollary 2.2 (Inverse formula of 𝐿^𝛼𝑛(𝑥)). For 𝑛∈Z₊, one has 𝑥^𝑛=𝑛!^𝑛_𝑘=0⎛⎜⎜⎝𝛼𝑛−+𝑘𝑛⎞⎟⎟⎠(−1)_𝑘𝐿^𝛼𝑘(𝑥).(2.7)

Let one takes Bernoulli polynomials of degree 𝑛 with 𝑝(𝑥)=𝐵_𝑛(𝑥)∈𝐏_𝑛. Then 𝐵_𝑛(𝑥) can be written as 𝐵_𝑛(𝑥)=^𝑛_𝑘=0𝐶_𝑘𝐿^𝛼𝑘(𝑥),(𝛼∈ℝwith𝛼>−1).(2.8) From Proposition 2.1, one has 𝐶_𝑘=1Γ(𝛼+𝑘+1)^∞0𝑑𝑑𝑥^𝑘𝑘𝑒^−𝑥𝑥^𝑘+𝛼𝐵^𝑛=(−1)_𝑘(𝑥)𝑑𝑥𝑛(𝑛−1)⋯(𝑛−𝑘+1)Γ(𝛼+𝑘+1)^∞0=(−1)_𝑘𝑒^−𝑥𝑥^𝑘+𝛼𝐵_𝑛−𝑘(𝑥)𝑑𝑥𝑛(𝑛−1)⋯(𝑛−𝑘+1)Γ(𝛼+𝑘+1)^𝑛−𝑘_𝑙=0⎛⎜⎜⎝𝑛−𝑙𝑘⎞⎟⎟⎠𝐵_{𝑛−𝑘−𝑙}^∞0=(−1)_𝑘𝑒^−𝑥𝑥^{𝑘+𝛼+𝑙}𝑑𝑥𝑛(𝑛−1)⋯(𝑛−𝑘+1)Γ(𝛼+𝑘+1)^𝑛−𝑘_𝑙=0⎛⎜⎜⎝𝑛−𝑙𝑘⎞⎟⎟⎠𝐵_{𝑛−𝑘−𝑙}Γ(𝛼+𝑘+𝑙+1).(2.9) By the fundamental property of gamma function, one gets Γ(𝛼+𝑘+𝑙+1)Γ(𝛼+𝑘+1)(𝑛−𝑘)!⎛⎜⎜⎝𝑛−𝑙𝑘⎞⎟⎟⎠=(𝛼+𝑙Γ+(𝛼𝑘)+⋯𝑘(+𝛼1+)(𝑘𝑛+−1𝑘))Γ!((𝛼𝑛+−𝑘𝑘+−1𝑙))!(𝑛𝑙!−𝑘)!=^{𝑙𝛼+𝑘+𝑙}(𝑛−𝑘−𝑙)!.(2.10) Therefore, by (2.8), (2.9), and (2.10), we obtain the following theorem.

Theorem 2.3. For 𝑛∈ℤ₊, 𝛼∈ℝ with 𝛼>−1, one has 𝐵_𝑛(𝑥)=𝑛!^𝑛_{𝑘=0𝑛−𝑘𝑙=0}(−1)^𝑘⎛⎜⎜⎝𝛼+𝑙𝑘+𝑙⎞⎟⎟⎠𝐵_{𝑛−𝑘−𝑙}(𝑛−𝑘−𝑙)!𝐿^𝛼𝑘(𝑥).(2.11)

As is known, relationships between Hermite and Laguerre polynomials are given by 𝐻𝐻_2𝑚(𝑥)=(−1)^𝑚2^2𝑚𝑚!𝐿^𝑚−1/2𝑥²,(2.12)_2𝑚+1(𝑥)=(−1)^𝑚2^2𝑚+1𝑚!𝐿^𝑚−1/2𝑥²,(2.13) see [1–6]. In the special case 𝛼=−1/2, by (2.12) and (2.13), we obtain the following corollary.

Corollary 2.4. For 𝑛∈ℤ₊, one has 𝐵_𝑛𝑥²=𝑛!^𝑛_{𝑘=0𝑛−𝑘𝑙=0}𝐻_2𝑘(𝑥)2^2𝑘𝑘!⎛⎜⎜⎝−12+𝑙𝑘+𝑙⎞⎟⎟⎠𝐵^{𝑛−𝑘−𝑙}(𝑛−𝑘−𝑙)!.(2.14)

By the same method as Theorem 2.3, one gets 𝐸_𝑛(𝑥)=𝑛!^𝑛_{𝑘=0𝑛−𝑘𝑙=0}(−1)^𝑘⎛⎜⎜⎝𝛼+𝑙𝑘+𝑙⎞⎟⎟⎠𝐸_{𝑛−𝑘−𝑙}(𝑛−𝑘−𝑙)!𝐿^𝛼𝑘(𝑥),(2.15) where 𝐸_𝑛(𝑥) are the 𝑛th Euler polynomials. In the special case, 𝑥=0, 𝐸_𝑛(0)=𝐸_𝑛 are called the 𝑛th Euler numbers.

