Les Annales RECITS http://www.lrecits.usthb.dz Vol. 01, 2014, pages 1-10 On the expansion of Fibonacci Lucas polynomials, revisited Hacène Belbachir a Athmane Benmezai b,c a USTHB, Faculty of Mathematics, RECITS Laboratory, DG-RSDT BP 32, El Alia 16111, Bab Ezzouar, Algiers, Algeria b University of Dely Brahim, Fac. of Eco. & Manag. Sc., RECITS Lab., DG-RSDT Rue Ahmed Ouaed, Dely Brahim, Algiers, Algeria c University of Oran, Faculty of Sciences, ANGE Lab., DG-RSDT BP 1524, ELM Naouer, 31000, Oran, Algeria. hacenebelbachir@gmail.com or hbelbachir@usthb.dz athmanebenmezai@gmail.com Abstract: As established by Prodinger in On the Expansion of Fibonacci Lucas Polynomials, we give -analogue of identities established by Belbachir Bencherif in On some properties of bivariate Fibonacci Lucas polynomials. This is doing according to the recent Cigler s definition for the -analogue of Fibonacci polynomials, given in Some beautiful -analogues of Fibonacci Lucas polynomials, by the authors for the -analogues of Lucas polynomials, given in An Alternative approach to Cigler s -Lucas polynomials. Keywords: Fibonacci Polynomials; Lucas Polynomials; -analogue. Résumé : Comme établi par Prodinger dans On the Expansion of Fibonacci Lucas Polynomials, nous donnons le -analogue des identités établies par Belbachir et Bencherif dans On some properties of bivariate Fibonacci Lucas polynomials. Ces identités sont basées sur l approche de Cigler pour le -analogue des polynômes de Fibonacci, donnée dans Some beautiful - analogues of Fibonacci Lucas polynomials, et par les auteurs pour les -analogues de polynômes de Lucas, donnée dans An Alternative approach to Cigler s -Lucas polynomials. Mots clés : Polynômes de Fibonacci; Polynômes Lucas; -analogue.
2 Hacène Belbachir Athmane Benmezai 1 Introduction The bivariate polynomials of Fibonacci Lucas, denoted respectively by U n V n, are defined by { { U0 0, U 1 1, V0 2, V 1 t, U n tu n 1 +zu n 2 n 2, V n tv n 1 +zv n 2 n 2. It is established, see for instance [1], that n U n+1 0 t n 2 z, V n 0 n n n t n 2 z n 1. In [2], the first author Bencherif proved that, for n 2 n, the families x U n+1 x V n constitute two basis of the Q-vector space spanned by the free family x n 2 y, they found that the coordinates of the bivariate polynomials of Fibonacci Lucas, over appropriate basis, satisfies remarable recurrence relations. They established the following formulae V 2n 2U 2n+1 xu 2n, 1 n 2U 2n+1 a n, t V 2n, with a n, 2 1 + 1 n+, 2 0 n V 2n b n, t V 2n, with b n, 1 +1, 3 1 n V 2n 1 c n, t U 2n, with c n, 2 1 +1 [ 1], 4 1 2V 2n 1 d n, t V 2n 1, with d n, 1 +1 2n n, 5 n 1 2U 2n e n, t V 2n, with 6 1 e n, 1 +1 2n n n + 1 + 1 1 n 1 1 2 1n+. 1 A similar approach was done by the authors, see [3], for Chebyshev polynomials. As -analogue of Fibonacci Lucas polynomials, J. Cigler [6], considers the following expressions F n+1 x,y,m +1 2 +m 2 n x 2n y, 7 0 Luc n x,y,m m+1 2 n [n] x 2n y, 8 [n ] 0
On the expansion of Fibonacci Lucas polynomials, revisited 3 with the -notations n [n] 1+ + + n 1, [n]! [1] [2] [n], [n]! []![n ]!. Without loss the generality, we can suppose that t 1. We refer here to the modified polynomials given by H. Prodinger in the introduction of [8]. Using the -analogues of Fibonacci Lucas polynomials suggested by J. Cigler, H. Prodinger, see [8], give -analogues for relations 3 5, the authors, see [4], give -analogues for relations 1, 2, 4, 6. The -identities associated to relations 2 6, given in [4], do not give for 1 the initial relations. This is the motivation which conclude to this paper: we propose an alternative -analogue for all the former relations 1, 2, 3, 4, 5 6 based on Cigler s definition, see [6], for the Fibonacci polynomials, the definitions givn by the authors, see [5], for the Lucas polynomials. In [5], we have defined the -Lucas polynomials of the first ind Lz the -Lucas polynomials of the second ind Lz respectively by L n z,m : L n z,m : 0 0 2+m 2 n +1 2 +m 2 n 1+ [] z, [n ] 9 1+ n 2 [] z, [n ] 10 we have showed that the polynomials L n z L n z satisfy the recursions L n+1 z,m L n z,m+ n 1 zl n 1 m 1 z,m, 11 L n+1 z,m L n z,m+zl n 1 m+1 z,m. 12 These two recursions are satisfied by the -analogue of Fibonacci polynomials F n z,m, see [6]. 2 Main results In [5], the authors expressed the -Lucas polynomials of the both inds in terms of - Fibonacci polynomials by the identities L n z,m 2F n+1 z,m F n z,m, L n z,m 2F n+1 z,m F n z,m, which are considered as a -analogue of the euality 1.
