Abstract
In this paper we develop a novel matrix method for solving linear recurrence relations and present explicit formulae for the general solution of the third-order linear homogeneous recurrence relations with variable coefficients. We obtain a summatory formula for the general solution of the recurrence relation in the special case. We review some known results and then consider some particular cases of the recurrence and examples with applications to combinatorics, especially to number sequences and polynomials. Finally, we briefly discuss further generalization of the method for higher order linear recurrence relations.
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References
Comtet, L.: Advanced Combinatorics. Reidel, Dordrecht (1974)
Andrica, D., Bagdasar, O.: Recurrent Sequences. Springer, Berlin (2020)
Bagdasaryan, A.G., Bagdasar, O.: On an arithmetic triangle of numbers arising from inverses of analytic functions. Electron. Notes Discret. Math. 70, 17–24 (2018)
Elaydi, S.: Difference equations in combinatorics, number theory, and orthogonal polynomials. J. Differ. Equ. Appl. 5, 379–392 (1999)
Everest G., van der Poorten, A.J., Shparlinski, I., Ward, T.: Recurrence Sequences. American Mathematical Society (2003)
Elaydi, S.: An Introduction to Difference Equations. In: Undergraduate Texts in Mathematics, 3rd ed. Springer, New York (2005)
Gel’fond, A.O.: Calculus of Finite Differences. Hindustan Publ, Corp (1971)
Koshy, T.: Fibonacci and Lucas Numbers with Applications. Wiley, Hoboken (2001)
Kiliç, E., Stănică, P.: A matrix approach for general higher order linear recurrences. Bull. Malays. Math. Sci. Soc. 34(2), 51–67 (2011)
Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34(3), 596–615 (1987)
Kiefer, J.: Sequential minimax search for a maximum. Proc. Am. Math. Soc. 4, 502–506 (1953)
Knuth, D.E.: The Art of Computer Programming, vol. 3. Addison Wesley, Boston (2003)
Bešo, E., Kalabušić, S., Mujić, N., Pilav, E.: Stability of a certain class of a host-parasitoid models with a spatial refuge effect. J. Biol. Dyn. 14(1), 1–31 (2020)
Kalabušić, S., Pilav, E.: Global behaviour of a class of discrete epidemiological SI models with constant recruitment of susceptibles. J. Differ. Equ. Appl. 28, 259–288 (2022)
Grossman, G., Zeleke, A.: On linear recurrence relations. J. Concr. Appl. Math. 1(3), 229–245 (2003)
Shannon, A.G., Horadam, A.F.: Some properties of third-order recurrence relations. Fibonacci Quart. 10(2), 135–146 (1972)
Shannon, A.G.: Iterative formulas associated with generalized third-order recurrence relations. SIAM J. Appl. Math. 23(3), 364–368 (1972)
El Wahbi, B., Mouline, M., Rachidi, M.: Solving nonhomogeneous recurrence relations of order \(r\) by matrix methods. Fibonacci Quart. 40(2), 106–117 (2002)
Zhao, Yu-Kuang.: The solutions of a linear homogeneous recurrence relation with unit coefficients. Tamkang J. Math. 31(3), 229–237 (2000)
Mallik, R.K.: On the solution of a third-order linear homogeneous difference equation with variable coefficents. J. Differ. Equ. Appl. 4(6), 501–521 (1998)
Mallik, R.K.: On the solution of a linear homogeneous difference equation with variable coefficients. SIAM J. Math. Anal. 31(2), 375–385 (2000)
Mallik, R.: Solutions of linear difference equations with variable coefficients. J. Math. Anal. Appl. 222, 79–91 (1998)
Kittappa, R.: A representation of the solution of the \(n\) order linear difference equation with variable coefficients. Linear Algebr. Appl. 193, 211–222 (1993)
Apéry, R.: Irrationalité de \(\zeta (2)\) et \(\zeta (3)\). Soc. Math. Fr., Astérisque 61, 11–13 (1979)
Heim, B., Neuhauser, M.: Difference equations related to number theory. In: Baigent, S., et al. (eds.)Progress on Difference Equations and Discrete Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol. 341, pp. 245–255 (2020)
Heim, B., Luca, F., Neuhauser, M.: Recurrence relations for polynomials obtained by arithmetic functions. Int. J. Numb. Theory 15(6), 1291–1303 (2019)
Andrews, G.E.: The Theory of Partitions. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1998)
Cerda-Morales, G.: Quadratic approximation of generalized Tribonacci sequences. Discuss. Math. Gen. Algebr. Appl. 38(2), 227–337 (2018)
Jancić, M.: On linear recurrence equations arising from compositions of positive integer. J. Integer Seq. 18(15.4.7) (2015)
Chen, W.Y.C., Louck, J.D.: The combinatorial power of the companion matrix. Linear Algebr. Appl. 232, 261–278 (1996)
Lim, A., Dai, J.: On product of companion matrices. Linear Algebr. Appl. 435, 2921–2935 (2011)
Ramírez, J.L., Sirvent, V.F.: A note on the \(k\)-Narayana sequence. Ann. Math. Inform. 45, 91–105 (2015)
Acknowledgements
The author wishes to thank the organizers of the conference, especially Senada Kalabušić, for their invitation and their excellent work on the conference organization, and the anonymous reviewers for their careful reading of the manuscript and helpful remarks and suggestions toward improvement of the presentation of this paper.
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Bagdasaryan, A.G. (2023). Solving Third-Order Linear Recurrence Relations with Applications to Number Theory and Combinatorics. In: Elaydi, S., Kulenović, M.R.S., Kalabušić, S. (eds) Advances in Discrete Dynamical Systems, Difference Equations and Applications. ICDEA 2021. Springer Proceedings in Mathematics & Statistics, vol 416. Springer, Cham. https://doi.org/10.1007/978-3-031-25225-9_5
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