Generalized Fibonacci-Like Polynomials and Some Identities

Global Journal of Mathematical Analysis, 2 (4) (2014) 249-258 ©Science Publishing Corporation www.sciencepubco.com/index.php/GJMA doi: 10.14419/gjma.v2i4.3126 Research Paper Generalized Fibonacci-Like Polynomials and Some Identities Mamta Singh 1, Yogesh Kumar Gupta 2, Omprakash Sikhwal3* 1 Department of Mathematical Sciences and Computer Application, Bundelkhand University, Jhansi (U. P.) India 2 School of Studies in Mathematics, Vikram University Ujjain, India 3 Department of Mathematics, Mandsaur Institute of Technology Mandsaur (M. P.) India *Corresponding author E-mail: [email protected] Copyright © 2014 Mamta singh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The Fibonacci polynomials and Lucas polynomials are famous for possessing wonderful and amazing properties and identities. In this paper, Generalized Fibonacci-Like Polynomials are introduced and defined by mn ( x)  xmn1 ( x)  mn2 ( x), n  2. with m0 ( x)  2s and m1 ( x)  1  s , where s is integer. Further, some basic identities are generated and derived by standard methods. Keywords: Fibonacci Polynomials, Lucas Polynomials, Generalized Fibonacci-Like Polynomials, Binet’s formula. 1. Introduction The Fibonacci polynomials and Lucas polynomials are famous for possessing wonderful and amazing properties and identities. It is well-known that the Fibonacci polynomials and Lucas polynomials are closely related and widely investigated. Fibonacci polynomials appear in different frameworks. These polynomials are of great importance in the study of many subjects such as algebra, geometry, combinatorics, approximation theory, statistics and number theory itself. Moreover these polynomials have been applied in every branch of mathematics. Fibonacci polynomials are special cases of Chebyshev polynomials and have been studied on a more advanced level by many mathematicians. Basin, S. L. [1] show that Q matrix generates a set of Fibonacci Polynomials is defined by the recurrence formula (1.1) f n 1 ( x)  xf n ( x)  f n 1 ( x), n  2 with f 0 ( x)  0, f 1 ( x)  1. The Lucas Polynomials is defined by the recurrence formula ln 1 ( x)  xln ( x)  ln 1 ( x), n  2 with l0 ( x)  2, l1 ( x)  x. (1.2) Generating function of Fibonacci polynomial is   f  xt n 0  t 1  xt  t 2  n n 1 (1.3) Generating function of Lucas polynomial is  l  x t n0 n n   2  xt  1  xt  t 2  1 (1.4) Explicit sum formula for (1.1) is given by f n ( x)   n 1  2     n  k  1 n 1 2 k . x  k   k 0 n where   a binomial coefficient and [x] is define as the greatest integer less than or equal to x. k  Explicit sum formula for (1.3) is given by (1.5) 250 Global Journal of Mathematical Analysis n 2   n  n  k  n2k  x . k 0 n  k  k  ln ( x)   (1.6) n where   a binomial coefficient and [x] is defined as the greatest integer less than or equal to x. k  Fibonacci-Like polynomials [11] is defined by the recurrence relation: sn  x   xsn-1  x   sn  2  x  , n  2 with initial terms s0  x  = 2 and s1  x  =2x. (1.7) Generalized Fibonacci-Lucas Polynomials [12] is defined by recurrence relation. bn  x   xbn-1  x   b n-2 ,  x  , n  2 With initial conditions b0  x   2b and b1  x   s , (1.8) where b and s are integers. The Fibonacci and Lucas polynomials possess many fascinating properties which have been studied in [2] to [12]. In this paper, Fibonacci-Like Polynomials are introducing with some basic identities and derived by standard method. 