Some Identities on Laguerre Polynomials in Connection with Bernoulli and Euler Numbers

Abstract

We study some interesting identities and properties of Laguerre polynomials in connection with Bernoulli and Euler numbers. These identities are derived from the orthogonality of Laguerre polynomials with respect to inner product ⟨𝑓,𝑔⟩=∫^∞₀𝑒^−𝑥2𝑓(𝑥)𝑔(𝑥)𝑑𝑥.

1. Introduction/Preliminaries

As is well known, Laguerre polynomials are defined by the generating function as exp(−𝑥𝑡/(1−𝑡))=1−𝑡^∞_𝑛=0𝐿_𝑛(𝑥)𝑡^𝑛 (1.1) (see [1, 2]). By (1.1), we get ^∞_𝑛=0𝐿_𝑛(𝑥)𝑡^𝑛=exp(−𝑥𝑡/(1−𝑡))=1−𝑡^∞_𝑟=0(−1)^𝑟𝑥^𝑟𝑡^𝑟()𝑟!1−𝑡^−(𝑟+1)=^∞∞_𝑛=0𝑟=0(−1)^𝑟𝑥^𝑟(𝑟+𝑠)!𝑡𝑟!𝑟!𝑠!^𝑟+𝑠=^∞_𝑛=0^𝑛_𝑟=0(−1)^𝑟(^𝑛_𝑟)𝑥𝑟!^𝑟𝑡^𝑛. (1.2) Thus, from (1.2), we have 𝐿_𝑛(𝑥)=^𝑛_𝑟=0(−1)^𝑟(^𝑛_𝑟)𝑥𝑟!^𝑟. (1.3) By (1.3), we see that 𝐿_𝑛(𝑥) is a polynomial of degree 𝑛 with rational coefficients and the leading coefficient (−1)^𝑛/𝑛!. It is well known that Rodrigues' formula is given by 𝐿_𝑛(𝑥)=1𝑒𝑛!^𝑥𝑑^𝑛𝑑𝑥^𝑛𝑒^−𝑥𝑥^𝑛 (1.4) (see [1–27]). From (1.1), we can derive the following of Laguerre polynomials: 𝐿₀(𝑥)=1,𝐿₁(𝑥)()𝐿=−𝑥+1,𝑛+1_𝑛+1(𝑥)=(2𝑛+1−𝑥)𝐿_𝑛(𝑥)−𝑛𝐿_𝑛−1(𝑥),(),𝐿𝑛≥1_′𝑛(𝑥)=𝐿^′_𝑛−1(𝑥)−𝐿_𝑛−1(𝑥)=0,(𝑛≥1),𝑥𝐿^′_𝑛(𝑥)=𝑛𝐿_𝑛(𝑥)−𝑛𝐿_𝑛−1(𝑥)=0,(𝑛≥1). (1.5)(1.6)(1.7) By (1.7), we easily see that 𝑢=𝐿_𝑛(𝑥) is a solution of the following differential equation of order 2: 𝑥𝑢^′′(𝑥)+(1−𝑥)𝑢^′(𝑥)+𝑛𝑢(𝑥)=0. (1.8) The Bernoulli numbers, 𝐵_𝑛, are defined by the generating function as 𝑡𝑒^𝑡−1=𝑒^𝐵𝑡=^∞_𝑛=0𝐵_𝑛𝑡^𝑛𝑛! (1.9) (see [1–28, 28]), with the usual convention about replacing 𝐵^𝑛 by 𝐵_𝑛.

It is well known that Bernoulli polynomials of degree 𝑛 are given by 𝐵_𝑛(𝑥)=(𝐵+𝑥)^𝑛=^𝑛_𝑙=0𝑛𝑙𝐵_𝑛−𝑙𝑥^𝑙 (1.10) (see [2, 26]). Thus, from (1.10), we have 𝐵^′_𝑛(𝑥)=𝑑𝐵_𝑛(𝑥)𝑑𝑥=𝑛𝐵_𝑛−1(𝑥) (1.11) (see [3–12]). From (1.9) and (1.10), we can derive the following recurrence relation: 𝐵₀=1,(𝐵+1)^𝑛−𝐵_𝑛=𝛿_1,𝑛 (1.12) where 𝛿_𝑛,𝑘 is Kronecker's symbol.

