Simplifying Coefficients in a Family of Ordinary Differential Equations Related to the Generating Function of the Laguerre Polynomials

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Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Applications and Applied Mathematics: An International Journal (AAM Vol. 13, Issue 2 (December 2018, pp. 750 755 Simplifying Coefficients in a Family of Ordinary Differential Equations Related to the Generating Function of the Laguerre Polynomials Feng Qi Department of Mathematics College of Science Tianjin Polytechnic University Tianjin 300387, China College of Mathematics Inner Mongolia University for Nationalities Tongliao 028043, China Institute of Mathematics Henan Polytechnic University Jiaozuo 454010, China qifeng618@gmail.com, qifeng618@hotmail.com Received: June 10, 2018; Accepted: September 27, 2018 Abstract In the paper, by virtue of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the Lah inversion formula, the author simplifies coefficients in a family of ordinary differential equations related to the generating function of the Laguerre polynomials. Keywords: Simplifying; coefficient; Laguerre polynomial; generating function; Faà di Bruno formula; ordinary differential equation; Bell polynomial of the second kind; Lah inversion formula MSC 2010 No.: Primary 34A05; Secondary 05A15, 11A25, 11B73, 11B83, 11C08, 42C10 750

AAM: Intern. J., Vol. 13, Issue 2 (December 2018 751 1. Introduction In (Kim et al., 2016, Theorem 1, it was established inductively and recursively that the family of differential equations has a solution F (n (t = F (t where a 0 (n, = n!, a n (n, = ( n, n j a j (n, = ( j i j=0 n i j j i j 1=0 2 i=n a i n (n, ( i, n N, (1 F (t = 1 ( ep t, (2 n i j i 2 j i 1=0 n + j ij j n i j i k [j (2k 2] ik 1 (n i j i 1 j!, 2 j n 1, (3 k=2 the falling factorial { n 1 ( 1 ( n + 1, n 1, n = ( k = 1, n = 0, and the function F (t in (2 can be used to generate the Laguerre polynomials L n ( by F (t = 1 ( ep t = L n (t n. Hereafter, the epression (3 was employed in (Kim et al., 2016, Theorem 2. It is not difficult to see that (1 the epression (3 is too complicated to be remembered, understood, and computed easily; (2 the original proof of (Kim et al., 2016, Theorem 1 is long and tedious. In this paper, we will provide a nice and standard proof for (Kim et al., 2016, Theorem 1 and, more importantly, discover a simple, meaningful, and significant form for a i (n,. n=0 2. Main results Our main results can be stated as the following theorem. Theorem 2.1. For n 0, the function F (t defined by (2 and its derivatives satisfy [ ( ] F (n n! n 1 (t = ( n ( 1 k k 1 k( k 1 k! ( k F (t, (4

752 F. Qi and F (t = n!( n n 1 [ n(] where ( 1 = 1 and ( k 1 = 0 if k 0. Proof: ( 1 k ( n 1 ( k F (k (t, (5 The famous Faà di Bruno formula reads that d n d t n f h(t = f (k ( (h(t B n,k h (t, h (t,..., h (n k+1 (t, (6 where n 0 and the Bell polynomials of the second kind B n,k ( 1, 2,..., n k+1 for n k 0 are defined (Comtet, 1974, p. 134, Theorem A and (Comtet, 1974, p. 139, Theorem C by B n,k ( 1, 2,..., n k+1 = The function F (t in (2 can be rearranged as 1 i n k+1 l i {0} N il i=n n k+1 n k+1 l i=k F (t = e ep Applying u = h(t = 1 t and f(u = u e to (6 gives u e F (n d k ( u (t = d u k e u = = ( 1 d k d u k = ( u e u n! n k+1 l i! ( ( 1! B n,k ( 2, 2! ( k ( 1 ( 1 k u k n ep ( where we used the identities e u. n k+1 ( i i! li. (n k + 1!,..., 3 ( n k+2 n+k B n,k(1!, 2!,..., (n k + 1! k ( 1 n+k n! k! n! ( n 1 ( 1 k k 1 k! k ( n 1 k 1 ( k ( 1 k, B n,k ( ab1, ab 2 2,..., ab n k+1 n k+1 = a k b n B n,k ( 1, 2,..., n k+1, and B n,k (1!, 2!,..., (n k + 1! = n! ( n 1, in (Comtet, 1974, p. 135 and (Qi, 2016, Remark 3.5. The formula (4 is thus proved. The Lah inversion theorem in (Aigner, 1979, p. 96, Corollary 3.38 (iii and (Aigner, 2007, pp. 60 61, Eercise 2.9 reads that ( ( 1 n n! n 1 ( a n = b k ( 1 n n! n 1 b n = a k. (7

AAM: Intern. J., Vol. 13, Issue 2 (December 2018 753 Combining (7 with (4 arrives at n 1 [ n(] ( n = ( 1 k k 1 k! which can be rewritten as (5. The proof of Theorem 2.1 is complete. ( k F (k (t, F (t 3. Remarks Finally, we list several remarks on our main results and closely related things. The equations (4 and (5 can be regarded as inversion formulas each other. The equation (5 is a new one which cannot be obtained inductively and recursively, as was in Kim et al. (2016. The equation (4 can be rewritten as F (n (t = F (t ( [ n 1 n! ( n ( 1 k k ] k 1 k! ( k kk 1 ( k 1 [ = F (t ( n ( 1 k k k 1 k! ( k ( 1 k k 1 k! k=1 [ = F (t ( n ( 1 k k n 1 k 1 k! ( k + ( n (n 1! ( 1 k k + 1 k! [ = F (t ( 1 n n n 1 ( n ( n + ( ] n n! ( 1 k k k k! ( k = F (t ( n n! ( n ( 1 k k k k! ( k. The equation (1 can be reformulated as F (n (t = F (t i=0 k k 1 ( k 1 ] (k + 1 k ( k ] (8 a i (n,, n N. (9 ( n+i The motivations in the papers Qi (2018a,b,c, Qi and Guo (2018, 2017, Qi et al. (2019a, 2018a, 2019b, 2018b,c, Qi and Zhao (2018, Zhao et al. (2018 are same as the one in this paper. This paper is a slightly modified version of the preprint Qi (2017.