4 Hacène Belbachir Athmane Benmezai The following result gives two -analogues of relation 2, the first one is related to the -Lucas polynomials of the first ind the second one is related to the -Lucas polynomials of the second ind. Theorem 1 For every integer n 0, one has z 2F 2n+1,m 2 n 1 n+ L 2n z,m+ 0 2 0 2F 2n+1 z,m 2 0 0 2 2n 2 2+2n nn ++1 2 1 + L 2n 2 z,m, 13 1 + L 2n 2 z,m + n 1 L 2n n z,m. 14 The -analogue of relation3, found by Prodinger in[8], is given by the following Theorem which gives also a second -analogue identity. Theorem 2 For n 1, we have F 2n z,m F 2n z,m 2 n 1 +1 F 2n z,m, 15 n 2 n 1 +1 F 2n z,m. 16 1 1 The relation 4, admits two -analogues related to the -Lucas polynomials of the first ind Lz, two -analogues related to the -Lucas polynomials of the second ind Lz, given respectively by the following Theorem. Theorem 3 For n 1, the -Lucas polynomials of the first ind is developed by L 2n 1 z,m 2 2 n z 1 +1 F 2n F 1,m 2n 1 z,m, 17 L 2n 1 z,m 2 n 2 n 1 +1 F 2n 1 z,m F 2n 1 z,m. 18 1 the -Lucas polynomials of the second ind is developed by L 2n 1 z,m 2 2 n 1 +1 F 2n z,m F 2n 1 z,m, 19 1 L 2n 1 z,m 2 n 2 n 1 +1 F 2n z,m F 2n 1 z,m. 20 1
On the expansion of Fibonacci Lucas polynomials, revisited 5 Using the -Lucas polynomials of the both inds we find two -analogues of relation 5. Theorem 4 For n 1, we have 2L 2n 1 z,m 2 n 1 +1 +1 1 2L 2n 1 z,m n 2 n 1 +1 1 n 1 1+ [n ] [n] 1+ [n ] [] L 2n 2 z,m, 21 L 2n 1 z,m 22. In the following theorem, we give two -analogues of 6. Theorem 5 For every integer n 0, one has 2F 2n z,m 1 2 1 2 n 1 +1 +1 1+ [n ] L 2n 2 z,m+ 2 1 [n] 1 2 n 1 1 n 1+ L 2n 2 z,m+ 2 0 n 2 2 2n 2 1 + L 2n 2 2+1 z,m. 23 0 2F 2n z 1 2 1 0 n 2 n 1 +1 1 n 1+ [n ] [] n 1 n 1n ++1 2 0 2 2+2n 2 ] [ L 2n 1 z,m + 1 n 1+ L 2n 2 +1 n z,m + 1 + L 2n 2 2 z,m. 24 Remar 1 Notice that the coefficients appeared in the sums given in the different results do not depends on m. 3 Proof of the results We need the following Lemmas.
6 Hacène Belbachir Athmane Benmezai Lemma 6 For U n z satisfying 11 12 respectively, we have for 1 2 2 1 U n+ z,m m 2+ n 2 n 2 z U n m z,m, 1 U n+ z,m m 2 z U n m z,m. Proof. We use induction over, the case 1 is given by the recursions 11 12. We suppose the relation true for, m+1 2 + n 2 n 1 2 z +1 U n 1 m+m 1 z,m m 2+ n 2 n 2 z U n+1 m z,m U n m z,m m 2+ n 1 2 n 1 2 z U n+1 m z,m m 2+ n 2 n 2 z U n m z,m 2 1 U n+1 z,m 2 2 2 2 1 U n+ z,m + 1 2 1 U n++1 z,m 1 1 U n++1 z,m m+1 2 z +1 U n 1 m+m z,m m 2 z U n+1 m 1 z,m U n m z,m m 2 2 1 U n+ 1 z,m 2 1 U n+ z,m 2 [ ] + +1 2 +1 2 1 U n++1 1 z,m 1 U n+ 1 z,m
On the expansion of Fibonacci Lucas polynomials, revisited 7 Lemma 7 For every integer n 1, one has F 2n+1 z,m z n m+1n 2 + m+1 2 z L 2n m+ z,m, F 2n+1 z,m z n n+1 2 +m n 2 + 2n 1+m 3 2 z L 2n m z,m. Proof. We use induction over n, the case n 1 is given by the following relations, see [5] L n z,m F n+1 z,m +zf n 1 m z,m, L n z,m F n+1 z,m+ n 1 zf n 1 m 1 z,m. We suppose the identities true for n, then F 2n+3 z,m L 2n+2 z,m zf 2n+1 m z,m, L 2n+2 z,m+ z n+1 m+1n+1 2 z 2 m+1 z L2n m++m+1 z,m, z n+1 m+1n+1 2 + 2 z L 2n+1 m+ z,m. F 2n+3 z,m L 2n+2 z,m 2n+1 zf 2n+1 m 1 z,m, L 2n+2 z,m 2n+1 z z n n 2+m n+1 2 + 2n+1 z 2n+1+m 3+1 2 z L 2n m+m 1 z,m, L 2n+2 z,m+ z n+1 n+2 2 m n+1 2 2n+1 n 1 2n+1+m 3+1 2 z L 2n m+m 1 z,m, z n+1 n+2 2 m n+1 2 + 2n+1 m 3 2 z L 2n+1 m z,m.
