SOME FORMULAE FOR THE FIBONACCI NUMBERS Brian Curtin Department of Mathematics, University of South Florida, 4202 E Fowler Ave PHY4, Tampa, FL 33620 e-mail: bcurtin@mathusfedu Ena Salter Department of Mathematics, Manatee Community College, 8000 South Tamiami Tr, Office #67, Venice, FL 34293 e-mail: saltere@mccfledu David Stone Mathematical Sciences, Georgia Southern University, PO Box 8093, Statesboro, GA 30460-8093 e-mail: dstone@georgiasouthernedu (Submitted December 2005-Final Revision September 2006) ABSTRACT Using elementary methods, we derive formulae for the shifted summations d F n+jf m+j, d L n+jl m+j, and d F n+jl m+j and the shifted convolutions d F n+jf d m j, d L n+jl d m j, and d F n+jl d m j for positive integers d and arbitrary integers n and m INTRODUCTION We derive the following identities involving the Fibonacci numbers F n and the Lucas numbers L n For all positive integers d and for all integers n and m, Fd F n+m+d if d is even, F n+j F m+j = 5 (L dl n+m+d ( ) n L m n ) if d is odd, 5Fd F n+m+d if d is even, L n+j L m+j = L d L n+m+d + ( ) n L m n if d is odd, Fd L n+m+d if d is even, F n+j L m+j = L d F n+m+d + ( ) n+ F m n if d is odd, () (2) (3) F n+j F d+m j = 5 (dl n+m+d ( ) n L m n F d ), (4) L n+j L d+m j = dl n+m+d + ( ) n L m n F d, (5) F n+j L d+m j = df n+m+d + ( ) n+ F m n F d (6) 7
We also present a number of interesting consequences of these formulae We wish to emphasize three aspects of this paper which we find particularly satisfying First, our methods are elementary we believe that this paper is accessible to anyone who has completed an elementary linear algebra course Indeed, we use little more than the Binet formulae for the Fibonacci and Lucas numbers, the sum of a finite geometric series, and linearity of the dot product In fact, the standard linear algebraic derivation [2] of the Binet formulas fits well into the framework of Fibonacci vectors Second, our methods are productive That is to say, carrying out the proofs produces the formulae, in contrast to many inductions which require prior knowledge of the formula to be proved Finally, our results concerning Fibonacci and Lucas numbers closely parallel one another This further supports our faith in the orderliness of the subject We note that special cases of () (3) appear in [3] (modulo a few identities) as Equations (4) (7) For example, Equation (4) in [3] is Fd F m+d if d is even, F +j F m++j = (7) F d F m+d F m if d is odd The left side of (7) is the same as that of () with n = When d is even, the corresponding right sides are exactly the same, and when d is odd, a few elementary identities, such as L x L y 5F x F y = 2( ) y L x y, transform one right side into the other Conversely, subtracting from (7) the same equation with d = n leaves a sum like that on the left side of () Again various identities show that the corresponding right sides are equal We note that our methods are quite different from those of [3] and have also led us to (4) (6) We also believe that our methods both open up interesting avenues of investigation and can be pushed further than they have been here 2 SOME VECTORS In this section, we introduce some vectors For the rest of this paper, we fix the length of all vectors to be some positive integer d We suppress the dependence of the vectors on d in our notation Definition 2: For all integers n, define F n F n+ fn = F n+d and ln = L n L n+ L n+d We refer to f n and l n as the n-th Fibonacci and Lucas vectors (of length d), respectively The following two vectors play a fundamental role in the study of Fibonacci and Lucas vectors Definition 22: Define a = α α 2 α d and b = 72 β β 2 β d