2. Generalized Fibonacci-Like Polynomials: Generalized Fibonacci-Like Polynomials mn  x  are defined by recurrence relation mn ( x)  xmn1 ( x)  mn2 ( x), n  2. with m0 ( x)  2s and m1 ( x)  1  s , where s is integer. (2.1) The first few terms of generalized Fibonacci-Like Polynomials are as follows: m0 ( x)  2s , m1 ( x)  1  s , m2 ( x)  (1  s) x  2s , m3 ( x)  (1  s) x 2  2sx  (1  s) and so on. If x=1, then mn (1) is generalized Fibonacci-Like sequence. The characteristic equation of recurrence relation (2.1) is  x x2  4 2 and  x 2  x  1  0 x2  4 2 (2.2) Also,   1 , Binet’s formula of Generalized Fibonacci-Like sequence is defined by mn  x   An  Bn x mn  x   A   n  x2  4    B x    2   x2  4    2  n Here, A  2s  x    and B  2s   x  .   Also, AB  4s 2     (2.3)   2 , A  B  m0 ( x)  2s,     x 2  4, and  2   2  x 2  5. Generating function of Generalized Fibonacci-Like Polynomials is  2s 1  xt   1  s  t mn  x  t n  .  n 0 1  xt  t 2  Now Hyper geometric representation of generating function   m  xt n 0 n n   2s 1  xt   1  s  t  1  xt  t 2  , 1 1  2s 1  xt   1  s  t  1   x  t  t  , (2.4) (2.5) 251 Global Journal of Mathematical Analysis   m  xt n 0  n n   2s 1  xt   1  s  t    x  t  t n n n0  n n   2s 1  xt   1  s  t   .t n    x n  k .t k n 0 k 0  k   n n!   2s 1  xt   1  s  t   . x n  k .t k  n n  0 k  0 k ! n  k  !     2s 1  xt   1  s  t   .  n  k ! k!n ! n 0 k 0  xt n n 0 n!  2s1  xt   1  s t  .    2s 1  xt   1  s  t  e xt  n  k  k 0    2s 1  xt   1  s  t  e xt  k 0   2s 1  xt   1  s  t  e n! x n .t 2 k  n   n  k ! .t 2 k n! k! k 0 t  2 k . k! n  k  1 t  . k! nk 2 k 1k  t 2  1 . n     k 1k k ! k 0  xt k  2s 1  xt   1  s  t  e xt 2 F1 n  1:1;1; t 2  ,   m  x t n n n 0   2s 1  xt   1  s  t  e xt 2 F1 n  1:1;1; t 2  (2.6) 3. Some Identities of Generalized Fibonacci-Like Polynomials In this section, we present some identities like Catalan’s; Cassini’s; d’Ocagne’s identities etc. by Binet’s formula or explicit sum formula or generating function. Theorem 3.1: Prove that mn 1  x   mn 1  x   xmn  x  , n  1. (3.1) Proof: By Generating function of Fibonacci-Like polynomials, we have   m  xt n 0   2s 1  xt   1  s  t  1  xt  t 2  n n 1 Differentiating both sides with respect to t, we get 2 1  n1  2s 1  xt  1  s t x  2t 1  xt  t 2  1  s  2sx  1  xt  t 2    nmn  x  t        n 0   1  xt  t   nm  x  t 2 n 0  1  xt  t   nm  x  t 2 n 0   nm  x  t n0 n n 1 n 1 n n n 1      2s 1  xt   1  s  t   x  2t  1  xt  t 2   1  s  2sx  1    x  2t   mn  x  t n  1  s  2sx  n 0     n0 n0 n0 n0   xnmn t n   nmn  x  t n 1   xmn  x  t n   2mn t n 1  1  s  2sx  Now equating the coefficient of t n on both sides we get,  n  1 mn1  x   nxmn  x    n 1 mn1  x   xmn  x   2mn1  x  ,  n  1 mn1  x    n  1 mn1  x    n  1 xmn  x  , mn 1  x   mn 1  x   xmn  x  This is required results. 