Let one considers the 𝑛th Hermite polynomials with 𝑝(𝑥)=𝐻_𝑛(𝑥)∈𝐏_𝑛. Then 𝐻_𝑛(𝑥) can be written as 𝐻_𝑛(𝑥)=^𝑛_𝑘=0𝐶_𝑘𝐿^𝛼𝑘(𝑥),(𝛼∈ℝwith𝛼>−1).(2.16) From Proposition 2.1, one notes that 𝐶_𝑘=1Γ(𝛼+𝑘+1)^∞0𝑑𝑑𝑥^𝑘𝑘𝑒^−𝑥𝑥^𝑘+𝛼𝐻^𝑛=(−2𝑛)(𝑥)𝑑𝑥Γ(𝛼+𝑘+1)^∞0𝑑𝑑𝑥^{𝑘−1𝑘−1}𝑒^−𝑥𝑥^𝑘+𝛼𝐻^𝑛−1==(⋯−2𝑛)(−2(𝑛−1))⋯(−2(𝑛−𝑘+1))(𝑥)𝑑𝑥Γ(𝛼+𝑘+1)^∞0=(−1)_𝑘𝑒^−𝑥𝑥^𝑘+𝛼𝐻_𝑛−𝑘(𝑥)𝑑𝑥2^𝑘𝑛!Γ(𝛼+𝑘+1)(𝑛−𝑘)!^𝑛−𝑘_𝑙=0⎛⎜⎜⎝𝑛−𝑙𝑘⎞⎟⎟⎠𝐻_{𝑛−𝑘−𝑙}2^𝑙^∞0=(−1)_𝑘𝑒^−𝑥𝑥^{𝑘+𝛼+𝑙}𝑑𝑥2^𝑘𝑛!Γ(𝛼+𝑘+1)(𝑛−𝑘)!^𝑛−𝑘_𝑙=0⎛⎜⎜⎝𝑛−𝑙𝑘⎞⎟⎟⎠𝐻_{𝑛−𝑘−𝑙}2^𝑙Γ(𝛼+𝑘+𝑙+1).(2.17) It is not difficult to show that ^{𝑙𝑛−𝑘}Γ(𝛼+𝑘+𝑙+1)Γ(𝛼+𝑘+1)(𝑛−𝑘)!=(𝑛−𝑘()𝑛!−(𝛼𝑘+−𝑘𝑙+)!𝑙𝑙)!⋯Γ(𝛼(𝛼++𝑘𝑘++11))(𝑛Γ−(𝛼𝑘+)!𝑘+1)=^{𝑙𝛼+𝑘+𝑙}(𝑛−𝑘−𝑙)!.(2.18) Therefore, by (2.16), (2.17), and (2.18), we obtain the following theorem.

Theorem 2.5. For 𝑛∈ℤ₊, 𝛼∈ℝ with 𝛼>−1, one has 𝐻_𝑛(𝑥)=𝑛!^𝑛_{𝑘=0𝑛−𝑘𝑙=0}(−1)^𝑘2^𝑘+𝑙⎛⎜⎜⎝𝛼+𝑙𝑘+𝑙⎞⎟⎟⎠𝐻_{𝑛−𝑘−𝑙}(𝑛−𝑘−𝑙)!𝐿^𝛼𝑘(𝑥).(2.19)

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

Corollary 2.6. For 𝑛∈ℤ₊, one has 𝐻_𝑛𝑥²=𝑛!^𝑛_{𝑘=0𝑛−𝑘𝑙=0}2^𝑙−𝑘𝐻_2𝑘(𝑥)𝑘!⎛⎜⎜⎝−12+𝑙𝑘+𝑙⎞⎟⎟⎠𝐻^{𝑛−𝑘−𝑙}(𝑛−𝑘−𝑙)!.(2.20)