The Euler polynomials 𝐸_𝑛(𝑥) are also defined by the generating function as 2𝑒^𝑡𝑒+1^𝑥𝑡=𝑒^{𝐸(𝑥)𝑡}=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛𝑛! (1.13) (see [27, 28]), with the usual convention about replacing 𝐸^𝑛(𝑥) by 𝐸_𝑛(𝑥).

In this special case, 𝑥=0, 𝐸_𝑛(0)=𝐸_𝑛 are called the 𝑛th Euler numbers. From (1.13), we note that the recurrence formula of 𝐸_𝑛 is given by 𝐸₀=1,(𝐸+1)^𝑛+𝐸_𝑛=2𝛿_0,𝑛 (1.14) (see [24]). Finally, we introduce Hermite polynomials, which are defined by 𝑒^{2𝑥𝑡−𝑡2}=𝑒^{𝐻(𝑥)𝑡}=^∞_𝑛=0𝐻_𝑛(𝑥)𝑡^𝑛𝑛! (1.15) (see [29]). In the special case, 𝑥=0, 𝐻_𝑛(0)=𝐻_𝑛 is called the 𝑛-th Hermite number. By (1.15), we get 𝐻_𝑛(𝑥)=(𝐻+2𝑥)^𝑛=^𝑛_𝑙=0𝑛𝑙𝐻_𝑛−𝑙2^𝑙𝑥^𝑙 (1.16) (see [29]). It is not difficult to show that ^∞₀𝑒^−𝑥𝐿_𝑚(𝑥)𝐿_𝑛(𝑥)𝑑𝑥=𝛿_𝑚,𝑛,𝑚,𝑛∈ℤ₊=ℕ∪{0}. (1.17) In the present paper, we investigate some interesting identities and properties of Laguerre polynomials in connection with Bernoulli, Euler, and Hermite polynomials. These identities and properties are derived from (1.17).

2. Some Formulae on Laguerre Polynomials in Connection with Bernoulli, Euler, and Hermite Polynomials

Let 𝐏_𝑛=𝑝(𝑥)∈ℚ[𝑥]∣deg𝑝(𝑥)≤𝑛. (2.1) Then 𝐏_𝑛 is an inner product space with the inner product 𝑝₁(𝑥),𝑝₂(𝑥)=^∞₀𝑒^−𝑥𝑝₁(𝑥)𝑝₂(𝑥)𝑑𝑥,𝑝₁(𝑥),𝑝₂(𝑥)∈𝐏_𝑛. (2.2) By (1.17), (2.1), and (2.2), we see that 𝐿₀(𝑥),𝐿₁(𝑥),…,𝐿_𝑛(𝑥) are orthogonal basis for 𝐏_𝑛.

For 𝑝(𝑥)∈𝐏_𝑛, it is given by 𝑝(𝑥)=^𝑛_𝑘=0𝐶_𝑘𝐿_𝑘(𝑥), (2.3) where 𝐶_𝑘=𝑝(𝑥),𝐿_𝑘(𝑥)=^∞₀𝑒^−𝑥𝐿_𝑘(𝑥)𝑝(𝑥)1𝑑𝑥=𝑘!^∞₀𝑑^𝑘𝑑𝑥^𝑘𝑒^−𝑥𝑥^𝑘𝑝(𝑥)𝑑𝑥. (2.4) Let us take 𝑝(𝑥)=𝑥^𝑛∈𝐏_𝑛. From (2.3) and (2.4), we note that 𝐶_𝑘=1𝑘!^∞₀𝑑^𝑘𝑑𝑥^𝑘𝑒^−𝑥𝑥^𝑘𝑥^𝑛𝑑𝑥=−𝑛𝑘!^∞₀𝑑^𝑘−1𝑑𝑥^𝑘−1𝑒^−𝑥𝑥^𝑘𝑥^𝑛−1=(−(𝑑𝑥−𝑛)(𝑛−1))𝑘!^∞₀𝑑^𝑘−2𝑑𝑥^𝑘−2𝑒^−𝑥𝑥^𝑘𝑥^𝑛−2=()𝑑𝑥=⋯−1^𝑘𝑛(𝑛−1)⋯(𝑛−𝑘+1)𝑘!^∞₀𝑒^−𝑥𝑥^𝑛𝑑𝑥=(−1)^𝑘𝑛𝑘𝑛!. (2.5) Therefore, by (2.3), (2.4), and (2.5), we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℤ₊, one has 𝑥^𝑛=𝑛!^𝑛_𝑘=0(−1)^𝑘𝑛𝑘𝐿_𝑘(𝑥). (2.6)