The key facts relating a and b to the Fibonacci and Lucas vectors are the following generalizations of the Binet formulae Theorem 23: For all integers n, fn = ( α n a β n ) b, (8) ln = α n a + β n b (9) Proof: Compare the j-th entry on both sides of the equations Recall the sum of a finite geometric series: For all real numbers γ, Lemma 24: γ j = γd γ (0) a Fd ()α d a = L d α d Fd ()β d b b = L d β d if d is even, if d is odd, if d is even, if d is odd, a b = 0 if d is even, if d is odd Proof: By definition of dot product and Equation (0), and since αβ =, ( ) j a α a = α 2j = = ( α/β)d β ( α/β) = ( ) d α d β d β d = α d ( α d ( ) d β d) Suppose d is even Then by (8) ( a α a = α d d β d ) () = α d ()F d Now suppose d is odd Then by (9) a a = α d ( α d + β d) = α d L d The computation of b b is similar A direct computation using αβ = gives a b to be claimed 73
Consider the map which reverses each vector Definition 25: Define ρ : R d R d by ρ( v v 2 v d ) = v d v d Observe that ρ is a vector space isomorphism Hence, for all integers n, v ρ( f n ) = ( α n ρ( a ) β n ρ( ) b ), () Observe that ρ( a ) = α d ρ( l n ) = α n ρ( a ) + β n ρ( b ) (2) β β 2 ( β) d 2 ( β) d and ρ( b ) = β d α α 2 ( α) d 2 ( α) d Note that ρ is an isometry: For all v, w R d : ρ( v ) ρ( w ) = d ρ( v )[i]ρ( w )[i] = i= d v [d i] w [d i] = i= d v [i] w [i] = v w i= In particular, ρ( a ) ρ( a ) = a a, and ρ( a ) ρ( b ) = a b, ρ( b ) ρ( b ) = b b are given by Lemma 24 Note that ρ is an involution, so v ρ( w ) = ρ( v ) w The following are direct results of the dot product and other definitions Lemma 26: a ρ( a ) = dα d, b ρ( b ) = dβ d, a ρ( b ) = Fd 3 MAIN FIBONACCI AND LUCAS IDENTITIES In this section we prove the shifted summation and convolution identities for the Fibonacci and Lucas numbers which were stated in the introduction We shall only present the details 74
for the proofs of () and (4) since those of (2), (3), (5), and (6) proceed similarly We shall need the following generalizations of the Binet formulae: For all integers n and m, α n β m + α m β n = ( ) n L m n, (3) α n β m α m β n = ( ) n+ ()F m n (4) Proof of Equation : Observe that the left side is equal to f n f m by the definition of the dot product Now expand the dot product with (8) to obtain: fn f m = ( α n a β n ) ( b α m a β m ) b = ( α n+m a a + β n+m b b (α n β m + α m β n ) a ) b 5 First assume d is even Then by Lemma 24 and (4) fn f m = F d ( α n+m+d β n+m+d ) = F d F n+m+d Now assume d is odd Then by Lemma 24 and (3) fn f m = ( Ld (α n+m+d + β n+m+d ) (α n β m + α m β n ) ) 5 = 5 (L dl n+m+d ( ) n L m n ) Proof of Equation (4): Observe that the left side is equal to f n ρ( f m ) by the definition of the dot product Now expand the dot product with (8) and () and simplify using α β = 5, Lemma 26, and (3) to obtain: fn ρ( f m ) = ( α n a β n ) ( b α m ρ( a ) β m ρ( ) b ) = ( α n+m a ρ( a ) + β n+m b ρ( b ) (α n β m + α m β n ) a ρ( ) b ) 5 = 5 ( dα n+m+d + dβ n+m+d ( ) n L m n F d ) = 5 (dl n+m+d ( ) n L m n F d ) 4 FURTHER FIBONACCI AND LUCAS IDENTITIES We derive a few samples of consequences of Equations () (6) Although many of these results are known or consequences of known identities, Equations () (6) provide very simple 75