For 𝛽∈ℝ with 𝛽>−1, let one takes 𝑝(𝑥)=𝐿^𝛽𝑛(𝑥)∈𝐏_𝑛.(2.21) Then 𝐿^𝛽𝑛(𝑥) is also written as 𝐿^𝛽𝑛(𝑥)=^𝑛_𝑘=0𝐶_𝑘𝐿^𝛼𝑘(𝑥).(2.22) From Proposition 2.1, one can determine the coefficients of (2.22) as follows: 𝐶_𝑘=1Γ(𝛼+𝑘+1)^∞0𝑑𝑑𝑥^𝑘𝑘𝑒^−𝑥𝑥^𝑘+𝛼𝐿^𝛽𝑛=1(𝑥)𝑑𝑥Γ(𝛼+𝑘+1)^∞0𝑑𝑑𝑥^{𝑘−1𝑘−1}𝑒^−𝑥𝑥^𝑘+𝛼𝐿^{𝛽+1𝑛−1}==⋯1(𝑥)𝑑𝑥Γ(𝛼+𝑘+1)^∞0=1𝑒^−𝑥𝑥^𝑘+𝛼𝐿^{𝛽+𝑘𝑛−𝑘}(𝑥)𝑑𝑥Γ(𝛼+𝑘+1)^𝑛−𝑘_𝑟=0(−1)^𝑟^{𝑛+𝛽𝑛−𝑘−𝑟}𝑟!^∞0=1𝑒^−𝑥𝑥^{𝑘+𝛼+𝑟}𝑑𝑥Γ(𝛼+𝑘+1)^𝑛−𝑘_𝑟=0(−1)^𝑟^{𝑛+𝛽𝑛−𝑘−𝑟}𝑟!Γ(𝑘+𝛼+𝑟+1).(2.23) By the fundamental property of gamma function, one gets Γ(𝑘+𝛼+𝑟+1)𝑟!Γ(𝛼+𝑘+1)=(𝑘+𝛼+𝑟)⋯𝑟!Γ(𝛼(𝛼++𝑘𝑘++1)1Γ)(𝛼+𝑘+1)=⎛⎜⎜⎝𝑘+𝛼𝑟+𝑟⎞⎟⎟⎠.(2.24) Therefore, by (2.22), (2.23), and (2.24), we obtain the following theorem.

Theorem 2.7. For 𝛽∈ℝ with 𝛽>−1, and 𝑛∈ℤ₊, one has 𝐿^𝛽𝑛(𝑥)=^𝑛_{𝑘=0𝑛−𝑘𝑟=0}(−1)^𝑟⎛⎜⎜⎝𝑛−𝑛+𝑘𝛽−𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝛼+𝑟𝑘+𝑟⎞⎟⎟⎠𝐿_𝛼𝑘(𝑥).(2.25)

In the special case, 𝛼=𝛽, one has ^𝑛−1_𝑘=0⎛⎜⎜⎝^𝑛−𝑘_𝑟=0(−1)^𝑟⎛⎜⎜⎝𝑛−𝑛+𝑘𝛼−𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝛼+𝑟𝑘+𝑟⎞⎟⎟⎠⎞⎟⎟⎠𝐿_𝛼𝑘(𝑥)=0.(2.26) Thus, by (2.26), we obtain the following corollary.

Corollary 2.8. For 0≤𝑘≤𝑛−1, 𝛼∈ℝ with 𝛼>−1, one has ^𝑛−𝑘_𝑟=0(−1)^𝑟⎛⎜⎜⎝𝑛−𝑛+𝑘𝛼−𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝛼+𝑟𝑘+𝑟⎞⎟⎟⎠=0.(2.27)

Let one assumes that 𝑝(𝑥)=^𝑛_𝑙=0𝐵_𝑙(𝑥)𝐵_𝑛−𝑙(𝑥)∈𝐏_𝑛.(2.28) Then 𝑝(𝑥) can be rewritten as a linear combination of 𝐿^𝛼0(𝑥),𝐿^𝛼1(𝑥),…,𝐿^𝛼𝑛(𝑥) as follows: 𝑝(𝑥)=^𝑛_𝑙=0𝐵_𝑙(𝑥)𝐵_𝑛−𝑙(𝑥)=^𝑛_𝑘=0𝐶_𝑘𝐿^𝛼𝑘(𝑥).(2.29) By Proposition 2.1, one can determine the coefficients of (2.29) as follows: 𝐶_𝑘=1Γ(𝛼+𝑘+1)^𝑛_𝑙=0^∞0𝑑𝑑𝑥^𝑘𝑘𝑒^−𝑥𝑥^𝑘+𝛼𝐵^𝑙(𝑥)𝐵_𝑛−𝑙(𝑥)𝑑𝑥.(2.30) It is known that ^𝑛_𝑙=0𝐵_𝑙(𝑥)𝐵_𝑛−𝑙(𝑥)=2𝑛+2^𝑛−2_𝑙=0⎛⎜⎜⎝𝑛+𝑙2⎞⎟⎟⎠𝐵_𝑛−𝑙𝐵_𝑙(𝑥)+(𝑛+1)𝐵_𝑛(𝑥),(2.31) see [25].

