Abstract
The aim of the paper is to present the binomial transformation formulae of Fibonacci numbers scaled by complex multipliers. Many of these new and nontrivial relations follow from the fundamental properties of the so-called delta-Fibonacci numbers defined by Wituła and Słota. The paper contains some original relations connecting the values of delta-Fibonacci numbers with the respective values of Chebyshev polynomials of the first and second kind.
1 lntroduction
Wituła and Słota in [1] defined the δ-Fibonacci numbers an(δ), bn(δ), n ∈ ℕ0 := ℕ ∪ {0} in the following way
or equivalently by the relation
From this, one can easily conclude that
and next, the following recurrence relations hold true
for every n ∈ ℕ0. We note that a1(δ) = 1 and b1(δ) = δ.
We note also that δ-Fibonacci numbers are the simplest members of the family of the quasi-Fibonacci numbers of (k-th, δ-as) order (see [1–3] and the references therein). So the following natural question arises: does there exist some algebraic relation connecting an(δ) and bn(δ), n ∈ ℕ0, δ ∈ ℂ, with the Fibonacci numbers? The answer is positive. There exist two basic relations of such type resulting from two following fundamental properties of an(δ) and bn(δ) (see formulae (1. 1), (3. 14), (3. 15), (5.6), (5.7) in [1]):
and
for every n ∈ ℕ0. We note that one can easily verify all relations (6)-(8) by induction (on the basis of recurrence relations (4) and (5)).
Moreover, it follows from (7) and (8) that the sequences
and
We note also that, after [4], from (7) and (8) one can derive the generating functions A(x; δ) and B(x; δ) of
where
Reflections
The δ-Fibonacci numbers [1] and the δ-Lucas numbers [5] represent the simplest ”quasi-Fibonacci” numbers of any order defined by R. Wituła and D. Słota (see [1, 2, 5, 6] and the references therein), which in fact are the recursively defined polynomials of the respective order. The above numbers have been introduced so that their definitions indicate an easy way for generating many standard identities for these numbers (including the sums of powers, the sums of the respective scalar products). Relations defining the quasi-Fibonacci numbers constitute the ”platform” for discussing the recursively defined sequences, alternative for Binet’s formulae or generating functions — very effective platform, according to us. It is worth to emphasize that all the quasi-Fibonacci numbers seem to exist independently of the background of the recurrence sequences discussed in literature.
Since the δ-Fibonacci numbers and the δ-Lucas numbers are in fact the binomial transformations of the scaled sequences of Fibonacci and Lucas numbers, thus in some moment we realized that it would be welcome, for emphasizing the meaning of these numbers, to have at our disposal the general formulae for δ-Fibonacci numbers and δ-Lucas numbers for “the most generally expressed” parameters δ from the set of complex numbers. So, these are the roots of this work.
2 Main results
First we will show relations between the special cases of δ-Fibonacci numbers for complex values of δ and the Fibonacci and Lucas numbers.
Theorem 2.1
Let
where Tn and Un are the n-th Chebyshev polynomials of the first and second kind, respectively.
Indeed, we get
where we set (and these notations hold in the entire paper):
(α denotes the golden ratio) and where we used the Binet’s formulae for the Fibonacci and Lucas numbers, respectively
Similarly we obtain the following theorem.
Theorem 2.2
Let
where again Tn and Un denote the Chebyshev polynomials.
Indeed, one can deduce that
Now we focus on the special cases of φ. One may note that if
Thus, as a consequence we obtain
for
where
Furthermore, we have (see [7, 8]):
since
Similar formulae hold for the values of Chebyshev polynomials of the second kind
because of identity
Moreover, we observe that all four numbers
belong to the number class
for
where
for
where
Now, from (4) and (5) one may also obtain the result given below (in fact, one of the authors deduced these formulae by “observing” the numerical values of an(−i) and bn(−i) for n = 0, 1, …, 20, and then he verified them easily by induction).
Lemma 2.3
The following recurrent identities hold
for every n ∈ ℕ0.
Corollary 2.4
For the Fibonacci numbers the following identities hold
for every n = 0, 1, ….
3 Some general recurrence relations
The above formulae can be used to generate the interesting general relations for Fibonacci numbers, but to this aim we need some technical result which holds true for both the Fibonacci and Lucas numbers (and even for the Gibonacci numbers).
Lemma 3.1
Let αk,n, βk,n ∈ ℂ, k = 0, 1, …, b n + c, b, c, n ∈ ℕ0, a, r0 ∈ ℤ. Let X, y, Z ∈ {F, L} (in fact, one can assume that X, y, Z are the Gibonacci numbers - see the respective definition at the end of this section). If there exists an r0 ∈ ℤ such that the following equalities
hold for r = r0, r0+1 and for every n ∈ ℕ0, then these equalities hold for all r ∈ ℤ and n ∈ ℕ0.
Proof
We discuss here only the case r0 = 0.
We will prove by induction that for every m ∈ ℕ and r ∈ ℤ, −m+1≤r≤m, equality (16) is an identity with respect to n ∈ ℕ0. For m = 1 the above thesis follows from the assumption in Lemma. Thus, let us suppose that the statement is true for some m ∈ ℕ and for all r ∈ ℤ, −m+1 ≤ r ≤m. Then we find
Hence, in view of the inductive assumption, relation (16) holds for every n ∈ ℕ0 and r ∈ ℤ, −m ≤ r ≤ m+1. Therefore, by virtue of the mathematical induction rule, the Lemma is true. □
From the above Lemma and Corollary 2.4 we get the next result.
Corollary 3.2
The following identities hold
for every n ∈ ℕ0 and r ∈ ℤ.
We note also that formulae (7) and (8) are not the only technical tools for generating the binomials transform formulae of scaled Fibonacci numbers. One can also apply the following formulae (see (5.1) and (5.2) in [1]).
Lemma 3.3
Let δ, γ ∈ ℂ and γ ≠ 0. Then
As examples of the previous lemma we may also deduce the additional relations between Fibonacci numbers. Let us note
So, if we set δ = −1, γ = − i, then by using (19), (10) and (11) we obtain
which implies, by Lemma 3.1, that
for every n ∈ ℕ0 and r ∈ ℤ. Similarly, if we set δ = 2, γ = 2i, then we can generate the identities (we note that especially for the left hand side the subtle calculations are needed-more precisely, one should use the auxiliary fact given in the footnote[1] ):
for every n ∈ ℕ0 and r ∈ ℤ since an(2) = 5⌊n/2⌋ and bn(2) = (1 − (−1)n)5⌊n/2⌋ (see [1]).
Finally, let us note that most of the presented results can be reformulated for the general Gibonacci numbers, i.e., the recurrent sequences
(only the recurrence relation is important, the initial conditions are not). It follows easily from the result given below and being the counterpart of Lemma 3.1 for more general case of Gibonacci numbers.
Theorem 3.4
Let αk,n ∈ ℂ, k = 0, 1, …, b n + c, b, c, n ∈ ℕ0, a ∈ ℤ. If the following identity holds with respect to n ∈ ℕ0 and r ∈ ℤ
then for every Gibonacci sequence
is satisfied for n ∈ ℕ0 and r ∈ ℤ.
Proof
Since Gn = G0 Fn − 1 + G1 Fn, n ∈ ℤ, thus we obtain
□
Acknowledgement
The Authors want to express sincere thanks to the Reviewer for many essential remarks and advices. We had a fortune that our paper was investigated by a careful and understanding Reviewer who not only showed us the failings but also influenced the final form of this paper. We are very grateful.
References
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