252 Global Journal of Mathematical Analysis Theorem 3.2: Prove that mn'  x   xmn' 1  x   mn 1  x   mn'  2  x  , n  2. (3.2) Proof: By generating function of Fibonacci-Like polynomials, we have   m  xt n 0   2s 1  xt   1  s  t  1  xt  t 2  n n 1 Differentiating both sides with respect to x, we get   m  xt n 0 n   2s 1  xt   1  s  t  1  xt  t 2  , n   2s 1  xt   1  s  t   1 1  xt  t 2  1 n   m  xt n0 ' n  1  xt  t   m  x  t 2 ' n n 0 2 ' n n 0     2s 1  xt   1  s  t   1 1  xt  t 2  n  t  mn  x  t n  2st , n 0 n n 0 ' n 1 1  t   2st   m  x  t   xm  x  t ' n  t   1  xt  t 2   2st  n  1  xt  t   m  x  t 2 n 0 n 1   n 0 n 0   mn'  x  t n  2  t  mn  x  t n 1  2st , Now equating the coefficient of t n , we get mn'  x   xmn' 1  x   mn' 2  x   mn 1  x  . mn'  x   xmn' 1  x   mn 1  x   mn'  2  x  . Theorem 3.3: Prove that mn' 1  x   xmn'  x   mn  x   mn' 1  x  , n  1. (3.3) Proof: By (3.1), we have mn 1  x   mn 1  x   xmn  x  , n  1. By differentiating with respect to x, we get mn' 1  x   mn' 1  x   xmn'  x   mn  x  , mn' 1  x   xmn'  x   mn  x   mn' 1  x  . Theorem 3.4: Prove that nmn  x   xmn'  x   2mn' 1  x  . , n  1 and xmn' 1  x    n  1 mn 1  x   2mn'  x  , n  1. Proof: By generating function of Fibonacci-Like polynomials, we have   m  xt n 0 n n   2s 1  xt   1  s  t  1  xt  t 2  . 1 Differentiating both sides with respect to t, we get   nm  x  t n 1 n n0  1  s  2sx  1  xt  t 2   2s 1  xt   1  s  t   x  2t  1  xt  t 2  1 2 . (3.4) Differentiating both sides with respect to x, we get   m  xt n0 ' n  n   2st  1  xt  t 2   2s 1  xt   1  s  t  .t 1  xt  t 2  1 ' n n 1   2s  1  xt  t 2   2s 1  xt   1  s  t  1  xt  t 2   m  xt n 1  2s 1  xt  t 2    2s 1  xt   1  s  t  1  xt  t 2   m  xt n0  n 0 ' n 1 1 2 2 2 (3.5) 253 Global Journal of Mathematical Analysis Using (3.5) in (3.4), we get   ' 2 1 2 1 n 1 n 1 nm x t 1 s 2 sx 1 xt t x 2 t                 mn  x  t  2s 1  xt  t  ,  n n0  n0    nm  x  t n0   nm  x  t n0 n  n 1  1  s  2sx  1  xt  t 2    x  2t   mn'  x  t n 1  2s  x  2t  1  xt  t 2  n 1  1  s  2sx  1  xt  t n 1 1 n0  2 1   n0 n0  x mn'  x  t n 1   2mn'  x  t n  2s  x  2t  1  xt  t 2  1 Equating the coefficient of t n 1 on both sides, we get nm n  x   xm n'  x   2m n' 1  x  . (3.6) n Again equating the coefficient of t , we get  n  1 mn1  x   xm'n1  x   2m'n  x  , xmn' 1  x    n  1 mn 1  x   2mn'  x  . . (3.7) Theorem 3.5: Prove that  n  1 mn  x   m'n1  x   m'n1  x  . , n  1 . Proof: By (3.1), we have mn 1  x   mn 1  x   xmn  x  , n  1. By differentiating with respect to x, we get m'n 1  x   m'n 1  x   xm'n  x   mn  x  , xm'n  x   mn  x   m'n 1  x   m'n 1  x  . (3.8) Using equation (3.5) in equation (3.8) we get nmn  x   2m'n 1  x   mn  x   m'n 1  x   m'n 1  x  , nmn  x   mn  x   m'n 1  x   2m'n 1  x   m'n 1  x  ,  n  1 mn  x   m'n1  x   m'n1  x  . (3.9) Theorem 3.6: Prove that xmn'  x   2mn' 1  x    n  2 mn  x  , n  0. Proof: Using equation (3.7) in equation (3.9), we get 1  n  1 mn  x   mn' 1  x   nmn  x   xmn'  x  , 2 2  n  1 mn  x   2mn' 1  x   nmn  x   xmn'  x  , xmn'  x   2mn' 1  x   nmn  x    2n  2 mn  x  , xmn'  x   2mn' 1  x    n  2n  2 mn  x  , xmn'  x   2mn' 1  x    n  2 mn  x  . (3.10) Theorem 3.7:  n  1 xm'n  x   nm'n 1  x    n  2 m'n 1  x  , n  1. Proof: Using equation (2.3) we get  n  1{m'n1  x   xm'n  x   m'n1  x }  m'n1  x   m'n1  x   n  1 m'n1  x    n  1 xm'n  x    n  1 m'n1  x   m'n1  x   m'n1  x  ,  n  1 m'n1  x    n  1 m'n1  x   m'n1  x   m'n1  x    n  1 xm'n  x  , nm'n 1  x    n  2 m'n 1  x    n  1 xm'n  x  .  n  1 xm'n  x   nm'n1  x    n  2 m'n1  x  . (3.11) 254 Global Journal of Mathematical Analysis Theorem 3.8 (Explicit sum formula): For Generalized Fibonacci-Like polynomials are given by n   2  n  k  n2k . mn  x   2s   x k 0  k  (3.12) Proof: By generating function (2.5), we have   m  xt n 0   2s 1  xt   1  s  t  1  xt  t 2  n n 1  2s 1  xt   1  s  t  1  xt  t 2   1   2s 1  xt   1  s  t    x  t  t n n n 0  n  n   2s 1  xt   1  s  t   .t n    x n  k .t k n 0 m0  k   n n!   2s 1  xt   1  s  t   . x n  k .t k  n n  0 m  0 k! n  k  !  n   2s 1  xt   1  s  t   . n 0 m0     2s 1  xt   1  s  t   . n 0 m0 n 2   mn  x   2s   n  k ! k  0 k ! n  2k !  n  k ! k!n !  n  k ! k!n ! x n .t 2 k  n x n .t 2 k  n , xn2k . Equating coefficients of t n on both sides, we get required explicit formula. Theorem 3.9 For positive integer n  0 , prove that 4   n n  1 mn  x   2sx n 2 F1  : ; n; 2  . 2 x   2 . Proof: By explicit sum formula (3.13), it follows that (3.13) n 2    n  k  n2k , mn  x   2s   x k 0  k  mn  x   2sx mn  x   2sx n 2   n  1 1n  n 2k 2k k  0   n   1 1n k k  n 2   mn  x   2sx n   1 k 0 mn  x   2sx 2 k n 2   n , k 0 n 2   n  n  k !  k ! n  2k ! x  k 0  1 k k x 2 k , k! 2 k  n   n  1  x 22 k      2 k  2 k k! , 2k  n k  1  n   n  1   4  22 k      2  2 k  2 k  x  .  n k k ! Hence, 4   n n  1 mn  x   2sx n 2 F1  , ; n, 2  . 2 2 x   Theorem 3.10: For positive integer n  0 , prove that   c c 1 tn n 1 n  2 t2 c , n  1, , ,  c n mn  x   2s 1  xt  3 F1  ,  n! 2 2 1  xt 2 n0 2 2  .   (3.14) 255 Global Journal of Mathematical Analysis Proof: Multiplying both sides of (3.13) by  c n    c n mn  x  t n 0 n 2 n  k! tn  c n x n  2 k , n n  0 k  0 k ! n  2k !   n  k!  2s   c n  2 k x n t n  2 k k ! n ! n  2 k ! n 0 k 0   n  k! n  2s   c  2k n  c 2k  xt  t 2k n  0 k  0 k !n ! n  2k  !  n tn and summing between the limit n=0 to n   , we obtain n!  2s  n   xt    n  k !   2 s    c  2k  n  c 2 k t 2 k  ! ! 2 ! n k n k     n0  n 0  n  k!  c  2 k   2s 1  xt   c 2 k t 2 k  k  0 k ! n  2k !  n  k!  c  2 k   2s  1  xt   c  t 2k k !n  2k ! 2 k k 0  2s 1  xt  n  k!  c 2 k  k  0 k ! n  2k !  c  2s1  xt  c  2s 1  xt  c  2s 1  xt  c  2s1  xt  c  t n k 2   n  k! 22 k  c   c  1   t 2    2  k  2  k  1  xt   k  0 k ! n  2 k!  n  k! k 2   n! 22 k  c   c  1   t 2    2  k  2  k  1  xt   k  0 k!n  2k!   n  1k c c 1  t2 2k   2        2  2  k  2  k  1  xt   k  0  n  1 2 k  n  1k   c k  k 0  2 s 1  xt   t2     1  xt    n 1  n  2  2 2k      2 k  2 k 2 2k k 2   c   c 1  t      2   2  k  2  k  1  xt   k  c   c 1 k      n  1 k   t2  2 k  2 k .   2   n 1   n  2  s 0  1  xt          2 k  2 k  Hence,   c n mn  x   2s 1  xt  n! n 0 c  n 1 n  2  c c 1 3 F1  2 , 2 , n  1, 2 , 2 ,   . 2 xt  1    t2 Theorem 3.11 (Explicit sum formula): For Generalized Fibonacci-Like polynomials n 2    n  k  n2k mn 1  x   1  s  2sx    x . k 0  k  (3.15) Proof: Generating function (2.5), we have   m  xt n 0 n n   2s 1  xt   1  s  t  1  xt  t 2  , 1  2s 1  xt   1  s  t  1  xt  t 2   1   2s 1  xt   1  s  t    x  t  t n , n 0 n 256 Global Journal of Mathematical Analysis  n n   2s 1  xt   1  s  t   .t n    x n  k t k , n 0 m 0  k   n n!   2s 1  xt   1  s  t   . x n  k .t k  n n  0 m  0 k! n  k  !   