Let us consider 𝑝(𝑥)=𝐵_𝑛(𝑥)∈𝐏_𝑛. Then, by (2.3) and (2.4), we get 𝐶_𝑘=1𝑘!^∞₀𝑑^𝑘𝑑𝑥^𝑘𝑒^−𝑥𝑥^𝑘𝐵_𝑛(𝑥)𝑑𝑥=−𝑛𝑘!^∞₀𝑑^𝑘−1𝑑𝑥^𝑘−1𝑒^−𝑥𝑥^𝑘𝐵_𝑛−1(𝑥)=(−(𝑑𝑥−𝑛)(𝑛−1))𝑘!^∞₀𝑑^𝑘−2𝑑𝑥^𝑘−2𝑒^−𝑥𝑥^𝑘𝐵_𝑛−2(𝑥)=()𝑑𝑥=⋯−1^𝑘𝑛(𝑛−1)⋯(𝑛−𝑘+1)𝑘!^𝑛−𝑘_𝑙=0𝑙𝐵𝑛−𝑘_{𝑛−𝑘−𝑙}^∞₀𝑒^−𝑥𝑥^𝑘+𝑙=()𝑑𝑥−1^𝑘𝑛𝑘^{𝑛−𝑘𝑙=0}𝑙𝐵𝑛−𝑘_{𝑛−𝑘−𝑙}(𝑘+𝑙)!=𝑛!(−1)^{𝑘𝑛−𝑘}_𝑙=0𝑘𝑘+𝑙𝐵_{𝑛−𝑘−𝑙}()!.𝑛−𝑘−𝑙 (2.7) Therefore, by (2.3), (2.4), and (2.7), we obtain the following theorem.

Theorem 2.2. For 𝑛∈ℤ₊, one has 𝐵_𝑛(𝑥)=𝑛!^𝑛_𝑘=0^𝑛−𝑘_𝑙=0(−1)^𝑘𝑘𝑘+𝑙𝐵_{𝑛−𝑘−𝑙}()!𝐿𝑛−𝑘−𝑙^𝑘(𝑥). (2.8)

Let us take 𝑝(𝑥)=𝐸_𝑛(𝑥)∈𝐏_𝑛. By the same method, we easily see that 𝐸_𝑛(𝑥)=𝑛!^𝑛_𝑘=0^𝑛−𝑘_𝑙=0(−1)^𝑘𝑘𝑘+𝑙𝐸_{𝑛−𝑘−𝑙}()!𝐿𝑛−𝑘−𝑙^𝑘(𝑥). (2.9)