8 Hacène Belbachir Athmane Benmezai Proof of relations 13 14.. Replacing U n z by L n z L n z respectively, in Lemma 6, we get 2 n 1 L 2n z,m m+1n 2 z n 2, 0 2 1 L 2n 2 z,m m+1 2+2n z L 2n m+ z,m. 0 n 2 n 1 L 2n n z,m mn 2 z n 2, 2 1 L 2n 2 z,m m 2 2 z L 2n m z,m. 0 0 using these relations in Lemma 7, we draw z 2F 2n+1,m 2 n 1 n+ L 2n z,m+ 0 2 ] 2n 2[ 1 L 2n 2 z,m, 0 2F 2n+1 z,m 0 0 n 2 + n+1 2 2n 2 2 n 1 n+ L 2n n z,m + 1 + L 2n 2 z,m Proof of relations 15 16.. For n U n z F n z in Lemma 6, we have 2 n 2 n [ n ] 1 F 2n z,m m 2+ n 2 z n F 0 mn n z,m 0, 1 F 2n n z,m mn 2 z n F 0 mn z,m 0. Proof of relations 17, 18, 19 20.. It suffices to replace relations 15 16 in L 2n 1 z,m 2F 2n z,m F 2n 1 z,m, L 2n 1 z,m 2F 2n z,m F 2n 1 z,m.
On the expansion of Fibonacci Lucas polynomials, revisited 9 Proof of relations 21 22.. According to Lemma 6, we have 0 n 1 0 2 n 1 0 2 n 1 0 n 1 2 n 1 2 n 1 n 1 1 L 2n 2 z,m 2z n 1 mn 1 2 + n 1 2, 1 L 2n 1 z,m z n 1 mn 1 2 + n 2, 1 L 2n 2 +2 n z,m 2 mn 1 2 z n 1 1 L 2n 1 +1 n z,m mn 1 2 z n 1 Then 2L 2n 1 z,m n 1 1 2 n 1 1 +1 L 2n 1 z,m 1 1 2 2 n 1 1 L 2n 1 z,m, 1 2 n +1 n [] +2 [n ] L 2n 1 z,m, 1 [n] [n] 2 n +1 1+ [n ] L 2n 2 z,m. [n] 1 2L 2n 1 1 n z,m [ 1 n n 2 n 1 1 n 1 2 1 1 1 n 1 2 n 2 n 2 ] 1 +1 L 2n 1 +1 n z,m 1 n 1 1 L 2n 1 +1 n z,m, n 1 +1 [] 1 n +2 [n ] L 2n 1 +1 n z,m, [n] [n] n 1 +1 1 n L 2n 1 +1 n z,m. 1+ [n ] [] Proof of the relations 23, 24.. Using relations 13, 14, 21, 22 in 2F 2n z,m L 2n 1 z,m+f 2n 1 z,m, we draw the results. 2F 2n z,m L 2n 1 z,m+f 2n 1 z,m.
10 Hacène Belbachir Athmane Benmezai Remar 2 Considering these results, we obtain a duality between the -Lucas polynomials of the first ind the -Lucas polynomials of the second ind. References [1] H. Belbachir, F. Bencherif, Linear recurrent seuences powers of a suare matrix. Integers 6, 2006, A12. [2] H. Belbachir, F. Bencherif, On some properties of bivariate Fibonacci Lucas polynomials, J. Integer Seuences 11, Article 08.2.6, 2008. [3] H. Belbachir, F. Bencherif, On some properties of Chebyshev polynomials, Discuss. Math. Gen. Algebra Appl., 28 1, 2008, 121 133. [4] H. Belbachir, A. Benmezai, Expansion of Fibonacci Lucas polynomials: An answer to Prodinger s uestion, J. Integer Seuences, Vol. 15 2012,.Article12.7.6 [5] H. Belbachir, A. Benmezai, An Alternative approach to Cigler s -Lucas polynomials, Applied Mathematics Computation, 226 2014, 691 698. [6] J. Cigler, Some beautiful -analogues of Fibonacci Lucas polynomials, arxiv:11042699. [7] J. Cigler, A new class of -Fibonacci polynomials, Electronic J. Combinatorics 10, Article R19 2003. [8] H. Prodinger, On the Expansion of Fibonacci Lucas Polynomials, J. Integer Seuences 12, Article 09.1.6, 2009.