Some identities of Laguerre polynomials arising from differential equations

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1 Kim et a. Advances in Difference Equations 2016) 2016:159 DOI /s R E S E A R C H Open Access Some identities of Laguerre poynomias arising from differentia equations Taekyun Kim 1,2,DaeSanKim 3, Kyung-Won Hwang 4* andjongjinseo 5 * Correspondence: khwang@dau.ac.kr 4 Department of Mathematics, Dong-A University, Busan, 49315, Repubic of Korea Fu ist of author information is avaiabe at the end of the artice Abstract In this paper, we derive a famiy of ordinary differentia equations from the generating function of the Laguerre poynomias. Then these differentia equations are used in order to obtain some properties and new identities for those poynomias. MSC: 05A19; 33C45; 11B37; 35G35 Keywords: Laguerre poynomias; differentia equations 1 Introduction The Laguerre poynomias, L n x)n 0), are defined by the generating function e 1 t xt 1 t L n x)t n see [1, 2]). 1) n0 Indeed, the Laguerre poynomia L n x) is a soution of the second order inear differentia equation xy +1 x)y + ny 0 see[2 5]). 2) From 1), we can get the foowing equation: L n x)t n n0 xt e 1 t 1 t m0 1) m x m t m 1 t) m 1 m! 1) m x m t m ) m + t m! m0 0 n 1) m ) ) n m x m t n. 3) m! n0 m0 Thus by 3), we get immediatey the foowing equation: L n x) n m0 1) m ) n m x m m! n 0) see [2, 6 8] ). 4) 2016 Kim et a. This artice is distributed under the terms of the Creative Commons Attribution 4.0 Internationa License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the origina authors) and the source, provide a ink to the Creative Commons icense, and indicate if changes were made.

2 Kim eta. Advances in Difference Equations 2016) 2016:159 Page 2 of 9 Aternativey, the Laguerre poynomias are aso defined by the recurrence reation as foows: L 0 x)1, L 1 x)1 x, n +1)L n+1 x)2n +1 x)l n x) nl n 1 x) n 1). 5) The Rodrigues formua for the Laguerre poynomias is given by L n x) 1 dn ex e x x n) n 0). 6) n! dx n The first few of L n x)n 0) are L 0 x)1, L 1 x)1 x, L 2 x) 1 2 x 2 4x +2 ), L 3 x) 6 1 x 3 +9x 2 18x +6 ), L 4 x) 1 x 4 16x 3 +72x 2 96x +24 ). 24 The Laguerre poynomias arise from quantum mechanics in the radia part of the soution of the Schrödinger equation for a one-eectron action. They aso describe the static Wigner functions of osciator system in the quantum mechanics of phase space. They further enter in the quantum mechanics of the Morse potentia and of the 3D isotropic harmonic osciator see [4, 5, 9]). A contour integra that is commony taken as the definition of the Laguerre poynomia is given by L n x) 1 e xt 1 t 2πi C 1 t t n 1 dt see [4, 5, 10, 11] ), 7) where the contour encoses the origin but not the point z 1. FDEs fractiona differentia equations) have wide appications in such diverse areas as fuid mechanics, pasma physics, dynamica processes and finance, etc. Most FDEs do not have exact soutions and hence numerica approximation techniques must be used. Spectra methods are widey used to numericay sove various types of integra and differentia equations due to their high accuracy and empoy orthogona systems as basis functions. It is remarkabe that a new famiy of generaized Laguerre poynomias are introduced in appying spectra methods for numerica treatments of FDEs in unbounded domains. They can aso be used in soving some differentia equations see [12 17]). Aso, it shoud be mentioned that the modified generaized Laguerre operationa matrix of fractiona integration is appied in order to sove inear muti-order FDEs which are important in mathematica physics see [12 17]). Many authors have studied the Laguerre poynomias in mathematica physics, combinatorics and specia functions see [1 30]). For the appications of specia functions and poynomias, one may referred to the papers see [18, 19, 28]).