For 𝑝(𝑥)=𝐻_𝑛(𝑥)∈𝐏_𝑛, we have 𝐻_𝑛(𝑥)=^𝑛_𝑘=0𝐶_𝑘𝐿_𝑘(𝑥), (2.10) where 𝐶_𝑘=1𝑘!^∞₀𝑑^𝑘𝑑𝑥^𝑘𝑒^−𝑥𝑥^𝑘𝐻_𝑛(𝑥)𝑑𝑥=−2𝑛𝑘!^∞₀𝑑^𝑘−1𝑑𝑥^𝑘−1𝑒^−𝑥𝑥^𝑘𝐻_𝑛−1(𝑥)=((𝑑𝑥−2𝑛)(−2𝑛−1))𝑘!^∞₀𝑑^𝑘−2𝑑𝑥^𝑘−2𝑒^−𝑥𝑥^𝑘𝐻_𝑛−2(𝑥)=((⋯((𝑑𝑥=⋯−2𝑛)(−2𝑛−1))−2𝑛−𝑘+1))𝑘!^∞₀𝑒^−𝑥𝑥^𝑘𝐻_𝑛−𝑘(𝑥)=()𝑑𝑥−1^𝑘2^𝑘𝑛(𝑛−1)⋯(𝑛−𝑘+1)𝑘!^𝑛−𝑘_𝑙=0𝑙𝐻𝑛−𝑘_{𝑛−𝑘−𝑙}2^𝑙∞₀𝑒^−𝑥𝑥^𝑘+𝑙=()𝑑𝑥−1^𝑘𝑛𝑘^{𝑛−𝑘𝑙=0}𝑙2𝑛−𝑘_𝑘+𝑙𝐻_{𝑛−𝑘−𝑙}(𝑘+𝑙)!=𝑛!(−1)^{𝑘𝑛−𝑘}_𝑙=02^𝑘+𝑙𝑘𝑘+𝑙𝐻_{𝑛−𝑘−𝑙}()!.𝑛−𝑘−𝑙 (2.11) Therefore, by (2.10) and (2.11), we obtain the following theorem.

Theorem 2.3. For 𝑛∈ℤ₊, one has 𝐻_𝑛(𝑥)=𝑛!^𝑛_𝑘=0^𝑛−𝑘_𝑙=0(−1)^𝑘2^𝑘+𝑙𝑘𝑘+𝑙𝐻_{𝑛−𝑘−𝑙}()!𝐿𝑛−𝑘−𝑙^𝑘(𝑥). (2.12)

Let 𝑝(𝑥)=∑^𝑛_𝑘=0𝐵_𝑘(𝑥)𝐵_𝑛−𝑘(𝑥)∈𝐏_𝑛. Then we have 𝑝(𝑥)=^𝑛_𝑘=0𝐵_𝑘(𝑥)𝐵_𝑛−𝑘(𝑥)=^𝑛_𝑘=0𝐶_𝑘𝐿_𝑘(𝑥), (2.13) where 𝐶_𝑘=1𝑘!^∞₀𝑑^𝑘𝑑𝑥^𝑘𝑒^−𝑥𝑥^𝑘𝑝(𝑥)𝑑𝑥. (2.14) In [15], it is known that ^𝑛_𝑘=0𝐵_𝑘(𝑥)𝐵_𝑛−𝑘(𝑥)=2𝑛+2^𝑛−2_𝑙=0𝑙𝐵𝑛+2_𝑛−𝑙𝐵_𝑙(𝑥)+(𝑛+1)𝐵_𝑛(𝑥). (2.15) By (2.14) and (2.15), we get 𝐶_𝑘=12𝑘!𝑛+2^𝑛−2_𝑙=0𝑙𝐵𝑛+2_𝑛−𝑙^∞₀𝑑^𝑘𝑑𝑥^𝑘𝑒^−𝑥𝑥^𝑘𝐵_𝑙(𝑥)+()𝑑𝑥𝑛+1^∞₀𝑑^𝑘𝑑𝑥^𝑘𝑒^−𝑥𝑥^𝑘𝐵_𝑛(𝑥)𝑑𝑥. (2.16) From (2.16), we can derive the following equations ((2.17)-(2.18)): 𝐶_𝑛=(−1)^𝑛(𝑛+1)!,𝐶_𝑛−1=𝑛(𝑛+1)!(−1)^𝑛−1−12(−1)^𝑛−1(𝑛+1)!. (2.17) For 0≤𝑘≤𝑛−2, we have 𝐶_𝑘=2𝑛+2^𝑛−2_𝑙=𝑘^𝑙−𝑘_𝑚=0𝑙𝑘()𝑛+2𝑘+𝑚𝑙!−1^𝑘𝐵_𝑛−𝑙𝐵_{𝑙−𝑘−𝑚}()!+()𝑙−𝑘−𝑚−1^𝑘(𝑛+1)!^𝑛−𝑘_𝑚=0𝑘𝑘+𝑚𝐵_{𝑛−𝑘−𝑚}()!.𝑛−𝑘−𝑚 (2.18) Therefore, by (2.13), (2.17), and (2.18), we obtain the following theorem.