From (2.30) and (2.31), one notes that 𝐶_𝑛=𝑛+1Γ(𝛼+𝑛+1)^∞0𝑑𝑑𝑥^𝑛𝑛𝑒^−𝑥𝑥^𝑛+𝛼𝐵^𝑛=𝑛+1(𝑥)𝑑𝑥Γ(𝛼+𝑛+1)(−1)^𝑛𝑛!^∞0𝑒^−𝑥𝑥^𝑛+𝛼𝑑𝑥=(𝑛+1)!(−1)^𝑛=(𝑛+1)!(−1)_𝑛Γ(𝛼+𝑛+1)Γ(𝑛+𝛼+1)𝐶_𝑛−1,=𝑛+1Γ(𝛼+𝑛)^∞0𝑑𝑑𝑥^{𝑛−1𝑛−1}𝑒^−𝑥𝑥^{𝑛−1+𝛼}𝐵^𝑛=𝑛+1(𝑥)𝑑𝑥Γ(𝛼+𝑛)(−1)^𝑛−1=𝑛+1𝑛!^∞0𝑒^−𝑥𝑥^{𝑛−1+𝛼}𝐵₁(𝑥)𝑑𝑥Γ(𝛼+𝑛)(−1)^𝑛−1𝑛!Γ(𝛼+𝑛+1)−1=(𝑛+1)!(−1)_𝑛−12Γ(𝛼+𝑛)𝑛+𝛼−12.(2.32) For 0≤𝑘≤𝑛−2, one has 𝐶_𝑘=1Γ(𝛼+𝑘+1)⎧⎪⎨⎪⎩𝑛+22^𝑛−2_𝑙=0⎛⎜⎜⎝𝑛+𝑙2⎞⎟⎟⎠𝐵_𝑛−𝑙^∞0𝑑𝑑𝑥^𝑘𝑘𝑒^−𝑥𝑥^𝑘+𝛼𝐵^𝑙+(𝑛+1)_∞0(𝑥)𝑑𝑥𝑑𝑑𝑥^𝑘𝑘𝑒^−𝑥𝑥^𝑘+𝛼𝐵^𝑛=1(𝑥)𝑑𝑥⎫⎪⎬⎪⎭Γ(𝛼+𝑘+1)⎧⎪⎨⎪⎩𝑛+22^𝑛−2_𝑙=𝑘⎛⎜⎜⎝𝑛+𝑙2⎞⎟⎟⎠𝐵_𝑛−𝑙+(𝑛+1)(−1)_𝑘(−1)^𝑘𝑙(𝑙−1)⋯(𝑙−𝑘+1)^∞0𝑒^−𝑥𝑥^𝑘+𝛼𝐵_𝑙−𝑘(𝑥)𝑑𝑥=1𝑛(𝑛−1)⋯(𝑛−𝑘+1)^∞0𝑒^−𝑥𝑥^𝑘+𝛼𝐵_𝑛−𝑘(𝑥)𝑑𝑥⎫⎪⎬⎪⎭Γ(𝛼+𝑘+1)⎧⎪⎨⎪⎩𝑛+22^𝑛−2_𝑙=𝑘⎛⎜⎜⎝𝑛+𝑙2⎞⎟⎟⎠𝐵_𝑛−𝑙(−1)^𝑘𝑙!(𝑙−𝑘)!^𝑙−𝑘_𝑗=0⎛⎜⎜⎝𝑙−𝑗𝑘⎞⎟⎟⎠𝐵_{𝑙−𝑘−𝑗}^∞0+(𝑛+1)(−1)_𝑘𝑒^−𝑥𝑥^{𝑘+𝛼+𝑗}𝑑𝑥𝑛!(𝑛−𝑘)!^𝑛−𝑘_𝑗=0⎛⎜⎜⎝𝑛−𝑗𝑘⎞⎟⎟⎠𝐵_{𝑛−𝑘−𝑗}^∞0=1𝑒^−𝑥𝑥^{𝑘+𝛼+𝑗}𝑑𝑥⎫⎪⎬⎪⎭Γ(𝛼+𝑘+1)⎧⎪⎨⎪⎩𝑛+22^𝑛−2_{𝑙=𝑘𝑙−𝑘𝑗=0}⎛⎜⎜⎝𝑛+𝑙2⎞⎟⎟⎠𝐵_𝑛−𝑙(−1)^𝑘𝑙!(𝑙−𝑘)!⎛⎜⎜⎝𝑙−𝑗𝑘⎞⎟⎟⎠𝐵^{𝑙−𝑘−𝑗}+(𝑛+1)(−1)_𝑘Γ(𝛼+𝑘+𝑗+1)𝑛!(𝑛−𝑘)!^𝑛−𝑘_𝑗=0⎛⎜⎜⎝𝑛−𝑗𝑘⎞⎟⎟⎠𝐵_{𝑛−𝑘−𝑗}=2Γ(𝛼+𝑘+𝑗+1)⎫⎪⎬⎪⎭𝑛+2^𝑛−2_{𝑙=𝑘𝑙−𝑘𝑗=0}⎛⎜⎜⎝𝑛+𝑙2⎞⎟⎟⎠𝐵_𝑛−𝑙(−1)^𝑘𝑙!(𝛼+𝑘+𝑗)⋯(𝛼+𝑘+1)𝐵^{𝑙−𝑘−𝑗}+(𝑛+1)(−1)_𝑘𝑗!(𝑙−𝑘−𝑗)!𝑛!^𝑛−𝑘_𝑗=0(𝛼+𝑘+𝑗)⋯(𝛼+𝑘+1)𝑗!(𝑛−𝑘−𝑗)!𝐵^{𝑛−𝑘−𝑗}=2𝑛+2^𝑛−2_{𝑙=𝑘𝑙−𝑘𝑗=0}⎛⎜⎜⎝𝑛+𝑙2⎞⎟⎟⎠𝐵_𝑛−𝑙(−1)^𝑘𝑙!⎛⎜⎜⎝𝛼+𝑗𝑘+𝑗⎞⎟⎟⎠𝐵^{𝑙−𝑘−𝑗}+(𝑛+1)(−1)_𝑘(𝑙−𝑘−𝑗)!𝑛!^𝑛−𝑘_𝑗=0⎛⎜⎜⎝𝛼+𝑗𝑘+𝑗⎞⎟⎟⎠𝐵_{𝑛−𝑘−𝑗}(𝑛−𝑘−𝑗)!.(2.33) Therefore, by (2.29) and (2.32), we obtain the following theorem.

