On Generalized Fibonacci Polynomials and Bernoulli Numbers 1

Transcription

1 Journal of Integer Sequences, Vol 8 005), Article 0553 On Generalized Fibonacci Polynomials and Bernoulli Numbers 1 Tianping Zhang Department of Mathematics Northwest University Xi an, Shaanxi P R China tpzhang@snnueducn Yuankui Ma Department of Mathematics and Physics Xi an Institute of Technology Xi an, Shaanxi P R China ykma@eyoucom Abstract In this paper we use elementary methods to study the relationship between the generalized Fibonacci polynomials and the famous Bernoulli numbers, and give several interesting identities involving them 1 Introduction and results As usual, the famous Fibonacci polynomials F x) = {F n x)} are defined by the second-order linear recurrence F n+ x) = xf n+1 x) + F n x) 1) for n 0 and F 0 x) = 0, F 1 x) = 1 These polynomials are of great importance in the study of many subjects such as algebra, geometry, and number theory itself Obviously, they 1 This work is supported by the N S F , ) and P N S F of P R China Author s current address: College of Mathematics and Information Science, Shannxi Normal University, Xi an, Shaanxi, P R China 1

2 have a deep relationship with the famous Fibonacci numbers F n ) n 0 That is, F n 1) = F n Many scholars have studied numerous properties of the Fibonacci numbers For example, R L Duncan [1] and L Kuipers [] proved that log F n ) is uniformly distributed mod 1 N Robbins [3] studied the Fibonacci numbers of the forms px ± 1, px 3 ± 1, where p is a prime Wenpeng Zhang [4] and Fengzhen Zhao [5] obtained some identities involving the Fibonacci numbers Moreover, Yuan Yi and Wenpeng Zhang [6] studied the calculation on the summation involving the Fibonacci polynomials, and obtained the following Proposition 1 Let F x) = {F n x)} be defined by 1) Then for all positive integers k and n, we have the formula a 1 +a + +a k =n F a1 +1x)F a +1x) F ak +1x) = n m=0 ) n + k 1 m n + k 1 m m k 1 ) x n m where the summation is over all k-dimension nonnegative integer coordinates a 1, a,, a k ) such that a 1 + a + + a k = n, and z denotes the greatest integer not exceeding z, and ) m n = m! m n)! On the other hand, the famous Bernoulli numbers are defined by x e x 1 = x n B n, x < π ) A recursion formula involving the Bernoulli numbers is n ) n B n = B k k for n and B 0 = 1, B 1 = 1, which successively yields the values k=0 B = 1 6, B k+1 = 0, k = 1,, ), B 4 = 1 30, B 6 = 1 4, B 8 = 1, B 10 = 5 66, B 1 = , B 14 = 7 6, Moreover, the Bernoulli numbers B k alternate in sign, and are related to ζk) as follows: ζk) = 1) k+1 π)k B k k)! Other important results involving the Bernoulli numbers can be found in references [7, 8, 9] Now we consider the polynomial sequence Hx) = {H n x)} defined by H 0 x) = 0, H 1 x) = 1, and H n+ x) = P x)h n+1 x) + Qx)H n x), 3) where P x) and Qx) are polynomials with x) = P x) + 4Qx) > 0 It is easy to see that 3) is a generalization of 1)

3 It is well known that the Fibonacci numbers and Lucas numbers are closely related to the Chebyshev polynomials Yuankui Ma and the first author [10] studied the relationships between the Chebyshev polynomials of the first kind and the famous Euler numbers, and obtained an interesting identity involving them But no similar relationship between the generalized Fibonacci polynomials and the Bernoulli numbers was previously known In this paper, we use elementary methods to study the relationship between the generalized Fibonacci polynomials and the famous Bernoulli numbers, and give several interesting identities involving them That is, we shall prove the following Theorem 1 For all positive integers k and n with k n, we have the formula where β = P x) x) H a1 x) Ha k x) B b1 ) b1 + +b k kβ) n k x) = n k)!, If we take P x) = x and Qx) = 1 in Theorem 1, then we have Corollary 1 For all positive integers k and n with k n, we have where βx) = x x +4 F a1 x) Fa k x) B b1 b1 + +b k kβx)) x + 4) n k =, n k)! If P x) and nonzero Qx) in Theorem 1 are integers with P + 4Q > 0, then we immediately obtain the following Corollary For all positive integers k and n with k n, we have the identity where β = P P +4Q H a1 Ha k B b1 a k! b 1! Bb b1 + +b k k P kβ + 4Q) ) n k = n k)!, Taking x = 1 in Corollary 1, or P = Q = 1 in Corollary, we immediately deduce the following Corollary 3 For all positive integers k and n with k n, we have the identity where β1) = 1 5 F a1 Fa k B b1 3 ) b1 + +b k kβ1)) n k 5 =, n k)!