Theorem 2.4. For 𝑛∈ℤ₊, one has ^𝑛_𝑘=0𝐵_𝑘(𝑥)𝐵_𝑛−𝑘(𝑥)=^𝑛−2_𝑘=02𝑛+2^𝑛−2_𝑙=𝑘^𝑙−𝑘_𝑚=0(−1)^𝑘𝑙𝑘𝑙!𝐵𝑛+2𝑘+𝑚_𝑛−𝑙𝐵_{𝑙−𝑘−𝑚}()!+()𝑙−𝑘−𝑚−1^𝑘(𝑛+1)!^𝑛−𝑘_𝑚=0𝑘𝑘+𝑚𝐵_{𝑛−𝑘−𝑚}()!𝐿𝑛−𝑘−𝑚^𝑘(𝑥)+𝑛(𝑛+1)!(−1)^𝑛−1−12(−1)^𝑛−1(𝑛+1)!𝐿_𝑛−1(𝑥)+(−1)^𝑛(𝑛+1)!𝐿_𝑛(𝑥). (2.19)

Let us take 𝑝(𝑥)=∑^𝑛_𝑘=0𝐸_𝑘(𝑥)𝐸_𝑛−𝑘(𝑥)∈𝐏_𝑛. By (2.3) and (2.4), we get 𝑝(𝑥)=^𝑛_𝑘=0𝐸_𝑘(𝑥)𝐸_𝑛−𝑘(𝑥)=^𝑛_𝑘=0𝐶_𝑘𝐿_𝑘(𝑥), (2.20) where 𝐶_𝑘=1𝑘!^∞₀𝑑^𝑘𝑑𝑥^𝑘𝑒^−𝑥𝑥^𝑘𝑝(𝑥)𝑑𝑥. (2.21) It is known (see [15]) that ^𝑛_𝑘=0𝐸_𝑘(𝑥)𝐸_𝑛−𝑘(𝑥)=^𝑛−1_𝑘=0(𝑛+1)(^𝑛_𝑘)𝑛−𝑘+1^𝑛_𝑙=𝑘𝐸_𝑙−𝑘𝐸_𝑛−𝑙−2𝐸_𝑛−𝑘𝐸_𝑘(𝑥)+2(𝑛+1)𝐸_𝑛(𝑥). (2.22) From (2.20), (2.21), and (2.22), we can derive the following equations ((2.23)-(2.24)): 𝐶_𝑛=(−1)^𝑛2()𝑛!𝑛+1)(𝑛!²=2(−1)^𝑛(𝑛+1)!. (2.23) For 0≤𝑘≤𝑛−1, we have 𝐶_𝑘=^𝑛−1_𝑙=𝑘(𝑛+1)(^𝑛_𝑙)(𝑛−𝑙+1)!^𝑛_𝑚=𝑙𝐸_𝑚−𝑙𝐸_𝑛−𝑚−2𝐸_𝑛−𝑙^𝑙−𝑘_𝑝=0(−1)^𝑘𝑘𝐸𝑙!𝑘+𝑝_{𝑙−𝑘−𝑝}!()!()𝑙−𝑘−𝑝+2𝑛+1−1^{𝑘𝑛−𝑘}_𝑝=0𝑘𝑘+𝑝𝐸_{𝑛−𝑘−𝑝}!.𝑛−𝑘−𝑝 (2.24) Therefore, by (2.20) and (2.24), we obtain the following theorem.

