Some identities of Laguerre polynomials arising from differential equations

Advances in Difference Equations, Jun 2016

In this paper, we derive a family of ordinary differential equations from the generating function of the Laguerre polynomials. Then these differential equations are used in order to obtain some properties and new identities for those polynomials. MSC: 05A19, 33C45, 11B37, 35G35.

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Some identities of Laguerre polynomials arising from differential equations

Kim et al. Advances in Difference Equations Some identities of Laguerre polynomials arising from differential equations Taekyun Kim 0 3 4 Dae San Kim 0 2 Kyung-Won Hwang 0 1 6 Jong Jin Seo 0 5 0 Dong-A University , Busan, 49315 , Republic of Korea 1 Department of Mathematics 2 Department of Mathematics, Sogang University , Seoul, 04107 , Republic of Korea 3 Department of Mathematics, Kwangwoon University , Seoul, 139-701 , Republic of Korea 4 Department of Mathematics, College of Science, Tianjin Polytechnic University , Tianjin, 300387 , China 5 Department of Applied mathematics, Pukyong National University , Busan, 48513 , Republic of Korea 6 Department of Mathematics, Dong-A University , Busan, 49315 , Republic of Korea In this paper, we derive a family of ordinary differential equations from the generating function of the Laguerre polynomials. Then these differential equations are used in order to obtain some properties and new identities for those polynomials. Laguerre polynomials; differential equations - 1 Introduction e– x–tt  – t = ∞ n= equation Ln(x)tn (see [, ]). xy + ( – x)y + ny =  (see [–]). From (), we can get the following equation: ∞ n= Ln(x)tn = = = e– x–tt  – t = m= ∞ n= m= m= m! ∞ (–)mxmtm ∞ n (–)m n m x m m! ∞ (–)mxmtm ( – t)–m– m! l= m + l l tn. tl Thus by (), we get immediately the following equation: Ln(x) = m= n (–)m n m x m m! (n ≥ ) see [, –] . () () () () Alternatively, the Laguerre polynomials are also defined by the recurrence relation as follows: () () L(x) = , L(x) =  – x, (n + )Ln+(x) = (n +  – x)Ln(x) – nLn–(x) (n ≥ ). The Rodrigues’ formula for the Laguerre polynomials is given by Ln(x) = n! ex ddxnn e–xxn (n ≥ ). The first few of Ln(x) (n ≥ ) are L(x) = , L(x) =  – x, L(x) =  x – x +  , L(x) =  –x + x – x +  , L(x) =  x – x + x – x +  . The Laguerre polynomials arise from quantum mechanics in the radial part of the solution of the Schrödinger equation for a one-electron action. They also describe the static Wigner functions of oscillator system in the quantum mechanics of phase space. They further enter in the quantum mechanics of the Morse potential and of the D isotropic harmonic oscillator (see [, , ]). A contour integral that is commonly taken as the definition of the Laguerre polynomial is given by Ln(x) = π i C e –––xttt t–n– dt see [, , , ] , () where the contour encloses the origin but not the point z = . FDEs (fractional differential equations) have wide applications in such diverse areas as fluid mechanics, plasma physics, dynamical processes and finance, etc. Most FDEs do not have exact solutions and hence numerical approximation techniques must be used. Spectral methods are widely used to numerically solve various types of integral and differential equations due to their high accuracy and employ orthogonal systems as basis functions. It is remarkable that a new family of generalized Laguerre polynomials are introduced in applying spectral methods for numerical treatments of FDEs in unbounded domains. They can also be used in solving some differential equations (see [–]). Also, it should be mentioned that the modified generalized Laguerre operational matrix of fractional integration is applied in order to solve linear multi-order FDEs which are important in mathematical physics (see [–]). Many authors have studied the Laguerre polynomials in mathematical physics, combinatorics and special functions (see [–]). For the applications of special functions and polynomials, one may referred to the papers (see [, , ]). F = F(t, x) =   – t –xt e –t . From (), we note that F() = dFd(tt, x) = ( – t)– – x( – t)– F. Thus, by (), we get and So we are led to put F() = dF() F() = dF() dt dt N i=N = ( – t)– – x( – t)– + x( – t)– – x( – t)– F. F(N+) = ai–N (N , x)i( – t)–i– F + ai–N (N , x)( – t)–i F() In [], Kim studied nonlinear differential equations arising from Frobenius-Euler polynomials and gave some interesting identities. In this paper, we derive a family of ordinary differential equations from the generating function of the Laguerre polynomials. Then these differential equations are used in order to obtain some properties and new identities for those polynomials. 2 Laguerre polynomials arising from linear differential equations Let () () () () () () = = = N Replacing N by N +  in (), we get F(N+) = i=NN++ ai–N–(N + , x)( – t)–i F. Comparing the coefficients on both sides of () and (), we have a(N + , x) = (N + )a(N , x), and aN+(N + , x) = –xaN (N , x), ai–N–(N + , x) = iai–N–(N , x) – xai–N–(N , x) (N +  ≤ i ≤ N + ). We note that F = F() = a(, x)F. Thus, by (), we get a(, x) = . From () and (), we note that ( – t)– – x( – t)– F = F() = a(, x)( – t)– + a(, x)( – t)– F. Thus, by comparing the coefficients on bot (...truncated)


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Taekyun Kim, Dae Kim, Kyung-Won Hwang, Jong Seo. Some identities of Laguerre polynomials arising from differential equations, Advances in Difference Equations, 2016, pp. 159, 2016, DOI: 10.1186/s13662-016-0896-1