3 Kim eta. Advances in Difference Equations 2016) 2016:159 Page 3 of 9 In [22], Kim studied noninear differentia equations arising from Frobenius-Euer poynomias and gave some interesting identities. In this paper, we derive a famiy of ordinary differentia equations from the generating function of the Laguerre poynomias. Then these differentia equations are used in order to obtain some properties and new identities for those poynomias. 2 Laguerre poynomias arising from inear differentia equations Let F Ft, x) 1 1 t e xt 1 t. 8) From 8), we note that F 1) dft, x) dt Thus, by 3), we get 1 t) 1 x1 t) 2) F. 9) and F 2) df1) dt 21 t) 2 4x1 t) 3 + x 2 1 t) 4) F 10) F 3) df2) dt So we are ed to put F N) 2N 61 t) 3 18x1 t) 4 +9x 2 1 t) 5 x 3 1 t) 6) F. 11) a i N N, x)1 t) i )F, 12) where N 0,1,2,... From 12), we can get equation 13): F N+1) 2N 2N + 2N 2N 2N+1 +1 a i N N, x)i1 t) i 1 )F + a i N N, x)i1 t) i 1 )F 2N a i N N, x)1 t) i ) a i N N, x)1 t) i ) 1 t) 1 x1 t) 2) F i +1)a i N N, x)1 t) i 1 x ia i N 1 N, x)1 t) i x 2N N F 1) a i N N, x)1 t) i 2 )F a i N 2 N, x)1 t) i )F. 13)

4 Kim eta. Advances in Difference Equations 2016) 2016:159 Page 4 of 9 Repacing N by N +1in12), we get F N+1) 2N+2 +1 a i N 1 N +1,x)1 t) i )F. 14) Comparing the coefficients on both sides of 13)and14), we have and a 0 N +1,x)N +1)a 0 N, x), 15) a N+1 N +1,x)a N N, x), 16) a i N 1 N +1,x)ia i N 1 N, x) xa i N 2 N, x) N +2 i 2N + 1). 17) We note that F F 0) a 0 0, x)f. 18) Thus, by 18), we get a 0 0, x)1. 19) From 9)and12), we note that 1 t) 1 x1 t) 2) F F 1) a 0 1, x)1 t) 1 + a 1 1, x)1 t) 2) F. 20) Thus, by comparing the coefficients on both sides of 20), we get a 0 1, x)1, a 1 1, x). 21) From 15), 16), we get and a 0 N +1,x)N +1)a N N, x)n +1)Na N 1 N 1,x) N +1)NN 1) 2a 0 1, x)n + 1)! 22) a N+1 N +1,x) x)a N N, x) x) 2 a N 1 N 1,x) x) N a 1 1, x) x) N+1. 23) We observe that the matrix [a i j, x)] 0 i,j N is given by 1 1! 2! N! 0 x) 0 0 x) x) N

5 Kim eta. Advances in Difference Equations 2016) 2016:159 Page 5 of 9 From 17), we can get the foowing equations: and a 1 N +1,x)a 0 N, x)+n +2)a 1 N, x) { a 0 N, x)+n +2)a 0 N 1,x) } +N +2)N +1)a 1 N 1,x) N 1 N +2) i a 0 N i, x)+n +2)N +1) 3a 1 1, x) N 1 N +2) i a 0 N i, x)+n +2)N +1) 3 x) N N +2) i a 0 N i, x), 24) a 2 N +1,x)a 1 N, x)+n +3)a 2 N, x) { a 1 N, x)+n +3)a 1 N 1,x) } +N +3)N +2)a 2 N 1,x) N 2 N +3) i a 1 N i, x)+n +3)N +2) 5a 2 2, x) N 2 N +3) i a 1 N i, x)+n +3)N +2) 5 x) 2 N 1 N +3) i a 1 N i, x), 25) a 3 N +1,x)a 2 N, x)+n +4)a 3 N, x) { a 2 N, x)+n +4)a 2 N 1,x) } +N +4)N +3)a 3 N 1,x) N 3 N +4) i a 2 N i, x)+n +4)N +3) 7a 3 3, x) N 3 N +4) i a 2 N i, x)+n +4)N +3) 7 x) 3 N 2 N +4) i a 2 N i, x), 26) where x) n xx 1) x n +1)n 1), and x) 0 1. Continuing this process, we have N j+1 a j N +1,x) N + j +1) i a j 1 N i, x), 27)

