Title: About Solving Some Functional Equations related to the Lagrange Inversion Theorem
Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814
Article ID: MTJPAM-D-20-00011; Volume 3 / Issue 1 / Year 2021, Pages 62-69
Document Type: Research Paper
aTomsk State University of Control Systems and Radioelectronics, Tomsk, Russia
bTomsk State University of Control Systems and Radioelectronics, Tomsk, Russia
Received: 1 May 2020, Accepted: 9 October 2020, Available online: 7 January 2021.
Corresponding Author: Dmitry V. Kruchinin (Email address: firstname.lastname@example.org)
Full Text: PDF
Using the notion of the composita and the Lagrange inversion theorem, we present techniques for solving the functional equations B(x)=H(xB(x)r) and A(F(x)) = xH(A(x)), where H(x), B(x) and F(x) are known generating functions and A(x) is unknown, and r is a rational number. The first equation is a generalization of the Lagrange inversion formula for a rational power r in terms of compositae. For the second equation, the recurrent solution in terms of the compositae is obtained. Also we give some examples of the obtained results.
Keywords: Composita, Generating function, Lagrange inversion theorem, Functional equationReferences:
- Y. Simsek, Construction method for generating functions of special numbers and polynomials arising from analysis of new operators, Math. Methods Appl. Sci. 41 (16), 6934-6954, 2018.
- Y. Simsek, Explicit formulas for p-adic integrals: Approach to p-adic distributions and some families of special numbers and polynomials, Montes Taurus J. Pure Appl. Math. 1 (1), 1-76, 2019.
- N. Kilar and Y. Simsek, Identities for special numbers and polynomials involving Fibonacci-type polynomials and Chebyshev polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 30 (4), 493-502, 2020.
- T. Kim, D.S. Kim, H.Y. Kim and J. Kwon, A note on degenerate multi-poly-Genocchi polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 30 (3), 447-454, 2020.
- R.P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999.
- I.M. Gessel, Lagrange inversion, J. Combin. Theory Ser. A 144, 212-249, 2016.
- D. Merlini, R. Sprugnoli and M.C. Verri, Lagrange inversion: when and how, Acta Appl. Math. 94 (3), 233-249, 2006.
- D.V. Kruchinin, On solving some functional equations, Adv. Differ. Equ. 2015, 1-7, 2015, Article 17.
- D.V. Kruchinin and V.V. Kruchinin, A method for obtaining generating functions for central coefficients of triangles, J. Integer Seq. 15 (9), 1–10, 2012, Article 12.9.3.
- D.V. Kruchinin and V.V. Kruchinin, Application of a composition of generating functions for obtaining explicit formulas of polynomials, J. Math. Anal. Appl. 404 (1), 161-171, 2013.
- V.V. Kruchinin and D.V. Kruchinin, Composita and its properties, Journal of Analysis and Number Theory 2 (2), 37-44, 2014.
- D.V. Kruchinin, Y.V. Shablya and V.V. Kruchinin, Solution for a functional equation B(x)=H(xB(x)r), Proceedings Book of The Mediterranean International Conference of Pure and Applied Mathematics and related areas (MICOPAM2019), 87-89, 2019.
- D. Kruchinin and V. Kruchinin, Powers of Generating Functions and Applications, Tomsk, TUSUR, 2013 (in rus).
- L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company, 1974.
- N.J.A. Sloane, The on-line encyclopedia of integer sequences, http://oeis.org.