4 In particular, taking k = 1,, 3 in Corollary 3, we easily get Corollary 4 For all positive integers n, we have a+b=n F a B b ) b β1)) n 1 5 = a! b! n 1)! Corollary 5 For all positive integers n, we have a+b+c+d=n F a F b B c B d a! b! c! d! Corollary 6 For all positive integers n 3, we have a+b+c+d+e+f=n Proof of Theorem F a F b F c B d B e B f a! b! c! d! e! f! ) c+d β1)) n 5 = n )! ) d+e+f 3β1)) n 3 5 = n 3)! In this section, we shall complete the proof of Theorem First we let α = P x)+ x) and β = P x)+ x) denote the roots of characteristic polynomial λ P x)λ Qx) of the generalized Fibonacci polynomial sequence Hx), then the terms of the sequence Hx) can be expressed as see [11, 1]) H n x) = 1 x) P x) + ) n x) Then we easily deduce that the generating function of Ht, x) is P x) ) n ) x) Ht, x) = H n x) t n = α n β n α β) tn 4) That is, Therefore, we have Then from ) and 4), we have e βt = Ht, x) = eαt e βt α β m=0 e βt = Ht, x) t = eβt e t ) x) 1 x) t x) e t x) 1 ) H m x) t m 1 B n t ) ) n x) 5) m! 4

5 Note that for two absolutely convergent power series ) ) a n t n b n t n = so k times on the both sides of formula 5), we have RHS = LHS = e βt) k = e kβt = H a1 x) a n t n and u+v=n kβ) n t n, a u b v ) t n, b n t n, we have Ha k x) B b1 ) b1 + +b k x) t n k Comparing the coefficients of t n k on the above, we immediately obtain the following identity H a1 x) This completes the proof of Theorem 1 3 Acknowledgments Ha k x) B b1 ) b1 + +b k kβ) n k x) = n k)! The author wishes to thank his supervisor, Professor Wenpeng Zhang, who has been most generous with his advice and support Moreover, the author also thanks the anonymous referee for his very helpful and detailed comments on the original manuscripts References [1] R L Duncan, Application of uniform distribution to the Fibonacci numbers, Fibonacci Quart ), [] L Kuipers, Remark on a paper by R L Duncan concerning the uniform distrubution mod 1 of the sequence of the logarithms of the Fibonacci numbers, Fibonacci Quart ), [3] N Robbins, Fibonacci numbers of the forms px ± 1, px 3 ± 1, where p is prime, in Applications of Fibonacci Numbers, Kluwer Academic, 1986, pp [4] Wenpeng Zhang, Some identities involving the Fibonacci numbers, Fibonacci Quart ), 5 9 [5] Fengzhen Zhao and Tianming Wang, Generalizations of some identities involving the Fibonacci polynomials, Fibonacci Quart ),

6 [6] Yuan Yi and Wenpeng Zhang, Some identities involving the Fibonacci polynomials, Fibonacci Quart 40 00), [7] Gi-Sang Cheon, A note on the Bernoulli and Euler polynomials, Appl Math Lett ), [8] H M Srivastava and Á Pintér, Remarks on some relationships between the Bernoulli and Euler polynomials, Appl Math Lett ), [9] M Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976 [10] Yuankui Ma and Tianping Zhang, An identity involving the first-kind Chebyshev polynomials and the Euler numbers, J Ningxia University, to appear [11] G H Hardy and E M Wright, An Introduction to the Theory of Numbers, 4th edition, Oxford Universty Press, 196 [1] Peter Borwein and Tamás Erdélyi, Polynomials and Polynomial Inequalities, Springer- Verlag, Mathematics Subject Classification: Primary 11B37; Secondary 11B39 Keywords: generalized Fibonacci polynomials; Bernoulli numbers; Concerned with sequence A000045) Received July ; revised version received October Published in Journal of Integer Sequences, October Return to Journal of Integer Sequences home page 6