6 Kim eta. Advances in Difference Equations 2016) 2016:159 Page 6 of 9 where j 1,2,...,N. Now we give expicit expressions for a j N +1,x), j 1,2,...,N.From 22)and24), we note that N a 1 N +1,x) N +2) i1 a 0 N i 1, x) i 1 0 N N +2) i1 N i 1 )!. 28) i 1 0 By 25)and28), we get N 1 a 2 N +1,x) N +3) i2 a 1 N i 2, x) i 2 0 N 1 N i 2 1 x) 2 i 2 0 N +3) i2 N i 2 +1) i1 N i 2 i 1 1)!. 29) i 1 0 From 26)and29), we get N 2 a 3 N +1,x) N +4) i3 a 2 N i 3, x) i 3 0 x) 3 N 2 i 3 0 N i 3 2 i 2 0 N i 3 i 2 2 i 1 0 N +4) i3 N i 3 +2) i2 N i 3 i 2 ) i1 N i 3 i 2 i 1 2)!. 30) By continuing this process, we get N j+1 a j N +1,x) x) j i j 0 N i j j+1 i j 1 0 N i j i 2 j+1 i 1 0 N + j +1) ij j N i j i k j 2k 1) ) ) i k 1 k2 N i j i 1 j + 1)!. 31) Therefore, we obtain the foowing theorem. Theorem 1 The inear differentia equation 2N F N) a i N N, x)1 t) )F i N N)

7 Kim eta. Advances in Difference Equations 2016) 2016:159 Page 7 of 9 has a soution F Ft, x)1 t) 1 exp xt 1 t ), where a 0N, x)n!, a N N, x) x) N, N j a j N, x) x) j N i j j i j 0 i j 1 0 N i j i 2 j i 1 0 N + j) ij j N ij i k j 2k 2) )) ) N i i j i 1 j)!. k 1 k2 From 1), we note that xt e 1 t F Ft, x) 1 t L n x)t n. 32) Thus, by 32), we get n0 ) d N F N) Ft, x) dt L n x)n) N t n N L n+n x)n + N) N t n. 33) nn On the other hand, by Theorem 1,wehave F N) 2N 2N 2N a i N N, x)1 t) i )F a i N N, x) a i N N, x) 2N n0 ) i + 1 t 0 n0 0 k0 n0 L k x)t k n ) ) i + 1 L n x) 0 t n N ) ) i + 1 a i N N, x) L n x) t n. 34) Therefore, by comparing the coefficients on both sides of 33) and34), we have the foowing theorem. Theorem 2 For n N {0} and N N, we have 1 L n+n x) n + N) N 2N a i N N, x) N ) i + 1 L n x), 0 where a 0 N, x)n!, a N N, x) x) N, N j a j N, x) x) j N i j j i j 0 i j 1 0 N i j i 2 j i 1 0 N + j) ij j N ij i k j 2k 2) )) ) N i i j i 1 j)!. k 1 k2

