Fibonacci sums and divisibility properties

Based on a variant of Sury's polynomial identity we derive new expressions for various finite Fibonacci (Lucas) sums. We extend the results to Fibonacci and Chebyshev polynomials, and also to Horadam sequences. In addition to deriving sum relations, the main identities will be shown to be very useful in establishing and discovering divisibility properties of Fibonacci and Lucas numbers.


Introduction
As usual, we will use the notation F n for the nth Fibonacci number and L n for the nth Lucas number, respectively.Both number sequences are defined, for n ∈ Z, through the same recurrence relation x n = x n−1 + x n−2 , n ≥ 2, with initial values F 0 = 0, F 1 = 1, and L 0 = 2, L 1 = 1, respectively.For negative subscripts we have F −n = (−1) n−1 F n and L −n = (−1) n L n .They possess the explicit formulas (Binet forms) For more information about these famous sequences we refer, among others, to the books by Koshy [10] and Vajda [16].In addition, one can consult the On-Line Encyclopedia of Integer Sequences [14] where these sequences are listed under the ids A000045 and A000032, respectively.
In 2014, Sury [15] presented a polynomial identity in two variables u and v of the form ((2u) j + (2v) j )(u + v) n−j . (1) As it contains the relation as a special instance, Sury called (1) a polynomial parent to (2).It is obvious that identity (2) can be derived directly using the geometric series.A slightly more general result is or even where r is an integer and G n is a Gibonacci sequence, i.e., a sequence given by In this paper, we apply a variant of Sury's polynomial identity to derive new expressions for various finite Fibonacci (Lucas) sums.As will be seen immediately we can infer some divisibility properties for Fibonacci (Lucas) numbers from these sums.Some divisibility properties were also studied by Hoggatt and Bergum [8] and in the recent articles by Pongsriiam [13] and Onphaeng and Pongsriiam [12], among others.Extensions will be provided to Fibonacci (Lucas) polynomials, Chebyshev polynomials, and finally to Horadam sequences.

First Results
The next polynomial identity is a variant of Sury's identity and will be of crucial importance in this paper.Lemma 1.If x and y are any complex variables and n is any integer, then and In addition to deriving sum relations, identities ( 6) and ( 7) are going to be very useful in establishing and discovering divisibility properties of Fibonacci and Lucas numbers.
Proof.Set x = α r and y = β r in (6), and use (7) and the Binet formulas.
Theorem 1 offers a new simple proof of a well-known fact concerning the divisibility of Fibonacci numbers.
Proof.Write (6) as set x = α r and y = −β r and combine according to the Binet formulas; thereby proving (9).
To prove (10), write (6) as and set x = α r and y = −β r .Finish the proof in both cases using (7).
Corollary 4. If m is an odd integer, then L r divides L mr .Also, if m is an even integer, then L r divides F mr .
Corollary 8.If r is a non-zero integer and n is any positive integer, then In particular, 5 Remark.We also have that But this is obvious as and because F r |F r(2n+1) .
Theorem 9.If n and r are any integers, then Proof.Set x = L 2r and y = 2 in ( 6) and (7), respectively, and use r , r even.
Corollary 10.If r is a non-zero integer and n is any positive integer, then Theorem 11.If r and n are any integers, then Proof.Set x = 5F 2 r and y = (−1) r 4 in ( 6) and (7), respectively, and use the identity 5F 2 r + (−1) r 4 = L 2 r .
Theorem 12.If r, n and t are any integers, then Proof.Set x = αL r and y = L r−1 in (11), noting that Multiply through the resulting equation by α t .Use 2α s = L s + F s √ 5 to reduce the resulting equation.Finally, compare the coefficients of √ 5.
Corollary 13.If r, n and t are any integers, then In particular, Theorem 14.If r, k, s and n are any integers, then Proof.Set x = L 2k+r+s and y = (−1) k+s L r−s in ( 6) and ( 7), respectively, and use the identities [16] Corollary 15.If r, k, s are integers and n is any non-negative integer, then In particular, 3 Extension to Fibonacci polynomials Fibonacci (Lucas) polynomials are polynomials that can be defined by the Fibonacci-like recursion and generalizing Fibonacci (Lucas) numbers.They were already studied in 1883 by E. Catalan and E. Jacobsthal.For any integer n ≥ 0, the Fibonacci polynomials {F n (x)} n≥0 are defined by the second-order recurrence relation while the Lucas polynomials {L n (x)} n≥0 follow the rule Their Binet forms are given by Theorem 16.For any non-negative integer n we have Proof.Apply Lemma 1 inserting x = α(x) and y = β(x).
Corollary 17.For any non-negative integer n we have where are the Pell and Pell-Lucas numbers, respectively.
Proof.Insert x = 2 and use F n (2) = P n and L n (2) = Q n , respectively.
Remark.We mention that a different proof of Theorem 1 can be provided by inserting x = L r , r odd, and x = iL r , r even, in Theorem 16 and making use of Theorem 18.For any non-negative integer n and any x = 0 we have Proof.Apply Theorem 16 with x = i(x 2 + 1), i = √ −1, and use Theorem 19.For any non-negative integer n, any positive integer r, and any x = 0 we have Corollary 20.For any n ≥ 0 and m ≥ 1 we have and 4 Extension to Chebyshev polynomials Recall that, for any integer n ≥ 0, the Chebyshev polynomials {T n (x)} n≥0 of the first kind are defined by the second-order recurrence relation [11] T while the Chebyshev polynomials {U n (x)} n≥0 of the second kind are defined by The sequences T n (x) and U n (x) have the exact (Binet) formulas More information about these polynomials can be found the book by Mason and Handscomb [11] and also in the recent articles by Frontczak and Goy [6], Fan and Chu [5] and Adegoke et al. [3].
Although (37) offers a very appealing relation we have learnt that it is not new.It was proved by completely other methods in 1985 by Boscarol [4].

Extension to the Horadam sequence
Lemma 1 in the form readily allows sum relations to be derived for the Horadam sequence and divisibility properties to be established.Let {w n (a, b; p, q)} n≥0 be the Horadam sequence [9] defined for all non-negative integers n by the recurrence where a, b, p and q are arbitrary complex numbers, with p = 0 and q = 0. Extension of the definition of w n (a, b; p, q) to negative subscripts is provided by writing the recurrence relation as w where, for brevity, we wrote (and will write) w n for w n (a, b; p, q).
Two important cases of w n are the Lucas sequences of the first kind, u n (p, q) = w n (0, 1; p, q), and of the second kind, v n (p, q) = w n (2, p; p, q).The most well-known Lucas sequences are the Fibonacci sequence F n = u n (1, −1) and the sequence of Lucas numbers L n = v n (1, −1).
The Binet formulas for sequences u n , v n and w n in the non-degenerate case, p 2 − 4q > 0, are are the distinct zeros of the characteristic polynomial x 2 − px + q of the Horadam sequence.
In this section, we will make use of the following known results.Lemma 3.For any integer s, q s + τ 2s = τ s v s , q s − τ 2s = −∆τ s u s , (41) q s + σ 2s = σ s v s , q s − σ 2s = ∆σ s u s .

Theorem 1 .
If r and n are any integers, then n j=0

Lemma 2 .
If a, b, c and d are rational numbers and λ is an irrational number, then a + b λ = c + d λ ⇐⇒ a = c, b = d.