Fibonacci and Lucas identities derived via the golden ratio

By expressing Fibonacci and Lucas numbers in terms of the powers of the golden ratio α = (1 + √ 5) / 2 and its inverse β = − 1 /α = (1 − √ 5) / 2 , a multitude of Fibonacci and Lucas identities have been developed in the literature. In this paper, the reverse course is followed: numerous Fibonacci and Lucas identities are derived by making use of the well-known expressions for the powers of α and β in terms of Fibonacci and Lucas numbers.

Carlitz [4] (also Hoggatt et al. [11]) derived the identity F k+t = α k F t + β t F k , which can be put in the form or equivalently, α s F k+t = α s+t F k + (−1) k α s−k F t .
As Koshy [14, p.93] noted, the two Binet formulas F n = α n − β n α − β , L n = α n + β n , expressing F n and L n in terms of α n and β n , can be used in tandem to derive an array of identities. The aim of this paper is to derive numerous Fibonacci and Lucas identities by emphasizing identities (1)- (6), expressing α n and β n in terms of F n and L n . The method used in this paper for deriving the mentioned identities relies on the fact that α and β are irrational numbers. The following fact is used frequently in the remaining part of this paper: if a, b, c, and d are rational numbers, and if γ is an irrational number, then aγ + b = cγ + d implies that a = c and b = d; an observation that was used by also Griffiths [8,9].
As a quick illustration of the above-mentioned method, take x = αF p and y = F p−1 in the binomial identity n j=0 n j x j y n−j = (x + y) n , which, after multiplying both sides by α q , can be written as which, by (1), gives Comparing the coefficients of α in (10), one finds which is valid for every non-negative integer n and for arbitrary integers p and q. Identity (11) contains many known identities as special cases. If one writes (9) as n j=0 n j α j+q √ 5F j p F n−j p−1 = α np+q √ 5 and applies (2), then the Lucas version of (11) is obtained; namely, Identities (11) and (12) are known in the literature; for example, see [19]. A more general identity that includes (11) and (12) as special cases is where (G k ) k∈Z is the Gibonacci sequence (a generalization of the Fibonacci sequence) whose initial terms G 0 and G 1 are given integers, not both zero, and Lemma 1.1. The following properties hold for the rational numbers a, b, c, and d: and Properties P3 to P6 follow from properties P1 and P2. Observe that P3 is a special case of P5 and P4 is a special case of P6. The next section aims to re-discover some known identities, using the above-mentioned method, and to discover some new results that may be easily deduced from the known ones. Presumably new results are developed in Section 3.

Preliminary results
In this section, the method described in the preceding section is utilized to re-discover some known identities and to discover some new identities that may be easily deduced from the known ones. In establishing some of such identities, one requires the fundamental relations F 2n = F n L n , L n = F n−1 + F n+1 , and 5F n = L n−1 + L n+1 .
By simplifying the right side of the last identity and then making use of (1), one gets By equating coefficients of α (property P1) from both sides of (14), one gets the well-known Fibonacci addition formula: A similar procedure using the identity produces the subtraction formula which may, of course, be obtained from (15) by changing q to −q. The Lucas counterpart of (15) is obtained by applying (1) and (2) to the identity α p+q √ 5 = (α p √ 5 )α q , and proceeding as in the Fibonacci case: Application of (1) to the right side and (2) to the left side of the identity

General Fibonacci and Lucas addition formulas and Catalan's identity
From (7), we can derive an addition formula that includes (15) as a particular case. Using (1) to write the left hand side (lhs) and the right hand side (rhs) of (7), we have lhs of (7) = αF s F k+t + F s−1 F k+t (24) and rhs of (7 Comparing the coefficients of α from (24) and (25), we find of which (15) is a particular case. Setting t = s − k in (26) produces Catalan's identity: Multiplying through (7) by √ 5 and performing similar calculations to above produces Identities (26) and (27)

Sums of Fibonacci and Lucas numbers with subscripts in arithmetic progression
and multiplying through by α q gives Thus, we have from which, with the use of (1) and property P5, we find valid for all integers p, q and n. The derivation here is considerably simpler than the one involving the direct use of Binet's formula; as done, for example, by Koshy [14, p. 104, Theorem 5.10] and Freitag [7] or by Siler [16] who first derived (30). Multiplying through (29) by from which, by identities (2), (1), and properties P5, P1, we find Identity (31) was first derived by Zeitlin [20] who established a generalization of Siler's result. Equations (30) and (31) can be summarized as which is a special case of (2.11) in [13] and (2) in [3].

Generating functions of Fibonacci and Lucas numbers with indices in arithmetic progression
Setting x = yα p in the identity ∞ j=0 x j = 1 1 − x and multiplying through by α q gives Application of (1) and properties P5 and P1 then produces To find the corresponding Lucas result, we write ∞ j=0 α pj+q √ 5y j = α q √ 5 1 − α p y and use (2) and properties P5 and P1, obtaining Identities (32) and (33) were first derived by Zeitlin [20]. Identity (32), but not (33)
Identity (34) appeared as problem H-4, proposed by Ruggles [15]. The identities (34) and (35) can be generalized as Similarly, identities (36) and (37) can be generalized as Theorem 3.2. The following identities hold for integers p, q, r, s and t: and Proof. Identity (38) is proved by applying (1) to the identity α p+q−r α t−s+r = α p−s α t+q , multiplying out the products and applying property P1. Identity (39) is derived by writing and applying identities (1) and (2) and property P1. Finally (40) is derived from
Identities related to or equivalent to some of the identities listed in Theorem 3.3 were also derived by Carlitz [5] and Griffiths [9].

Summation identities not involving binomial coefficients
. Let (X t ) and (Y t ) be any two sequences such that X t and Y t , t ∈ Z, are connected by a three-term recurrence relation where h, f 1 and f 2 are arbitrary non-vanishing complex functions, not dependent on t, and a and b are integers. Then, the following identity holds for integer n: Theorem 3.5. The following identities hold for integers n, k, s and t: Proof. Write (7) as Application of (1) to the resulting summation identity yields which, in view of the identity (see [18,Formula (20a) gives (76). Multiplying the α−sum by √ 5 and using (2) gives which with the use of the identity (see [18,Formula (19b) gives (77).
Lemma 3.4. The following identities hold for integers r and n and arbitrary x and y: x n j=0 y r−j (x + y) j = y r−n (x + y) n+1 − y r+1 and Proof. Identity (78) is obtained by replacing x with x/y in (28). Identity (79) is obtained by replacing x with x + y in (78). Finally, identity (80) is obtained by interchanging x and y in (78).