Theorem 2.9. For 𝑛∈ℤ₊, 𝛼∈ℝ with 𝛼>−1, one has ^𝑛_𝑘=0𝐵=_𝑘(𝑥)𝐵_𝑛−𝑘(𝑥)_𝑛−2_𝑘=0⎧⎪⎨⎪⎩2𝑛+2^𝑛−2_𝑙=𝑘_{0≤𝑗≤𝑛−𝑘}(−1)^𝑘⎛⎜⎜⎝𝑛+𝑙2⎞⎟⎟⎠𝑙!𝐵_𝑛−𝑙⎛⎜⎜⎝𝛼+𝑗𝑘+𝑗⎞⎟⎟⎠𝐵_{𝑙−𝑘−𝑗}+(𝑛+1)!_{0≤𝑗≤𝑛−𝑘}(𝑙−𝑘−𝑗)!(−1)^𝑘⎛⎜⎜⎝𝛼+𝑗𝑘+𝑗⎞⎟⎟⎠𝐵_{𝑛−𝑘−𝑗}(𝑛−𝑘−𝑗)!⎫⎪⎬⎪⎭𝐿^𝛼𝑘+(−1)_𝑛(𝑥)(𝑛+1)!𝐿^𝛼𝑛(𝑥)−𝑛+𝛼−12𝐿^𝛼𝑛−1(𝑥).(2.34)

