A Combinatorial Proof of a Result on Generalized Lucas Polynomials

Alexandre Laugier 1  and Manjil P. Saikia 2 , 3
  • 1 LYCÉE TRISTAN CORBIÈRE, 16 RUE DE KERVÉGUEN - BP 17149 - 29671, MORLAIX CEDEX, FRANCE
  • 2 DIPLOMA STUDENT, MATHEMATICS GROUP THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS 11, STRADA COSTIERA TRIESTE, ITALY
  • 3 current address: FACULTY OF MATHEMATICS UNIVERSITY OF VIENNA OSKAR-MORGENSTERN-PLATZ 1 1090 WIEN, AUSTRIA

Abstract

We give a combinatorial proof of an elementary property of generalized Lucas polynomials, inspired by [1]. These polynomials in s and t are defined by the recurrence relation 〈n〉 = s〈n-1〉+t〈n-2〉 for n ≥ 2. The initial values are 〈0〉 = 2; 〈1〉= s, respectively.

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  • [1] T. Amdeberhan, X. Chen, V. H. Moll, B. E. Sagan, Generalized Fibonacci polynomials and Fibonomial coefficients, Ann. Comb. 18(4) (2014), 541-562.

  • [2] S. Ekhad, The Sagan-Savage Lucas-Catalan polynomials have positive coefficients, preprint http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/bruce.html.

  • [3] B. E. Sagan, C. D. Savage, Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences, Integers, A52, 10 (2010), 697-703.

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