Theorem 2.5. For 𝑛∈ℤ₊, one has ^𝑛_𝑘=0𝐸_𝑘(𝑥)𝐸_𝑛−𝑘(𝑥)=^𝑛−1_𝑘=0^𝑛−1_𝑙=𝑘(𝑛+1)(^𝑛_𝑙)(𝑛−𝑙+1)!^𝑛_𝑚=𝑙𝐸_𝑚−𝑙𝐸_𝑛−𝑚−2𝐸_𝑛−𝑙^𝑙−𝑘_𝑝=0(−1)^𝑘𝑘×𝐸𝑙!𝑘+𝑝_{𝑙−𝑘−𝑝}!()!()𝑙−𝑘−𝑝+2𝑛+1−1^{𝑘𝑛−𝑘}_𝑝=0𝑘𝑘+𝑝𝐸_{𝑛−𝑘−𝑝}!𝐿𝑛−𝑘−𝑝^𝑘(𝑥)+2(−1)^𝑛(𝑛+1)!𝐿_𝑛(𝑥). (2.25)

It is known that ^𝑛_𝑘=0𝐸_𝑘(𝑥)𝐸_𝑛−𝑘(𝑥)(𝑥)4=𝑝=−𝑛+2^𝑛_𝑘=0𝑘𝐸𝑛+2_{𝑛−𝑘+1}𝐵_𝑘(𝑥) (2.26) (see [15]). From (2.20), (2.21), and (2.23), we have 𝐶_𝑘=1𝑘!^∞₀𝑑^𝑘𝑑𝑥^𝑘𝑒^−𝑥𝑥^𝑘𝑝(𝑥)4𝑑𝑥=−𝑛+2^𝑛_𝑙=𝑘𝑙𝐸𝑛+2_{𝑛−𝑙+1}1𝑘!^∞₀𝑑^𝑘𝑒^−𝑥𝑥^𝑘𝑑𝑥^𝑘𝐵_𝑙(𝑥)4𝑑𝑥=−𝑛+2^𝑛_𝑙=𝑘^𝑙−𝑘_𝑚=0𝑙()𝑛+2−1^𝑘𝐸_{𝑛−𝑙+1}𝐵_{𝑙−𝑘−𝑚}()!𝑘.𝑙−𝑘−𝑚𝑙!𝑚+𝑘 (2.27) Therefore, by (2.20) and (2.27), we obtain the following theorem.

Theorem 2.6. For 𝑛∈ℤ₊, one has ^𝑛_𝑘=0𝐸_𝑘(𝑥)𝐸_𝑛−𝑘(𝑥)4=−𝑛+2^𝑛𝑛_{𝑘=0𝑙=𝑘}^𝑙−𝑘_𝑚=0𝑙()𝑛+2−1^𝑘𝐸_{𝑛−𝑙+1}𝐵_{𝑙−𝑘−𝑚}()!𝑘𝐿𝑙−𝑘−𝑚𝑙!𝑚+𝑘_𝑘(𝑥). (2.28)

Remark 2.7. Laguerre's differential equation 𝑡𝑦^′′+(1−𝑡)𝑦^′+𝑛𝑦=0 (2.29) is known to possess polynomial solutions when 𝑛 is a nonnegative integer. These solutions are naturally called Laguerre polynomials and are denoted by 𝐿_𝑛(𝑡). That is, 𝑦=𝐿_𝑛(𝑡) are solutions of (2.29) which are given by 𝑦=𝐿_𝑛(𝑡)=^𝑛_𝑟=0(^𝑛_𝑟)(−1)^𝑟𝑡𝑟!^𝑟,𝐿₀(1)=1. (2.30) From (2.30), we note that Laplace transform of 𝑦=𝐿_𝑛(𝑡) is given by ℒ𝑦=ℒ𝐿_𝑛(𝑡)=1𝑠^𝑛_𝑟=0𝑛𝑟(−1)^𝑟1𝑠^𝑟=(𝑠−1)^𝑛𝑠^𝑛+1. (2.31) It is not difficult to show that ℒ𝑒^𝑡𝑑𝑛!𝑑𝑡^𝑛𝑛𝑒^−𝑡𝑡^𝑛𝑦=()=ℒ𝑠−1^𝑛𝑠^𝑛+1. (2.32) Thus, we conclude that 𝐿_𝑛(𝑡)=𝑒^𝑡𝑑𝑛!𝑑𝑡^𝑛𝑛𝑒^−𝑡𝑡^𝑛,for𝑛∈ℤ₊. (2.33)