Let one takes the polynomial 𝑝(𝑥) in 𝐏_𝑛 as follows: 𝑝(𝑥)=_{𝑖1+⋯+𝑖𝑟=𝑛}𝐵_𝑖1(𝑥)𝐵_𝑖2(𝑥)⋯𝐵_𝑖𝑟(𝑥)∈𝐏_𝑛.(2.35) From the orthogonality of {𝐿^𝛼0(𝑥),…,𝐿^𝛼𝑛(𝑥)}, one notes that 𝑝(𝑥)=_{𝑖1+⋯+𝑖𝑟=𝑛}𝐵_𝑖1(𝑥)𝐵_𝑖2(𝑥)⋯𝐵_𝑖𝑟(𝑥)=^𝑛_𝑘=0𝐶_𝑘𝐿^𝛼𝑘(𝑥),(2.36) where 𝐶_𝑘=1Γ(𝛼+𝑘+1)^∞0𝑑𝑑𝑥^𝑘𝑘𝑒^−𝑥𝑥^𝑘+𝛼𝑝(𝑥)𝑑𝑥.(2.37) It is known in [25] that _𝑖1+⋯_{+𝑖𝑟=𝑛}=1𝐵_𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)2^𝑛−2_𝑘=0⎛⎜⎜⎝𝑛+𝑘𝑟−1⎞⎟⎟⎠⎧⎪⎨⎪⎩_{max{0,𝑘+𝑟−𝑛}≤𝑎≤𝑟}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠_{𝑖1+⋯+𝑖𝑎=𝑛+𝑎−𝑘−𝑟}𝐵_𝑖1𝐵_𝑖2⋯𝐵_𝑖𝑎+_{𝑖1+⋯+𝑖𝑟=𝑛−𝑘}𝐵_𝑖1⋯𝐵_𝑖𝑟⎫⎪⎬⎪⎭𝐸_𝑘(𝑥)+⎛⎜⎜⎝𝑛+𝑛𝑟−1⎞⎟⎟⎠𝐸_𝑛(𝑥).(2.38) From (2.35), (2.37), and (2.38), one notes that 𝐶_𝑛=^{𝑛𝑛+𝑟−1}Γ(𝛼+𝑛+1)^∞0𝑑𝑑𝑥^𝑛𝑛𝑒^−𝑥𝑥^𝑛+𝛼𝐸^𝑛=(𝑥)𝑑𝑥_{𝑛𝑛+𝑟−1}Γ(𝛼+𝑛+1)(−1)^𝑛=𝑛!^∞0𝑥^𝑛+𝛼𝑒^−𝑥𝑑𝑥_{𝑛𝑛+𝑟−1}Γ(𝛼+𝑛+1)(−1)^𝑛𝐶𝑛!Γ(𝛼+𝑛+1)=⎛⎜⎜⎝𝑛+𝑛𝑟−1⎞⎟⎟⎠(−1)^𝑛𝑛!,_𝑛−1=^{𝑛𝑛+𝑟−1}Γ(𝛼+𝑛)^∞0𝑑𝑑𝑥^{𝑛−1𝑛−1}𝑒^−𝑥𝑥^{𝑛+𝛼−1}𝐸^𝑛=(𝑥)𝑑𝑥_{𝑛𝑛+𝑟−1}Γ(𝛼+𝑛)(−1)^𝑛−1=𝑛!^∞0𝑒^−𝑥𝑥^{𝑛+𝛼−1}𝐸₁(𝑥)𝑑𝑥_{𝑛𝑛+𝑟−1}Γ(𝛼+𝑛)(−1)^𝑛−1𝑛!Γ(𝛼+𝑛+1)−1=⎛⎜⎜⎝𝑛+𝑛𝑟−1⎞⎟⎟⎠(−1)_𝑛−12Γ(𝛼+𝑛)𝑛!𝑛+𝛼−12.(2.39) For 0≤𝑘≤𝑛−2, by (2.37) and (2.38), one gets 𝐶_𝑘=1Γ(𝛼+𝑘+1)⎧⎪⎨⎪⎩21^𝑛−2_𝑙=0⎛⎜⎜⎝𝑛+𝑟𝑙−1⎞⎟⎟⎠⎛⎜⎜⎝_{max{0,𝑙+𝑟−𝑛}≤𝑎≤𝑟}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠_{𝑖1+⋯+𝑖𝑎=𝑛+𝑎−𝑙−𝑟}𝐵_𝑖1𝐵_𝑖2⋯𝐵_𝑖𝑎+_{𝑖1+⋯+𝑖𝑟=𝑛−𝑙}𝐵_𝑖1⋯𝐵_𝑖𝑟⎞⎟⎟⎠^∞0𝑑𝑑𝑥^𝑘𝑘𝑒^−𝑥𝑥^𝑛+𝛼𝐸^𝑙+⎛⎜⎜⎝𝑛+𝑛𝑟−1⎞⎟⎟⎠_∞0(𝑥)𝑑𝑥𝑑𝑑𝑥^𝑘𝑘𝑒^−𝑥𝑥^𝑛+𝛼𝐸^𝑛=1(𝑥)𝑑𝑥⎫⎪⎬⎪⎭Γ(𝛼+𝑘+1)⎧⎪⎨⎪⎩21^𝑛−2_𝑙=𝑘⎛⎜⎜⎝𝑛+𝑟𝑙−1⎞⎟⎟⎠⎛⎜⎜⎝_{max{0,𝑙+𝑟−𝑛}≤𝑎≤𝑟}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠_{𝑖1+⋯+𝑖𝑎=𝑛+𝑎−𝑙−𝑟}𝐵_𝑖1𝐵_𝑖2⋯𝐵_𝑖𝑎+_{𝑖1+⋯+𝑖𝑟=𝑛−𝑙}𝐵_𝑖1⋯𝐵_𝑖𝑟⎞⎟⎟⎠(−1)_𝑘×𝑙(𝑙−1)⋯(𝑙−𝑘+1)_∞0+⎛⎜⎜⎝𝑛+𝑒𝑛𝑟^−𝑥−𝑥1⎞⎟⎟⎠^𝑘+𝛼(𝐸−1_𝑙−𝑘)_𝑘(𝑥)𝑑𝑥=1𝑛(𝑛−1)⋯(𝑛−𝑘+1)^∞0𝑒^−𝑥𝑥^𝑘+𝛼𝐸_𝑛−𝑘(𝑥)𝑑𝑥⎫⎪⎬⎪⎭Γ(𝛼+𝑘+1)⎧⎪⎨⎪⎩21^𝑛−2_𝑙=𝑘⎛⎜⎜⎝𝑛+𝑟𝑙−1⎞⎟⎟⎠⎛⎜⎜⎝_{max{0,𝑙+𝑟−𝑛}≤𝑎≤𝑟𝑖1+⋯+𝑖}_{𝑎=𝑛+𝑎−𝑙−𝑟}𝐵_𝑖1⋯𝐵_𝑖𝑎+_{𝑖1+⋯+𝑖𝑟=𝑛−𝑙}𝐵_𝑖1⋯𝐵_𝑖𝑟⎞⎟⎟⎠(−1)^𝑘𝑙!(𝑙−𝑘)!^𝑙−𝑘_𝑗=0⎛⎜⎜⎝𝑙−𝑗𝑘⎞⎟⎟⎠𝐸_{𝑙−𝑘−𝑗}×+⎛⎜⎜⎝𝑛^∞0+𝑒𝑛𝑟^−𝑥−𝑥1⎞⎟⎟⎠(^{𝑘+𝛼+𝑗}−𝑑1𝑥)_𝑘𝑛!(𝑛−𝑘)!^𝑛−𝑘_𝑗=0⎛⎜⎜⎝𝑛−𝑗𝑘⎞⎟⎟⎠𝐸_{𝑛−𝑘−𝑗}^∞0=1𝑒^−𝑥𝑥^{𝑘+𝛼+𝑗}𝑑𝑥⎫⎪⎬⎪⎭2^𝑛−2_𝑙=𝑘⎛⎜⎜⎝𝑛+𝑟𝑙−1⎞⎟⎟⎠⎛⎜⎜⎝_{max{0,𝑙+𝑟−𝑛}≤𝑎≤𝑟𝑖1+⋯+𝑖}_{𝑎=𝑛+𝑎−𝑙−𝑟}𝐵_𝑖1⋯𝐵_𝑖𝑎+_{𝑖1+⋯+𝑖𝑟=𝑛−𝑙}𝐵_𝑖1⋯𝐵_𝑖𝑟×(−1)_𝑘⎞⎟⎟⎠𝑙!^𝑙−𝑘_𝑗=0⎛⎜⎜⎝𝛼+𝑗𝑘+𝑗⎞⎟⎟⎠𝐸_{𝑙−𝑘−𝑗}+⎛⎜⎜⎝𝑛+𝑛𝑟−1⎞⎟⎟⎠(−1)_𝑘(𝑙−𝑘−𝑗)!𝑛!^𝑛−𝑘_𝑗=0⎛⎜⎜⎝𝛼+𝑗𝑘+𝑗⎞⎟⎟⎠𝐸_{𝑛−𝑘−𝑗}(𝑛−𝑘−𝑗)!.(2.40) Therefore, by (2.36), (2.39), and (2.40), we obtain the following theorem.