proofs The first corollary provides straightforward generalizations of Lucas classic result n j= F j 2 = F n F n+ The second and third corollaries, respectively, specify the sum and difference of two squares which may not be consecutive, and the sum and difference of any twwo Fibonacci numbers with even subscripts Corollary 4: For all integers n, Fn+j 2 Fd F 2n+d if d is even, = 5 (L dl 2n+d 2( ) n ) if d is odd, L 2 5Fd F 2n+d if d is even, n+j = L d L 2n+d + 2( ) n if d is odd, Fd L 2n+d if d is even, F n+j L n+j = L d F 2n+d if d is odd Proof: Take m = n in Eq (), (2), and (3), respectively Corollary 42: For all integers m and n, Fn 2 + Fm 2 Fn m F n+m if n m is odd, = 5 (L n ml n+m 4( ) m ) if n m is even, L 2 n + L 2 5Fn m F n+m if n m is odd, m = L n m L n+m + 4( ) m if n m is even, Fn m L n+m if n m is odd, F 2n + F 2m = F n L n + F m L m = L n m F n+m if n m is even Proof: Subtract each of the equations of Corollary 4, with d replaced by d 2 and n replaced by n +, from the same equation with no changes, and simplify Write m = n + d Corollary 43: For all integers m and n, Fn 2 Fm 2 Fn m F n+m if n m is even, = 5 (L n ml n+m + 4( ) m ) if n m is odd, L 2 n L 2 5Fn m F n+m if n m is even, m = L n m L n+m 4( ) m if n m is odd, Fn m L n+m if n m is even, F 2n F 2m = F n L n F m L m = L n m F n+m if n m is odd Proof: Subtract each of the equations of Corollary 4 from the same equation with n replaced by n +, and simplify Write m = n + d 76
Corollary 44: For all integers l, m, and n, Fl F 2n m+l 2 if l is even, F n+l 2 F n m+l + F n F n m = 5 (L l L 2n m+l 2 2( ) n m L m ) if l is odd, 5Fl F 2n m+l 2 if l is even, L n+l 2 L n m+l + L n L n m = L l L 2n m+l 2 + 2( ) n m L m if l is odd, Fl L 2n m+l 2 if l is even, F n+l 2 L n m+l + F n L n m = L l F 2n m+l 2 + 2( ) n m F m if l is odd Proof: To prove the first equation for postive l, simplify the result of subtracting () with d, n, and m replaced by l, n and n m, respectively, from () with d and m replaced by l 2 and n m + respectively For l 0, the result follows from the l 0 case and the identities F n = ( ) n+ F n and L n = ( ) n L n The other formulae are derived similarly Corollary 45: For all integers m and n, F n 2m + F n+2m 2 F 2m = L n, F n+2m + F n 2m L 2m = F n, L n 2m + L n+2m 2 F 2m = F n, L n 2m + L n+2m L 2m = L n In particular, for fixed n, each of the quantities on the the left-hand sides of these equations is integral and independent of m Proof: To prove the first equation, let l = m be even in the first equation of Corollary 44, yielding F n+m 2 F n + F n F n m = F m F 2n 2, so (F n+m 2 + F n m )/F m = F 2n 2 /F n = F 2(n ) /F n = L n Replacing m with 2m, we have the stated conclusion To prove the second equation, let l = m be odd in the first equation of Corollary 44, yielding F n+m 2 F n + F n F n m = (L m L 2n 2 2( ) n d L m )/5 Simplifying, we find that (F n+m 2 + F n m )/L m = (L 2n 2 + 2( ) n )/(5F n ) Now the left-hand side must be independent of m: Setting m = reveals the constant value to be F n Replacing m with 2m + gives the result Similar arguments based upon the second equation of Corollary 44 give the last two results 77
