Generating Functions For Two-Variable Polynomials Related To a Family of Fibonacci Type Polynomials and Numbers

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1 Filomat 30:4 2016, DOI /FIL O Published by Faulty of Sienes and Mathematis, University of Niš, Serbia Available at: Generating Funtions For Two-Variable Polynomials Related To a Family of Fibonai Type Polynomials and Numbers Gulsah Ozdemir a, Yilmaz Simsek b a Department of Mathematis, Faulty of Siene University of Akdeniz TR Antalya, Turkey b Department of Mathematis, Faulty of Siene University of Akdeniz TR Antalya, Turkey Abstrat. The purpose of this paper is to onstrut generating funtions for the family of the Fibonai and Jaobsthal polynomials. Using these generating funtions and their funtional equations, we investigate some properties of these polynomials. We also give relationships between the Fibonai, Jaobsthal, Chebyshev polynomials and the other well known polynomials. Finally, we give some infinite series appliations related to these polynomials and their generating funtions. 1. Introdution The Fibonai numbers, F n and the Jaobsthal numbers, J n are very famous in the array of integer sequenes. These sequenes have an enormous amount of information in the mathematial literature f. [1]-[12]; see also the referenes ited in eah of these earlier works. Similarly, the famous Fibonai and Jaobsthal polynomials have also reently investigated and studied by many authors in different methods. In this paper, therefore we give some generating funtions related to these sequenes and polynomials. By using these generating funtions, we derive various identities, formulas and relations, some old and some new, for the Fibonai and the Jaobsthal sequenes and polynomials. It is well-known that the lasses of the Fibonai and the Jaobsthal polynomials are related to the Chebyshev polynomials f. [1], [7]. We also onstrut new generating funtions related to new lasses of polynomials whih inlude the Fibonai polynomials, the Jaobsthal polynomials, the generalized Chebyshev polynomials, the Vieta-Fibonai and the Vieta-Luas polynomials, the Humbert polynomials, the Geganbauer polynomials, et. The well-known Fibonai polynomials and Jaobsthal polynomials are defined by the following definition: Definition 1.1. It is well known that the Fibonai polynomials, {F n x} n0 and the Jaobsthal polynomials, {J n x} n0 are series of polynomials that satisfy the reurrene relation, respetively: F n x xf n 1 x + F n 2 x 2010 Mathematis Subet Classifiation. Primary 11B39; Seondary 05A15, 11B83 Keywords. Fibonai numbers, Fibonai polynomials, Jaobsthal polynomials, Chebyshev polynomials, Vieta-Fibonai polynomials, Vieta-Luas polynomials, Humbert polynomials, Geganbauer polynomials, generating funtions Reeived: 01 July 2015; Aepted: 07 August 2015 Communiated by Gradimir Milovanović The present investigation was supported by the Sientifi Researh Proet Administration of Akdeniz University addresses: ozdemir.gulsah@hotmail.om Gulsah Ozdemir, ysimsek@akdeniz.edu.tr Yilmaz Simsek

2 for all n 3, with initials F 1 x 1 and F 2 x x and J n x J n 1 x + xj n 2 x for all n 3, with initials J 1 x J 2 x 1f. [1]-[11]. Ozdemir and Simsek / Filomat 30:4 2016, Throughout of this paper for generating funtions, we assume that t < 1. The Fibonai and Jaobsthal polynomials are given by means of the following generating funtions, respetively: t G F x, t 1 xt t F 2 n x t n 1 n0 and t G J x, t 1 t xt J 2 n x t n 2 n0 f. [1]-[11]. Remark 1.2. By 1, we easily see that the following results: F 2n+1 0 1, F 2n 0 0, F n 1 F n, F n 2 P n, where P n denotes the Pell numbers f. [1]-[11]. 2. Generating Funtions For The Family of Fibonai and Jaobsthal Polynomials In this setion, we onstrut novel generating funtions for various kind of the well-known polynomials. We investigate some properties of these funtions. By using derivative operators and other algebrai manipulations, we give some funtional equations and PDEs of these generating funtions. By using these funtions, we also derive some identities and relations assoiated with these polynomials. We define a new family of two variable polynomials, G x, y; k, m, n by means of the following generating funtions: Ht; x, y; k, m, n G x, y; k, m, n t 1, 3 1 x k t y m tn+m where k, m, n N 0 {0, 1, 2,...}. Note that there is one generating funtion for eah value of k, m and n. Theorem 2.1. [10, P. 69 Lemma 10-11], [12] For m N, we have n0 k0 A k, n m A k, n mk n0 [ ] n k0 and [ ] n m B k, n B k, n + mk. n0 k0 n0 k0

