Kim et al. Advances in Difference Equations (2016) 2016:159 DOI 10.1186/s13662-016-0896-1
R E S E A R C H Open Access Some identities of Laguerre polynomials arising from differential equations Taekyun Kim1,2,DaeSanKim3, Kyung-Won Hwang4* andJongJinSeo5
*Correspondence: [email protected] Abstract 4Department of Mathematics, Dong-A University, Busan, 49315, In this paper, we derive a family of ordinary differential equations from the generating Republic of Korea function of the Laguerre polynomials. Then these differential equations are used in Full list of author information is order to obtain some properties and new identities for those polynomials. available at the end of the article MSC: 05A19; 33C45; 11B37; 35G35 Keywords: Laguerre polynomials; differential equations
1 Introduction
The Laguerre polynomials, Ln(x)(n ≥ ), are defined by the generating function
xt ∞ e– –t = L (x)tn (see [, ]). () –t n n=
Indeed, the Laguerre polynomial Ln(x) is a solution of the second order linear differential equation
xy +(–x)y + ny = (see[–]). ()
From (), we can get the following equation: