1. INTRODUCTION AND RESULTS

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1 SOME IDENTITIES INVOLVING THE FIBONACCI NUMBERS AND LUCAS NUMBERS Wenpeng Zhang Research Center for Basic Science, Xi an Jiaotong University Xi an Shaanxi, People s Republic of China (Subitted August 00 Final Revision March 003). INTRODUCTION AND RESULTS As usual, the Fibonacci sequence {F n } the Lucas sequences {L n }(n = 0,,,...,) are defined by the second-order linear recurrence sequences F n+ = F n+ + F n L n+ = L n+ + L n for n 0, F 0 = 0, F =, L 0 = L =. These sequences play a very iportant role in the studied of the theory application of atheatics. Therefore, the various properties of F n L n were investigated by any authors. For exaple, R. L. Duncan [] L. Kuipers [5] proved that (log F n ) is uniforly distributed od. Neville Robbins [4] studied the Fibonacci nubers of the fors px ±, px 3 ±, where p is a prie. The author [6] Fengzhen Zhao [3] obtained soe identities involving the Fibonacci nubers. In this paper, as a generalization of [3] [6], we shall use eleentary ethods to study the calculating probles of the general suations F (a +) F (a +)... F (ak +) () a +a + +a k =n a +a + +a k =n L a L a... L ak, () give two exact calculating forulas, where the suation is taken over all k-diension nonnegative integer coordinates (a, a,..., a k ) such that a + a + + a k = n, k are any positive integers, n be any nonnegative integer. For convenience, we first define Chebyshev polynoials of the first second kind T (x) = {T n (x)} U(x) = {U n (x)}(n = 0,,,..., ) as follows: T n+ (x) = xt n+ (x) T n (x) (3) U n+ (x) = xu n+ (x) U n (x) (4) for n 0, T 0 (x) =, T (x) = x, U 0 (x) = U (x) = x. Let U n (k) (x) denote the k th derivative of U n (x) with respect to x. We will use generating functions for the sequences T n (x) U n (x) their partial derivatives to prove the following two theores. Theore : For any positive integer k, nonnegative integer n, we have the identity a +a + +a =n ( ) n F F (a +) F (a +)... F (a +) = ( i) k k! i n+k L, 49

2 where i is the square root of. Theore : For any positive integer k, nonnegative integer n, we have L a L a... L a a +a + +a =n+ where ( ) h = ()! h! ( h)!. = ( i) (n+) k! ( i + h=0 L ) h ( k + h ) n+ h ( i Fro these two theores we ay iediately deduce the following corollaries: L Corollary : For any positive integer nonnegative integer n, we have the identities a +a +a 3 =n F (a +) F (a +) F (a3 +) = 3 [ (n + )(n + 4) F (n+3) 3 (n + 3)L 4 ( ) L ( ) F 4 ( ) L In particular, for =, 3, 4 5, we have the identities a +a +a 3 =n a +a +a 3 =n a +a +a 3 =n a +a +a 3 =n ), F (n+) + (n + )( ) L ] 4 ( ) L F (n+3). F (a +) F (a +) F (a3 +) = 50 [8(n + 3)F n+4 + (n + )(5n 7)F n+6 ], F 3(a +) F 3(a +) F 3(a3 +) = 50 [(n + )(5n + 8)F 3n+9 6(n + 3)F 3n+6 ], F 4(a +) F 4(a +) F 4(a3 +) = 50 [(n + )(5n + )F 4(n+3) + 4(n + 3)F 4(n+) ] F 5(a +) F 5(a +) F 5(a3 +) = 50 [(n + )(5n + 37)F 5(n+3) 66(n + 3)F 5(n+) ]. Corollary : For any positive integer k nonnegative integer n, we have the identities a +a +a 3 =n+3 a +a +a 3 =n+3 L a L a L a3 = n + 5 [(n + 0)F n+3 + (n + 7)F n+ ], L a L a L a3 = n + 5 [3(n + 0)F n+5 + (n + 6)F n+4 ] 50

3 a +a +a 3 =n+3 L 3a L 3a L 3a3 = n + 5 [4(n + 0)F 3n+7 + 3(n + 9)F 3n+6 ]. Corollary 3: For any positive integer nonnegative integer n, we have the congruences (n + )(4n + 6 ( ) L ) F (n+3) 6(n + 3) L F (n+) od (4 ( ) L ) F. In particular, for = 3, 4 5, we have (n + )(5n + 8)F 3n+9 6(n + 3)F 3n+6 od 400; (n + )(5n + )F 4(n+3) + 4(n + 3)F 4(n+) 0 od 4050; (n + )(5n + 37)F 5(n+3) 66(n + 3)F 5(n+) od SEVERAL LEMMAS In this section, we shall give several leas which are necessary in the proofs of the theores. First we need two exact expressions generating functions on T n (x) U n (x) (see (..) of []). That is, U n (x) = T n (x) = x [( x + ) n ( x + x ) n ] x [ ( x + n+ ( x ) x ) ] n+ x. (6) So we can easily deduce that the generating function of T (x) nd U(x) are + xt G(t, x) = xt + t = T n (x) t n (7) F (t, x) = (5) + xt + t = U n (x) t n. (8) Applying these generating functions we can easily deduce the following Lea : For any positive integer k nonnegative integer n, we have the identity a +a + +a =n U a (x) U a (x)... U a (x) = k k! n+k (x). 5

4 Proof: Differentiating (8) we obtain F (t, x) x = t ( xt + t ) = U () n+ (x) tn+ ; F (t, x) x = k F (t, x) x k =! (t) ( xt + t ) 3 = k! (t) k ( xt + t ) = U () n+ (x) tn+ ; where we have used the fact that U n (x) is a polynoial of degree n. Therefore, fro (9) we obtain = a +a + +a =n U a (x) U a (x)... U a (x) t n = ( xt + t ) = k F (t, x) k!(t) k x k = k k! n+k (x) tn+k. (9) ( ) U n (x) t n n+k (x) tn. (0) Equating the coefficients of t n on both sides of equation (0) we obtain the identity This proves Lea. a +a + +a =n U a (x) U a (x)... U a (x) = k k! n+k (x). Lea : For any positive integer k nonnegative integer n, we have a +a + +a =n+ T a (x) T a (x) = ( k + k ( x) h k! h h=0 ) n+ h (x). Proof: To prove Lea, ultiplying ( xt) on both sides of (9) we have ( xt) ( xt + t ) = k k! n+k (x) tn ( xt). () Note that ( xt) = h=0 ( x)h t h( ) h. Coparing the coefficients of t n+ on both sides of equation () we obtain Lea. 5

5 Lea 3: For any positive integers n, we have the identities T n (T (x)) = T n (x) U n (T (x)) = U (n+) (x). U (x) Proof: For any positive integer, fro (5) we have T (x) = 4 [ (x + x ) + (x x ) ] or = 4 T (x) = [(x + x ) (x x ) ] [ (x + x ) (x x ) ]. Thus, T (x) + T (x) = (x + x ). () Cobining (6), () (3) we have U n (T (x)) = T (x) T (x) T (x) = (x x ). (3) [ ( T (x) + ) n+ ( T(x) T (x) ) ] n+ T(x) = (x + x ) (n+) (x x ) (n+) (x + x ) (x x ) = U (n+) (x). U (x) Siilarly, we can also deduce that T n (T (x)) = T n (x). This proves Lea PROOF OF THE THEOREMS Now we coplete the proofs of the theores. Let i be the square root of. Taking x = T ( i ) in Lea Lea, noting that U n( i ) = in F n+, T n ( i ) = in L n, T n (T ( i )) = in. L n, U n (T ( i )) = in F (n+), we ay iediately deduce Theore Theore F Proof of the Corollaries: First we note that U n (x) satisfies the differential equations ( x )U n(x) = (n + )U n (x) nxu n (x) (4) 53

6 So fro Lea 3, (4) (5) we obtain U n ( ( )) i T = ( ( )) i U n T = ( x )U n (x) = 3xU n(x) n(n + )U n (x), (5) 4 4 ( ) L 4i n F (4 ( ) L ) [ i (n ) (n + )F n i (n+) nl ] F (n+) F F [ 6(n + )L 4 ( ) L F n ( ) 3nL ] 4 ( ) L F (n+) n(n + )F (n+). (6) Now Corollary Corollary follows fro the recurrence forula F n+ = F n+ + F n, (6), Theore Theore (with k = ). Corollary 3 follows fro Corollary the fact that F F (a+) for all integer a 0. ACKNOWLEDGMENTS The author expresses his gratitude to the referee for his very helpful detailed coents. This work is supported by the N.S.F. (07093) P.N.S.F. (00A) of P.R. China. REFERENCES [] P. Borwein T. Erdélyi. Polynoials Polynoials Inequalities. Springer-Verlag, New York, 995. [] R.L. Duncan. Applications of Unifor Distribution to the Fibonacci Nubers. The Fibonacci Quarterly 5 (967): [3] Fengzhen Zhao Tianing Wang. Generalizations of Soe Identities Involving the Fibonacci Nubers. The Fibonacci Quarterly 39 (00): [4] L. Kuipers. Reark on a Paper by R. L. Duncan Concerning the Unifor Distribution Mod of the Sequence of the Logariths of the Fibonacci Nubers. The Fibonacci Quarterly 7 (969): [5] N. Robbins. Applications of Fibonacci Nubers. Kluwer Acadeic Publishers, 986, pp [6] Wenpeng Zhang. Soe Identities Involving the Fibonacci Nubers. The Fibonacci Quarterly 35 (997): 5-9 AMS Classification Nubers: B37, B39 54