Wituła and Słota in [1] defined the *δ*-Fibonacci numbers *a*_{n}(*δ*), *b*_{n}(*δ*), *n* ∈ ℕ_{0} := ℕ ∪ {0} in the following way
$$(1-{\displaystyle \frac{1+\sqrt{5}}{2}\delta {)}^{n}={a}_{n}(\delta )-\frac{1+\sqrt{5}}{2}{b}_{n}(\delta ),\phantom{\rule{1em}{0ex}}\delta \in \mathbb{C},\phantom{\rule{1em}{0ex}}n\in {\mathbb{N}}_{0},}$$(1)

or equivalently by the relation
$$(1+{\displaystyle \frac{\sqrt{5}-1}{2}\delta {)}^{n}={a}_{n}(\delta )+\frac{\sqrt{5}-1}{2}{b}_{n}(\delta ),\phantom{\rule{1em}{0ex}}\delta \in \mathbb{C},\phantom{\rule{1em}{0ex}}n\in {\mathbb{N}}_{0}.}$$(2)

From this, one can easily conclude that
$${a}_{0}(\delta )=1,\phantom{\rule{1em}{0ex}}{b}_{0}(\delta )=0,$$
$$\begin{array}{}\left[\begin{array}{l}{a}_{n+1}(\delta )\\ {b}_{n+1}(\delta )\end{array}\right]=\left[\begin{array}{l}1\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\delta \\ \delta \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}1-\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\delta \end{array}\right]\left[\begin{array}{l}{a}_{n}(\delta )\\ {b}_{n}(\delta )\end{array}\right],\phantom{\rule{1em}{0ex}}n\in {\mathbb{N}}_{0}\end{array}$$(3)

and next, the following recurrence relations hold true
$${a}_{n+2}(\delta )=(2-\delta ){a}_{n+1}(\delta )+({\delta}^{2}+\delta -1){a}_{n}(\delta ),$$(4)
$${b}_{n+2}(\delta )=(2-\delta ){b}_{n+1}(\delta )+({\delta}^{2}+\delta -1){b}_{n}(\delta ),$$(5)

for every *n* ∈ ℕ_{0}. We note that *a*_{1}(*δ*) = 1 and *b*_{1}(*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 *a*_{n}(*δ*) and *b*_{n}(*δ*), *n* ∈ ℕ_{0}, *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 *a*_{n}(*δ*) and *b*_{n}(*δ*) (see formulae (1. 1), (3. 14), (3. 15), (5.6), (5.7) in [1]):
$${a}_{n}(1)={F}_{n+1},\phantom{\rule{1em}{0ex}}{b}_{n}(1)={F}_{n},$$(6)

and
$${a}_{n}({\displaystyle \delta )=\sum _{k=0}^{n}\left(\begin{array}{l}n\\ k\end{array}\right){F}_{k-1}(-\delta {)}^{k}=\sum _{k=0}^{n}\left(\begin{array}{l}n\\ k\end{array}\right){F}_{k+1}(1-\delta {)}^{n-k}{\delta}^{k},}$$(7)
$${b}_{n}({\displaystyle \delta )=\sum _{k=1}^{n}\left(\begin{array}{l}n\\ k\end{array}\right)(-1{)}^{k-1}{F}_{k}{\delta}^{k}=\sum _{k=1}^{n}\left(\begin{array}{l}n\\ k\end{array}\right){F}_{k}(1-\delta {)}^{n-k}{\delta}^{k},}$$(8)

for every *n* ∈ ℕ_{0}. 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
$\{{a}_{n}(\delta ){\}}_{n=0}^{\mathrm{\infty}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\{{b}_{n}(\delta ){\}}_{n=0}^{\mathrm{\infty}}$ are the binomial transformation (denoted for shortness by *Binom*(⋅)) of sequences
$\{{F}_{k-1}(-\delta {)}^{k}{\}}_{k=0}^{\mathrm{\infty}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\{-{F}_{k}(-\delta {)}^{k}{\}}_{k=0}^{\mathrm{\infty}},$ respectively. Similarly, we have
$$\{\frac{{a}_{n}(\delta )}{(1-\delta {)}^{n}}{\}}_{n=0}^{\mathrm{\infty}}=Bi\mathit{n}\mathit{o}\mathit{m}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}(\{{F}_{k+1}(\frac{\delta}{1-\delta}{)}^{k}{\}}_{k=0}^{\mathrm{\infty}})$$

and
$$\{\frac{{b}_{n}(\delta )}{(1-\delta {)}^{n}}{\}}_{n=0}^{\mathrm{\infty}}=Bi\mathit{n}\mathit{o}\mathit{m}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}(\{{F}_{k}(\frac{\delta}{1-\delta}{)}^{k}{\}}_{k=0}^{\mathrm{\infty}}).$$

We note also that, after [4], from (7) and (8) one can derive the generating functions *A*(*x*; *δ*) and *B*(*x*; *δ*) of
$\{{a}_{n}(\delta ){\}}_{n=0}^{\mathrm{\infty}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\{{b}_{n}(\delta ){\}}_{n=0}^{\mathrm{\infty}}$, respectively. For example, we obtain
$$B(x;\delta )=\frac{1}{1-(1-\delta )x}F(\frac{\delta x}{1-(1-\delta )x})=\frac{\delta x}{(1-\delta -{\delta}^{2}){x}^{2}+(\delta -2)x+1},$$

where
$F(x)={\displaystyle \frac{x}{1-x-{x}^{2}}}$ 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*(*x*; *δ*) and *B*(*x*; *δ*) as the alternative sources for deriving the presented here formulae.

## 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.

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