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Some Identities of Laguerre Polynomials Arising from Differential Equations Taekyun Kim1,2,Daesankim3, Kyung-Won Hwang4* Andjongjinseo5

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Some Identities of Laguerre Polynomials Arising from Differential Equations Taekyun Kim1,2,Daesankim3, Kyung-Won Hwang4* Andjongjinseo5 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: ∞ xt ∞ e– –t (–)mxmtm L (x)tn = = ( – t)–m– n –t m! n= m= ∞ ∞ (–)mxmtm m + l = tl m! l m= l= ∞ n (–)m n xm = m tn.() m! n= m= Thus by (), we get immediately the following equation: n (–)m n xm L (x)= m (n ≥ ) see [, –] .() n m! m= © 2016 Kim et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Kim et al. Advances in Difference Equations (2016)2016:159 Page 2 of 9 Alternatively, the Laguerre polynomials are also defined by the recurrence relation as fol- lows: 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 dn L (x)= ex e–xxn (n ≥ ). () n n! dxn 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 solu- tion 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 defi- nition of the Laguerre polynomial is given by –xt –t e –n– Ln(x)= t dt see [, , , ] ,() πi C –t 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. Spec- tral 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 ap- plying 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, combi- natorics and special functions (see [–]). For the applications of special functions and polynomials, one may referred to the papers (see [, , ]). Kim et al. Advances in Difference Equations (2016)2016:159 Page 3 of 9 In [], Kim studied nonlinear differential equations arising from Frobenius-Euler poly- nomials 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 identi- ties for those polynomials. 2 Laguerre polynomials arising from linear differential equations Let –xt F = F(t, x)= e –t .() –t From (), we note that dF(t, x) F() = = ( – t)– – x( – t)– F.() dt Thus, by (), we get dF() F() = = ( – t)– –x( – t)– + x( – t)– F () dt and dF() F() = = ( – t)– –x( – t)– +x( – t)– – x( – t)– F.() dt So we are led to put N (N) –i F = ai–N (N, x)( – t) F,() i=N where N =,,,.... From (), we can get equation (): N N (N+) –i– –i () F = ai–N (N, x)i( – t) F + ai–N (N, x)( – t) F i=N i=N N –i– = ai–N (N, x)i( – t) F i=N N –i – – + ai–N (N, x)( – t) ( – t) – x( – t) F i=N N N –i– –i– = (i +)ai–N (N, x)( – t) – x ai–N (N, x)( – t) F i=N i=N N+ N+ –i –i = iai–N–(N, x)( – t) – x ai–N–(N, x)( – t) F. () i=N+ i=N+ Kim et al. Advances in Difference Equations (2016)2016:159 Page 4 of 9 Replacing N by N +in(), we get N+ (N+) –i F = ai–N–(N +,x)( – t) F.() i=N+ Comparing the coefficients on both sides of ()and(), we have a(N +,x)=(N +)a(N, x), () aN+(N +,x)=–xaN (N, x), () and 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 both sides of (), we get a(, x)=, a(, x)=–x.() From (), (), we get a(N +,x)=(N +)aN (N, x)=(N +)NaN–(N –,x) ··· =(N +)N(N –)···a(, x)=(N + )! () and aN+(N +,x)=(–x)aN (N, x)=(–x) aN–(N –,x) ··· N N+ =(–x) a(, x)=(–x) .() We observe that the matrix [ai(j, x)]≤i,j≤N is given by ⎡ ⎤ !!··· N! ⎢ ⎥ ⎢ ··· ⎥ ⎢(–x) ⎥ ⎢ ⎥ ⎢(–x) ⎥ ⎢ ⎥ . ⎢ . ⎥ ⎣ . ⎦ ··· (–x)N Kim et al. Advances in Difference Equations (2016)2016:159 Page 5 of 9 From (), we can get the following equations: a(N +,x)=–xa(N, x)+(N +)a(N, x) =–x a(N, x)+(N +)a(N –,x) +(N +)(N +)a(N –,x) = ··· N– =–x (N +)ia(N – i, x)+(N +)(N +)···a(, x) i= N– =–x (N +)ia(N – i, x)+(N +)(N +)···(–x) i= N =–x (N +)ia(N – i, x), () i= a(N +,x)=–xa(N, x)+(N +)a(N, x) =–x a(N, x)+(N +)a(N –,x) +(N +)(N +)a(N –,x) = ··· N– =–x (N +)ia(N – i, x)+(N +)(N +)···a(, x) i= N– =–x (N +)ia(N – i, x)+(N +)(N +)···(–x) i= N– =–x (N +)ia(N – i, x), () i= and a(N +,x)=–xa(N, x)+(N +)a(N, x) =–x a(N, x)+(N +)a(N –,x) +(N +)(N +)a(N –,x) = ··· N– =–x (N +)ia(N – i, x)+(N +)(N +)···a(, x) i= N– =–x (N +)ia(N – i, x)+(N +)(N +)···(–x) i= N– =–x (N +)ia(N – i, x), () i= where (x)n = x(x –)···(x – n +)(n ≥ ), and (x) =. Continuing this process, we have N–j+ aj(N +,x)=–x (N + j +)iaj–(N – i, x), () i= Kim et al. Advances in Difference Equations (2016)2016:159 Page 6 of 9 where j =,,...,N. Now we give explicit expressions for aj(N +,x), j =,,...,N.From ()and(), we note that N a(N +,x)=–x (N +)i a(N – i, x) i= N =–x (N +)i (N – i)!. () i= By ()and(), we get N– a(N +,x)=–x (N +)i a(N – i, x) i= N– N–i– – =(–x) (N +)i (N – i +)i (N – i – i – )!. () i= i= From ()and(), we get N– a(N +,x)=–x (N +)i a(N – i, x) i= N– N–i– N–i–i– – =(–x) (N +)i (N – i +)i (N – i – i)i i= i= i= × (N – i – i – i – )!. () By continuing this process, we get ··· N–j+ N–ij–j+ N–ij––i–j+ j ··· aj(N +,x)=(–x) (N + j +)ij ij= ij–= i= j × N – ij – ···– ik – j –(k –) ik– k= × (N – ij – ···– i – j + )!. () Therefore, we obtain the following theorem. Theorem The linear differential equation N (N) –i F = ai–N (N, x)( – t) F (N ∈ N) i=N Kim et al. Advances in Difference Equations (2016)2016:159 Page 7 of 9 – xt N has a solution F = F(t, x)=(–t) exp(– –t ), where a(N, x)=N!, aN (N, x)=(–x) , ··· N–j N–ij–j N–ij– –i–j j ··· aj(N, x)=(–x) (N + j)ij ij= ij–= i= j × N – ij – ···– ik – j –(k –) (N – ij – ···– i – j)!. ik– k= From (), we note that xt ∞ e– –t F = F(t, x)= = L (x)tn.() –t n n= Thus, by (), we get ∞ ∞ d N F(N) = F(t, x)= L (x)(n) tn–N = L (x)(n + N) tn.() dt n N n+N N n=N n= On the other hand, by Theorem ,wehave N (N) –i F = ai–N (N, x)( – t) F i=N ∞ ∞ N i + l – = a (N, x) tl L (x)tk i–N l k i=N l= k= ∞ N n i + l – = a (N, x) L (x) tn i–N l n–l i=N n= l= ∞ N N i + l – = a (N, x) L (x) tn.() i–N l n–l n= i=N l= Therefore, by comparing the coefficients on both sides of ()and(), we have the fol- lowing theorem.
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