3 Ozdemir and Simsek / Filomat 30:4 2016, By using 3 and Theorem 2.1, we give expliit formula for the polynomials, G by the following theorem: Theorem 2.2. [ ] m+n m + n 1 G x, y; k, m, n 0 y m x k mk nk, 4 where [a] is the largest integer a. Proof. By using 3, we get Ht; x, y; k, m, n 0 + x k t y m t m+n, where x k t < 1 and y m t m+n < 1. By using Theorem 2.1, after some elementary alulations in the above equation, we get Ht; x, y; k, m, n m+n m n + [ 0 ] x k t m n y m t m+n. Therefore G x, y; k, m, n t [ ] m+n m n + 0 x k mk nk y m t. Comparing the oeffiients of t on both sides of the above equation, we arrive at the desired result. A relation between the polynomials G and the Legendre polynomials P x is given by the following theorem: Theorem 2.3. G 2x, 1; 1, 1, 1 P xp x. 0 Proof. In [11, P. 83, Eq-8], the Legendre polynomials are given by 1 2xt + t P xt. From the above equation and 3, we get G 2x, 1; 1, 1, 1 t 0 P xp x t. Comparing the oeffiients of t on both sides of the above equation, we arrive at the desired result.

4 Ozdemir and Simsek / Filomat 30:4 2016, Two variable Fibonai type polynomials, W are defined by means of the following generating funtions: Rt; x, y; k, m, n Ht; x, y; k, m, nt n 5 W x, y; k, m, n t. Observe that if we substitute k m n 1 and y 1 into 5, we arrive at 1. That is F x W x, 1; 1, 1, 1. Also, if we substitute k m n 1 and x 1 into 5, we arrive at 2. That is J y W 1, y; 1, 1, 1. We an give a relation between the polynomials W and G as follows: G n x, y; k, m, n t n W x, y; k, m, n t. Remark 2.4. Substituting k m 1, y 1 and n l 1 l N into 3, we get the well-known first and seond kind generalized Chebyshev polynomials, respetively: and also V,l x G x, 1; 1, 1, l 1 Ht; x, 1; 1, 1, l 1 trt; x, 1; 1, 1, l 1 Ω,l x t. 1 see for details f. [1]. Remark 2.5. If we substitute k m n 1 and y 1 into 5, we obtain the following well-known Vieta-Fibonai polynomials, V x and Vieta-Luas polynomials, v x, respetively f. [6]: and W x, 1; 1, 1, 1 V x v x 2G x, 1; 1, 1, 1 2xW x, 1; 1, 1, 1. Remark 2.6. By setting k m 1, y 1 and n a 1, a is an integer with a > 1, into 5, we obtain the Humbert polynomials, h 1 n,ax f. [11, P. 86, Eq-26]: h 1 n,ax G ax, 1; k, 1, a 1. For a 2, we easily see that the Humbert polynomials, h v n,2 x redue to the Gegenbauer polynomials Cv nx f. [11, P. 86, Eq-26].

5 3. Partial Derivatives For The Generating Funtions Ozdemir and Simsek / Filomat 30:4 2016, In this setion, by applying x Ht; x, y; k, m, n, y Ht; x, y; k, m, n and tht; x, y; k, m, n partial derivative operators, we give some partial derivative equations. By using these equations with generating funtions, we derive two derivative formulas and a reurrene formula for our polynomials G x, y; k, m, n. Using partial derivative, with respet to x, we obtain the following higher-order partial differential equation: x Ht; x, y; k, m, n kxk 1 th 2 t; x, y; k, m, n. 6 Using partial derivative, with respet to y, we obtain the following higher-order partial differential equation: y Ht; x, y; k, m, n mym 1 t m+n H 2 t; x, y; k, m, n. 7 Using partial derivative, with respet to t, we obtain the following higher-order partial differential equation: t Ht; x, y; k, m, n x k + m + n y m t m+n 1 H 2 t; x, y; k, m, n Derivative Formulas By applying PDEs of the generating funtions, we give two derivative formulas for the polynomials G. By ombining 6 with 3, sine G 1, we get 1 x G x, y; k, m, n t 1 kx k 1 G l x, y; k, m, n G 1 l x, y; k, m, n t. 1 Comparing the oeffiients of t on both sides of the above equation, we arrive at the following theorem: Theorem 3.1. Let be a positive integer. Then we have x G 1 x, y; k, m, n kx k 1 G l x, y; k, m, n G 1 l x, y; k, m, n. By ombining 7 and 3 with the following evaluation: If < m + n, then [ m+n] 0. Thus, the polynomials G x, y; k, m, n don t inlude the term y. Therefore, x, y; k, m, n 0. We obtain y G m+n y G x, y; k, m, n t m+n m n my m 1 G l G m n l t. Comparing the oeffiients of t on both sides of the above equation, we arrive at the following theorem: Theorem 3.2. Let m + n. Then we have y G m n x, y; k, m, n my m 1 G l G m n l.