Theorem 2.10. For 𝑛∈ℤ₊, 𝑟∈ℕ, and 𝛼∈ℝ with 𝛼>−1, one has _𝑖1+⋯_{+𝑖𝑟=𝑛}=𝐵_𝑖1(𝑥)𝐵_𝑖2(𝑥)⋯𝐵_𝑖𝑟(𝑥)_𝑛−2_𝑘=0(−1)^𝑘2⎧⎪⎨⎪⎩^𝑛−2_𝑙=𝑘𝑙!⎛⎜⎜⎝𝑛+𝑟𝑙−1⎞⎟⎟⎠⎛⎜⎜⎝_{max{0,𝑙+𝑟−𝑛}≤𝑎≤𝑟}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠_{𝑖1+⋯+𝑖𝑎=𝑛+𝑎−𝑙−𝑟}𝐵_𝑖1⋯𝐵_𝑖𝑎+_{𝑖1+⋯+𝑖𝑟=𝑛−𝑙}𝐵_𝑖1⋯𝐵_𝑖𝑟×_𝑙−𝑘⎞⎟⎟⎠_𝑗=0⎛⎜⎜⎝𝛼+𝑗𝑘+𝑗⎞⎟⎟⎠𝐸_{𝑙−𝑘−𝑗}(𝑙−𝑘−𝑗)!+⎛⎜⎜⎝𝑛+𝑛𝑟−1⎞⎟⎟⎠(−1)^𝑘×𝑛!_𝑛−𝑘_𝑗=0⎛⎜⎜⎝𝛼+𝑗𝑘+𝑗⎞⎟⎟⎠𝐸_{𝑛−𝑘−𝑗}(𝑛−𝑘−𝑗)!⎫⎪⎬⎪⎭𝐿^𝛼𝑘×𝑛+𝛼−1(𝑥)+⎛⎜⎜⎝𝑛+𝑛𝑟−1⎞⎟⎟⎠(−1)^𝑛−1𝑛!2𝐿^𝛼𝑛−1(𝑥)+⎛⎜⎜⎝𝑛+𝑛𝑟−1⎞⎟⎟⎠(−1)^𝑛𝑛!𝐿^𝛼𝑛(𝑥).(2.41)

For 𝑚,𝑠∈ℤ₊ with 𝑚+𝑠=𝑛, let one assumes that 𝑝(𝑥)=𝐿^𝛼𝑠(𝑥)𝐿^𝛼𝑚(𝑥)∈𝐏_𝑛.