8 Kim eta. Advances in Difference Equations 2016) 2016:159 Page 8 of 9 3 Concusion It has been demonstrated that it is a fascinating idea to use differentia equations associated with the generating function or a sight variant of generating function) of specia poynomias or numbers. Immediate appications of them have been in deriving interesting identities for the specia poynomias or numbers. Aong this ine of research, here we derived a famiy of differentia equations from the generating function of the Laguerre poynomias. Then from these differentia equations we obtained interesting new identities for those poynomias. Competing interests The authors decare that they have no competing interests. Authors contributions A authors contributed equay to this work. A authors read and approved the fina manuscript. Author detais 1 Department of Mathematics, Coege of Science, Tianjin Poytechnic University, Tianjin, , China. 2 Department of Mathematics, Kwangwoon University, Seou, , Repubic of Korea. 3 Department of Mathematics, Sogang University, Seou, 04107, Repubic of Korea. 4 Department of Mathematics, Dong-A University, Busan, 49315, Repubic of Korea. 5 Department of Appied mathematics, Pukyong Nationa University, Busan, 48513, Repubic of Korea. Acknowedgements This work was supported by the Dong-A university research fund. The first author is appointed as a chair professor at Tianjin Poytechnic University by Tianjin City in China from August 2015 to August Received: 25 January 2016 Accepted: 12 June 2016 References 1. Kim, T: Identities invoving Laguerre poynomias derived from umbra cacuus. Russ. J. Math. Phys. 211), ) 2. Zi, DG, Cuen, MR: Advanced Engineering Mathematics. Jones & Bartett, Boston 2005) 3. Abramowitz, M, Stegun, IA: Handbook of Mathematica Functions with Formuas, Graphs, and Mathematica Tabes. Nationa Bureau of Standards Appied Mathematics Series, vo. 55. U.S. Government Printing Office, Washington 1964) 4. Arfken, G, Weber, H: Mathematica Methods for Physicists. Academic Press, San Diego 2000) 5. Bhrawy, AH, Aghamdi, MA: The operationa matrix of Caputo fractiona derivatives of modified generaized Laguerre poynomias and its appications. Adv. Differ. Equ. 2013, Artice ID ) 6. Srivastava, HM, Lin, S-D, Liu, S-J, Lu, H-C: Integra representations for the Lagrange poynomias, Shivey s pseudo-laguerre poynomias, and the generaized Besse poynomias. Russ. J. Math. Phys. 191), ) 7. Uspensky, JV: On the deveopment of arbitrary functions in series of Hermite s and Laguerre s poynomias. Ann. Math. 2) 281-4), /1927) 8. Watson, GN: An integra equation for the square of a Laguerre poynomia. J. Lond. Math. Soc. S1-114), ) 9. Karaseva, IA: Fast cacuation of signa deay in RC-circuits based on Laguerre functions. Russ. J. Numer. Ana. Math. Mode. 263), ) 10. Caritz, L: Some generating functions for Laguerre poynomias. Duke Math. J. 35, ) 11. Caritz, L: The product of severa Hermite or Laguerre poynomias. Monatshefte Math. 66, ) 12. Baeanu, D, Bhrawy, AH, Taha, TM: Two efficient generaized Laguerre spectra agorithms for fractiona initia vaue probems. Abstr. App. Ana. 2013,Artice ID ) 13. Bhrawy, AH, Abdekawy, MA, Azahrani, AA, Baeanu, D, Azahrani, EO: A Chebyshev-Laguerre-Gauss-Radau coocation scheme for soving a time fractiona sub-diffusion equation on a semi-infinite domain. Proc. Rom. Acad., Ser. A : Math. Phys. Tech. Sci. Inf. Sci. 16, ) 14. Bhrawy, AH, Ahamed, YA, Baeanu, D, A-Zahrani, AA: New spectra techniques for systems of fractiona differentia equations using fractiona-order generaized Laguerre orthogona functions. Fract. Cac. App. Ana. 17, ) 15. Bhrawy, AH, Aghamdi, MM, Taha, TM: A new modified generaized Laguerre operationa matrix of fractiona integration for soving fractiona differentia equations on the haf ine. Adv. Differ. Equ. 2012, Artice ID ) 16. Bhrawy, AH, Hafez, RM, Azahrani, EO, Baeanu, D, Azahrani, AA: Generaized Laguerre-Gauss-Radau scheme for first order hyperboic equations on semi-infinite domains. Rom. J. Phys. 60, ) 17. Bhrawy, AH, Taha, TM, Azahrani, EO, Baeanu, D, Azahrani, AA: New operationa matrices for soving fractiona differentia equations on the haf-ine. PLoS ONE 105),e ).doi: /journa.pone Chaurasia, VBL, Kumar, D: On the soutions of integra equations of Fredhom type with specia functions. Tamsui Oxf. J. Inf. Math. Sci. 28, ) 19. Chaurasia, VBL, Kumar, D: The integration of certain product invoving specia functions. Scientia, Ser. A, Math. Sci. 19, ) 20. Chen,Y,Griffin,J:Deformedq 1 -Laguerre poynomias, recurrence coefficients, and non-inear difference equations. ActaPhys.Po.A469), ) 21. Hegazi, AS, Mansour, M: Generaized q-modified Laguerre functions. Int. J. Theor. Phys. 419), )

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