Corollary 46: For all integers n, ( ) ( ) j n+ F d F d if d is even, F n j F n+j = ( ) ( ) n+ L d L d + L 2n if d is odd, 5 5( ) ( ) j n F d F d if d is even, L n j L n+j = ( ) n L d L d L 2n if d is odd, ( ) ( ) j n F d L d 2( ) n if d is even, F n+j L n j = ( ) n F d L d + F 2n if d is odd, ( ) ( ) j n+ F d L d if d is even, F n j L n+j = ( ) n+ L d F d + F 2n if d is odd Proof: Set m = n in Eqs () (3) and simplify with F n = ( ) n+ F n and L n = ( ) n L n We view the equations of Corollary 46 as generalizations of Cassini s identity F 2 n F n F n+ = ( ) n+, which we recover by taking d = 2 in the first equation When d = 3 and d = 4 the generalizations are F 2 n F n F n+ + F n 2 F n+2 = (( ) n+ 2 + L 2n )/5 and F 2 n F n F n+ + F n 2 F n+2 F n 3 F n+3 = 6( ) n+ Corollary 47: For all integers t, ( ) j Fd F t d if d is even, F j+ F t j = 5 (L dl t d + L t+ ) if d is odd, ( ) j 5Fd F t d if d is even, L j+ L t j = L d L t d + L t+ if d is odd, ( ) j Fd L t d if d is even, F j+ L t j = L d F t d + F t+ if d is odd, ( ) j Fd L t d if d is even, F t j L j+ = L d F t d + F t+ if d is odd Proof: Take n = and m = t in Eqs () (3) and simplify with F n = ( ) n+ F n and L n = ( ) n L n The last equation is derived by taking n = t and m = in Eq (3) We now give some sample consequences of (4) (6) Corollary 48: For all integers l, m, and n, F n F l+m + F l+n F m = 2L n+m+l ( ) n L m n L l, L n L l+m + L l+n L m = 2L n+m+l + ( ) n m L m n L l, F n L l+m + F l+n L m = 2F n+m+l ( ) n F m n L l 78
Proof: The proof is similar to that of Corollary 44 Here the equations F n +F n+ = L n and L n + L n+ = 5F n for all integers n (see [,2]) are useful in the simplification 5 LINEAR ALGEBRA AND FUTURE DIRECTIONS This paper is based upon part of the Master s thesis of the second author (Salter) [4] performed under the joint supervision of the other two authors This thesis dealt with the linear algebra of the Fibonacci and Lucas vectors Observe that we have in fact proven the following dot product formulae Theorem 5: For all integers m and n, fn Fd F n+m+d if d is even, f m = 5 (L dl n+m+d ( ) n L m n ) if d is odd, ln 5Fd F n+m+d if d is even, l m = L d L n+m+d + ( ) n L m n if d is odd, fn Fd L n+m+d if d is even, l m = L d F n+m+d + ( ) n+ F m n if d is odd, fn ρ( f m ) = 5 (dl n+m+d ( ) n L m n F d ), ln ρ( l m ) = dl n+m+d + ( ) n L m n F d, fn ρ( l m ) = df n+m+d + ( ) n+ F m n F d Corollary 52: For all integers n, f 2 Fd F 2n+d if d is even, n = 5 (L dl 2n+d 2( ) n ) if d is odd, l 2 5Fd F 2n+d if d is even, n = L d L 2n+d + 2( ) n if d is odd Corollary 53: For all integers m and n, ln 5fn f m l m = 5 f n f m + 2( ) n L m n if d is even, if d is odd By Theorem 23, all Fibonacci and Lucas vectors of a given length lie in a plane In future work we shall further develop the linear algebraic and geometric results concerning the Fibonacci and Lucas vectors studied in [4] 79
REFERENCES [] VE Hoggatt Fibonacci and Lucas Numbers, Houghton Mifflin Company, New York, NY, 969 [2] R Honsberger Mathematical Gems III, Math Assoc America, Washington, DC, 985 [3] CT Long Discovering Fibonacci identities The Fibonacci Quarterly 23 (986): 60-67 [4] E Salter Fibonacci Vectors, Masters thesis, University of South Florida, 2005 AMS Classification Numbers: B39 80