m  xt n 0 n n  n   2s 1  xt   1  s  t   .  n  k ! k!n ! n 0 m0     2s 1  xt   1  s  t   .  n  k ! k!n ! n 0 m0 x n .t 2 k  n x n .t 2 k  n . Equating coefficients of t n on both sides, we get required explicit formula n 2   mn  x   2s   n  k ! k  0 k ! n  2k ! xn2k . Equating coefficients of t n 1 on both sides, we get required explicit formula n 2    n  k ! mn 1  x   1  s  2s   xn2k . k  0 k ! n  2k ! Theorem 3.12 (Catalan’s Identity): Let mn ( x) be the nth term of generalized Fibonacci-Like polynomials, then mn2  x   mn  r  x  mn r  x    1 nr 1  s  mr  x   2smr 1  x   , n  r  1. 1  5s 2  (3.16) Proof. Using Binet’s formula (2.5), we have mn2  x   mn  r  x  mn r  x    An  Bn    A n  r  Bn  r  A n  r  Bn  r  2   AB  1   AB  2   r   r    r  r n nr r    2 4s  1nr  r   r     2  4 s 2  1 nr Since  r 2  r   r         2 2  r   r 1  s mr x   2smr 1 x  1  s mr x   2smr 1 x  , we obtain     1  5s 2 1  s 2  2s1  s   4s 2 mn2  x   mn  r  x  mn r  x    1 nr 1  s  mr  x   2smr 1  x   , n  r  1. 1  5s 2  Corollary 3.13 (Cassini’s Identity): Let mn ( x) be the nth term of generalized Fibonacci-Like polynomials, then mn2 x   mn 1 x mn 1 x    1 n 1 1  5s , n  1 2 (3.17) Proof: If r = 1 in the Catalan’s identity, then obtained required result. Theorem 3.14 (d’Ocagne’s Identity): Let mn(x) be the nth term of generalized Fibonacci-Like polynomials, then n  1  s  m p  n  x   2 sm p  n 1  m p  x  mn1  x   m p 1  x  mn  x    1   , p  1, n  0, p  n. 1  5s 2   (3.18) 257 Global Journal of Mathematical Analysis Proof. Using the Binet’s formula (2.5), we have mp  x  mn 1  x   mp 1  x  mn  x    A p  B p  An 1  Bn 1    A p 1  B p 1  An  Bn  ,   AB  p  n1   n1 p   n  p1   p1 n    AB    n   AB q       n  4s 2  p n p n  p n p n ,     p n   p n   1      p n   p n  , n     2 p n   pn  n   4s 2  1  .     Using subsequent results of Binet’s formula, we get Since  p  n   p  n 1  s  m p  n  x   2sm p  n 1 1  s  m p  n  x   2sm p  n 1 we obtain ,,   2   1  5s 2 1  s   2s 1  s   4s 2 n  1  s  m p  n  x   2 sm p  n 1  m p  x  mn 1  x   m p 1  x  mn  x    1   , p  1, n  0, p  n. 1  5s 2   Theorem 3.15 (Generalized Identity): Let mn ( x) be the nth term of generalized Fibonacci-Like polynomials, then p r mp  x  mn  x   mp r  x  mnr  x    1 1  s  mr  2smr 1  1  s  mn p r  2smn p r  , n  m  r  1. Proof. Using the Binet’s formula (2.5), we have      m p x mn1 x   m p1 x mx n  A p  B p A n  B n  A pr  B pr A nr  B nr   p  n  n  p   AB  r   r   r  r      AB   r r   r    n r   n r  p  p  p  n  n  p   AB  r   r  r   r       AB 1 r  r  AB 1   r p   AB  1   r p    r  p  n r   n r  p p p   r r   r r   n p  r n p r    n p  r n p r .  Using subsequent results of Binet’s formula, we get Since r  r 1  1  s  mr  2smr 1  .   1  5s 2    n  p  r   n  p  r 1  s  mn  p  r  2smn  p  r 1  , 1  5s 2   m p x mn x   m pr x mnr x    1 p r 1  s mr  2smr 1 1  s mn pr  2smn pr , n  m  r  1 (3.19) The identity (3.15) provides Catalan’s identity, Cassini and d’Ocagne and other identities: 258 Global Journal of Mathematical Analysis 4. 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