Binomials transformation formulae for scaled Fibonacci numbers

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.

We note also that, after [4], from (7) and (8) one can derive the generating functions A.xI ı/ and B.xI ı/ of fa n .ı/g 1 nD0 and fb n .ı/g 1 nD0 , respectively. For example, we obtain where F .x/ D x 1 x x 2 is the generating function of the Fibonacci numbers. However, with regard to the length of our paper, we will not make use of the functions A.xI ı/ and B.xI ı/ as the alternative sources for deriving the presented here formulae.

Reflections
The ı-Fibonacci numbers [1] and the ı-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 ı-Fibonacci numbers and the ı-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 ı-Fibonacci numbers and ı-Lucas numbers for "the most generally expressed" parameters ı from the set of complex numbers. So, these are the roots of this work.

Main results
First we will show relations between the special cases of ı-Fibonacci numbers for complex values of ı and the Fibonacci and Lucas numbers.
where T n and U n are the n-th Chebyshev polynomials of the first and second kind, respectively.
Indeed, we get h p 5F n cos n' iL n sin n' i ; for even n 2 N 0 ; h L n cos n' i p 5F n sin n' i ; for odd n 2 N; where we set (and these notations hold in the entire paper): (˛denotes the golden ratio) and where we used the Binet's formulae for the Fibonacci and Lucas numbers, respectively F n D˛n ˇn p 5 ; L n D˛n Cˇn; n D 0; 1; 2; ::: Similarly we obtain the following theorem.
where again T n and U n denote the Chebyshev polynomials.
Indeed, one can deduce that h p 5F n 1 cos .n'/ C iL n 1 sin .n'/ i ; for even n 2 N: Now we focus on the special cases of '. One may note that if ' WD arctan Thus, as a consequence we obtain -for ' 0 D 3 : . Furthermore, we have (see [7,8]): Similar formulae hold for the values of Chebyshev polynomials of the second kind because of identity  where ' WD arctan 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 a n . i / and b n . i / for n D 0; 1; :::; 20; and then he verified them easily by induction).

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˛k ;n ;ˇk ;n 2 C; k D 0; 1; :::; b n C c, b; c; n 2 N 0 ; a; r 0 2 Z: Let X; Y; Z 2 fF; Lg (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 r 0 2 Z such that the following equalities hold for r D r 0 ; r 0 C 1 and for every n 2 N 0 , then these equalities hold for all r 2 Z and n 2 N 0 .
Proof. We discuss here only the case r 0 D 0. We will prove by induction that for every m 2 N and r 2 Z, m C 1 Ä r Ä m, equality (16) is an identity with respect to n 2 N 0 . For m D 1 the above thesis follows from the assumption in Lemma. Thus, let us suppose that the statement is true for some m 2 N and for all r 2 Z, m C 1 Ä r Ä m. Then we find X a n m D X a n mC2 X a n mC1 D b nCc Hence, in view of the inductive assumption, relation (16) holds for every n 2 N 0 and r 2 Z; m Ä r Ä m C 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.
for every n 2 N 0 and r 2 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 formulae (5.1) and (5.2) in [1]).
As examples of the previous lemma we may also deduce the additional relations between Fibonacci numbers. Let us note a n . 1/ D F 2n 1 ; b n . 1/ D F 2n : So, if we set ı D 1; D i; then by using (19), (10) and (11) we obtain . . .
which implies, by Lemma 3.1, that .
for every n 2 N 0 and r 2 Z. Similarly, if we set ı D 2; D 2 i; 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 ): .1 2 i / n .F r 1 F n C F r F n 2 / D 2 2n 2n X kD0 2n k ! .2 i 1/ 2n k .F rC2 F rC1 . 1/ k / 5 bk=2c ; (23) .1 2 i / n .F r 1 F nC1 C F r F n 1 i.F r 1 F n C F r F n 2 // D 2 2n 1 2nC1 X kD0 2n C 1 k ! .2 i 1/ 2nC1 k .F rC2 F rC1 . 1/ k / 5 bk=2c ; (24) for every n 2 N 0 and r 2 Z since a n .2/ D 5 bn=2c and b n .2/ D .1 . 1/ n /5 bn=2c (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 fG n g 1 nD 1 satisfying the recurrent relation G nC2 D G nC1 C G n ; n 2 Z (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. ;n G kCr is satisfied for n 2 N 0 and r 2 Z.
Proof. Since G n D G 0 F n 1 C G 1 F n ; n 2 Z, thus we obtain b nCc X kD0˛k ;n G kCr D G 0 b nCc X kD0˛k ;n F kCr 1 C G 1 b nCc X kD0˛k ;n F kCr D G 0 F a nCr 1 C G 1 F a nCr D G a nCr :