By Proposition 2.1, one sees that 𝑝(𝑥) can be written as 𝑝(𝑥)=𝐿^𝛼𝑠(𝑥)𝐿^𝛼𝑚(𝑥)=^𝑛_𝑘=0𝐶_𝑘𝐿^𝛼𝑘(𝑥),𝛼∈ℝwith𝛼>−1.(2.42) From the orthogonality of {𝐿^𝛼0(𝑥),𝐿^𝛼1(𝑥),…,𝐿^𝛼𝑛(𝑥)}, one has 𝐶_𝑘=1Γ(𝛼+𝑘+1)^∞0𝑑𝑑𝑥^𝑘𝑘𝑒^−𝑥𝑥^𝑘+𝛼𝑝(𝑥)𝑑𝑥.(2.43) By (1.2), (1.3), and (1.8), one gets 𝐿^𝛼𝑠(𝑥)𝐿^𝛼𝑚(𝑥)=⎛⎜⎜⎝^𝑠_𝑟₁₌₀(−1)^𝑟1𝑟₁!⎛⎜⎜⎝𝑠𝑠−+𝑟𝛼₁⎞⎟⎟⎠𝑥_𝑟1⎞⎟⎟⎠⎛⎜⎜⎝^𝑚_𝑟₂₌₀(−1)^𝑟2𝑟₂!⎛⎜⎜⎝𝑚𝑚−+𝑟𝛼₂=_𝑛⎞⎟⎟⎠𝑥_𝑟2⎞⎟⎟⎠_𝑟=0⎛⎜⎜⎝^𝑟_𝑟₁₌₀(−1)^𝑟⎛⎜⎜⎝𝑟𝑟₁⎞⎟⎟⎠⎛⎜⎜⎝𝑠𝑠−+𝑟𝛼₁⎞⎟⎟⎠⎛⎜⎜⎝𝛼𝑛+−𝑟𝑠−+𝑟𝛼₁⎞⎟⎟⎠⎞⎟⎟⎠𝑥_𝑟𝑟!.(2.44) Thus, from (2.44), one has 𝐿^𝛼𝑠(𝑥)𝐿^𝛼𝑚(𝑥)=^𝑛_𝑟=0⎛⎜⎜⎝^𝑟_𝑙=0(−1)^𝑟⎛⎜⎜⎝𝑟𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑠𝛼++𝛼𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑛𝛼−+𝑠𝑟+−𝛼𝑙⎞⎟⎟⎠⎞⎟⎟⎠𝑥_𝑟𝑟!.(2.45) By (2.44) and (2.45), one gets 𝐶_𝑘=1Γ(𝛼+𝑘+1)^𝑛_𝑟=0(−1)^𝑟⎧⎪⎨⎪⎩^𝑟_𝑙=0⎛⎜⎜⎝𝑟𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑠𝛼++𝛼𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑛𝛼−+𝑠𝑟+−𝛼𝑙⎞⎟⎟⎠⎫⎪⎬⎪⎭1×𝑟!_∞0𝑑𝑑𝑥^𝑘𝑘𝑒^−𝑥𝑥^𝑘+𝛼𝑥^𝑟=1𝑑𝑥Γ(𝛼+𝑘+1)^𝑛_𝑟=𝑘(−1)^𝑟⎧⎪⎨⎪⎩^𝑟_𝑙=0⎛⎜⎜⎝𝑟𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑠𝛼++𝛼𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑛𝛼−+𝑠𝑟+−𝛼𝑙⎞⎟⎟⎠1×(−1)_𝑘𝑟!⎫⎪⎬⎪⎭=1𝑟(𝑟−1)⋯(𝑟−𝑘+1)^∞0𝑒^−𝑥𝑥^𝑟+𝛼𝑑𝑥Γ(𝛼+𝑘+1)^𝑛_𝑟=𝑘(−1)^𝑟⎧⎪⎨⎪⎩^𝑟_𝑙=0⎛⎜⎜⎝𝑟𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑠𝛼++𝛼𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑛𝛼−+𝑠𝑟+−𝛼𝑙⎞⎟⎟⎠1×(−1)_𝑘𝑟!⎫⎪⎬⎪⎭𝑟!=(𝑟−𝑘)!Γ(𝛼+𝑟+1)_𝑛_𝑟=𝑘(−1)^𝑟+𝑘⎛⎜⎜⎝^𝑟_𝑙=0⎛⎜⎜⎝𝑟𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑠𝛼++𝛼𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑛𝛼−+𝑠𝑟+−𝛼𝑙⎞⎟⎟⎠(𝛼+𝑟)(𝛼+𝑟−1)⋯(𝛼+𝑘+1)=(𝑟−𝑘)!_𝑛⎞⎟⎟⎠_𝑟=𝑘(−1)^𝑟+𝑘⎛⎜⎜⎝^𝑟_𝑙=0⎛⎜⎜⎝𝑟𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑠𝛼++𝛼𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑛𝛼−+𝑠𝑟+−𝛼𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝛼𝑟−+𝑘𝑟⎞⎟⎟⎠⎞⎟⎟⎠.(2.46) Therefore, by (2.42) and (2.46), we obtain the following theorem.

Theorem 2.11. For 𝑠,𝑚∈ℤ₊ with 𝑠+𝑚=𝑛, 𝛼∈ℝ with 𝛼>−1, one has 𝐿^𝛼𝑠(𝑥)𝐿^𝛼𝑚(𝑥)=^𝑛_𝑛𝑘=0_𝑟=𝑘(−1)^𝑟+𝑘⎛⎜⎜⎝^𝑟_𝑙=0⎛⎜⎜⎝𝑟𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑠𝛼++𝛼𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑛𝛼−+𝑠𝑟+−𝛼𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝛼𝑟−+𝑘𝑟⎞⎟⎟⎠⎞⎟⎟⎠𝐿_𝛼𝑘(𝑥).(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.