754 F. Qi 4. Conclusions Comparing two equalities (8 and (9 reveals that a k (n, = ( 1 k ( n k n! k! k, for n k 0. This form for a k (n, is apparently simpler, more meaningful, and more significant than the one (3 obtained in (Kim et al., 2016, Theorem 1. Acknowledgement: The author thanks the anonymous referees for their careful comments on and valuable suggestions to the original version of this paper. REFERENCES Aigner, M. (1979. Combinatorial Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 234, Springer-Verlag, Berlin-New York. Aigner, M. (2007. Discrete Mathematics, translated from the 2004 German original by David Kramer, American Mathematical Society, Providence, RI. Comtet, L. (1974. Advanced Combinatorics: The Art of Finite and Infinite Epansions, Revised and Enlarged Edition, D. Reidel Publishing Co., Dordrecht and Boston. Available online at https://doi.org/10.1007/978-94-010-2196-8. Kim, T., Kim, D.S., Hwang, K.-W., and Seo, J.J. (2016. Some identities of Laguerre polynomials arising from differential equations, Advances in Difference Equations, 2016, 2016:159, 9 pages. Available online at https://doi.org/10.1186/s13662-016-0896-1. Qi, F. (2016. Diagonal recurrence relations for the Stirling numbers of the first kind, Contributions to Discrete Mathematics, Vol. 11, No. 1, pp. 22 30. Available online at http://hdl.handle.net/10515/sy5wh2d6 and https://doi.org/10515/sy5wh2d6. Qi, F. (2017. Simplifying coefficients in a family of ordinary differential equations related to the generating function of the Laguerre polynomials, ResearchGate Preprint, available online at https://doi.org/10.13140/rg.2.2.13602.53448. Qi, F. (2018a. A simple form for coefficients in a family of nonlinear ordinary differential equations, Advances and Applications in Mathematical Sciences, Vol. 17, No. 8, pp. 555 561. Qi, F. (2018b. Notes on several families of differential equations related to the generating function for the Bernoulli numbers of the second kind, Turkish Journal of Analysis and Number Theory, Vol. 6, No. 2, pp. 40 42. Available online at https://doi.org/10.12691/tjant-6-2-1. Qi, F. (2018c. Simple forms for coefficients in two families of ordinary differential equations, Global Journal of Mathematical Analysis, Vol. 6, No. 1, pp. 7 9. Available online at https://doi.org/10.14419/gjma.v6i1.9778. Qi, F. and Guo, B.-N. (2017. Eplicit formulas and recurrence relations for higher order Eulerian

AAM: Intern. J., Vol. 13, Issue 2 (December 2018 755 polynomials, Indagationes Mathematicae, Vol. 28, No. 4, pp. 884 891. Available online at https://doi.org/10.1016/j.indag.2017.06.010. Qi, F. and Guo, B.-N. (2018. A diagonal recurrence relation for the Stirling numbers of the first kind, Applicable Analysis and Discrete Mathematics, Vol. 12, No. 1, pp. 153 165. Available online at https://doi.org/10.2298/aadm170405004q. Qi, F., Lim, D. and Guo, B.-N. (2019a. Eplicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations, Revista de la Real Academia de Ciencias Eactas, Físicas y Naturales Serie A: Matemáticas, Vol. 113, in press. Available online at https://doi.org/10.1007/s13398-017-0427-2. Qi, F., Lim, D. and Guo, B.-N. (2018a. Some identities related to Eulerian polynomials and involving the Stirling numbers, Applicable Analysis and Discrete Mathematics, Vol. 12, No. 2, in press. Available online at https://doi.org/10.2298/aadm171008014q. Qi, F., Niu, D.-W. and Guo, B.-N. (2019b. Some identities for a sequence of unnamed polynomials connected with the Bell polynomials, Revista de la Real Academia de Ciencias Eactas, Físicas y Naturales Serie A: Matemáticas, Vol. 113, in press. Available online at https://doi.org/10.1007/s13398-018-0494-z. Qi, F., Wang, J.-L. and Guo, B.-N. (2018b. Notes on a family of inhomogeneous linear ordinary differential equations, Advances and Applications in Mathematical Sciences, Vol. 17, No. 4, pp. 361 368. Available online at https://doi.org/10.1007/s13398-018-0494-z. Qi, F., Wang, J.-L. and Guo, B.-N. (2018c. Simplifying differential equations concerning degenerate Bernoulli and Euler numbers, Transactions of A. Razmadze Mathematical Institute, Vol. 172, No. 1, pp. 90 94. Available online at https://doi.org/10.1016/j.trmi.2017.08.001. Qi, F. and Zhao, J.-L. (2018. Some properties of the Bernoulli numbers of the second kind and their generating function, Bulletin of the Korean Mathematical Society, Vol. 55, in press. Available online at https://doi.org/10.4134/bkms.b180039. Zhao, J.-L. and Qi, F. (2018. Derivative polynomials of a function related to the Apostol Euler and Frobenius Euler numbers, Journal of Nonlinear Sciences and Applications, Vol. 10, No. 4, pp. 1345 1349. Available online at https://doi.org/10.22436/jnsa.010.04.06.

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