6 3.2. Reurrene Formula Ozdemir and Simsek / Filomat 30:4 2016, Here, using PDE with generating funtions, we derive a reurrene formula for our polynomials G. By ombining 8 with 3, we get + 1 G+1 x, y; k, m, n t x k m+n 1 m n+1 G l x, y; k, m, n G l x, y; k, m, n t m + n y m G l G m n l+1 t. Comparing the oeffiients of t on both sides of the above equation, we arrive at the following theorem: Theorem 3.3. If < m + n 1, then we have + 1 G+1 x k G l x, y; k, m, n G l x, y; k, m, n 0. If m + n 1, then we have + 1 G+1 x k m n+1 m + n y m G l x, y; k, m, n G l x, y; k, m, n G l G m n l Appliations of The Generating Funtions, Rt; x, y; k, m, n In this setion, we give some series inluding our polynomials and also the Fibonai type polynomials and numbers. In 1953, F. Stanliff found the following sum inluding the Fibonai numbers: n0 F n n+1 F 11 f. [7, P. 424]. Setting t 1 with > 1 in 5, we get the following series: W x, y; k, m, n m m+n x k n+m 1 y. 10 m Remark 4.1. If we set 10, x y 1, k m n 1 in 10, we get 9. And also, by substituting 2, x y 1, k m n 1 into in 10, we get F 2 2. f. [7, P. 437].

7 Ozdemir and Simsek / Filomat 30:4 2016, Remark 4.2. For 2, y 1, k m n 1, 10 redues to the following sum: F x x. Similarly, setting 3, x 1, k m n 1 in 10, we get the following sum: J d y 3 3 d 6 y. d0 Remark 4.3. For t 1 5, y 1, k 1, m 1, n l 1, 3 redues to the following sum reletad to the generalized Chebyshev polynomials of the first kind: V,l x 5 l 5 5 l 5 l 1 x + 1. Referenes [1] G. B. Dordevi, Polynomials related to generalized Chebyshev polynomials, Filomat [2] G. B. Dordevi and H. M. Srivastava, Some generalizations of the inomplete Fibonai and the inomplete Luas polynomials, Adv. Stud. Contemp. Math , [3] G. B. Dordevi and H. M. Srivastava, Inomplete generalized Jaobsthal and Jaobsthal-Luas numbers, Math. Comput. Modelling , [4] G. B. Dordevi and H. M. Srivastava, Some generalizations of ertain sequenes assooiated with the Fibonai numbers, J. Indonesian Math. So , [5] S. Falon and A. Plaza, On k-fibonai sequenes and polynomials and their derivatives, Chaos, Solitons and Fratals [6] A. F. Horadam, Vieta polynomials, Fibonai Quart. 403, [7] T. Koshy, Fibonai and Luas numbers with appliations, John Wiley Sons, In [8] I. Mezo, Several Generating Funtions for Seond-Order Reurrene Sequenes, J. Integer Sequenes, 12, Artile [9] H. Prodinger, Sums of powers of Fibonai polynomials, Pro. Indian Aad. Si. Math. Si [10] E. D. Rainville, Speial Funtions, The Mamillan Company, [11] H. M. Srivastava and H. L. Manoha, A Treatise on Generating Funtions, Ellis Horwood Limited Publisher, Chihester [12] H. M. Srivastava, M. A. Özarslan and C. Kaanoğlu, Some families of generating funtions for a ertain lass of three-variable polynomials, Integral Transforms Spe. Funt ,