Generalized Fibonacci Polynomials

• Generalized Fibonacci Polynomials are polynomial sequences defined by linear recurrences that extend Fibonacci and Lucas numbers through variable coefficients, higher order recurrences, and multivariate frameworks.
• They are analyzed via explicit Binet-type formulas, generating functions, and determinant identities to reveal structural identities, divisibility, and irreducibility properties.
• Applications of GFP span coding theory, combinatorial tiling, finite-field periodicity, and cryptography, demonstrating both theoretical depth and practical relevance.

Searching arXiv for recent and foundational papers on generalized Fibonacci polynomials to ground the article in the literature. Generalized Fibonacci polynomials (GFP) are polynomial sequences defined by linear recurrences that extend the classical Fibonacci and Lucas numbers to polynomial settings, variable coefficients, higher order recurrences, and multivariate frameworks. In the literature summarized here, the term encompasses several closely related constructions: second-order polynomial sequences of Fibonacci type and Lucas type, two-parameter polynomial families $\{n\}_{s,t}$, generalized bivariate and $k$-generalized recurrences, and multivariate $r$-Fibonacci polynomials. Across these settings, GFP theory studies explicit Binet-type formulas, generating functions, divisibility and irreducibility, determinant and matrix realizations, geometric and combinatorial models, and applications ranging from coding theory to finite-field periodicity, orthogonality, and Markov processes (Flórez et al., 2017, Amdeberhan et al., 2013, Park et al., 2023).

1. Foundational definitions and principal classes

A central second-order definition treats a generalized Fibonacci polynomial sequence $\{G_n(x)\}$ as a recurrence

$$G_0(x)=p_0(x),\quad G_1(x)=p_1(x),\quad G_n(x)=d(x)G_{n-1}(x)+g(x)G_{n-2}(x)\quad \text{for } n\ge 2,$$

with $p_0(x)$ a constant, $p_1(x),d(x),g(x)\in \mathbb{Z}[x]$, and $\gcd(d(x),g(x))=1$ (Flórez et al., 2017). Within this framework, the literature separates GFP into two canonical types. A sequence is of Fibonacci type if $p_0(x)=0$ and $p_1(x)$ is a nonzero constant; it is of Lucas type if $k$0 with $k$1 (Flórez et al., 2017). The same classification is used in work on identities, strong divisibility, irreducibility, and orthogonality (Flórez et al., 2017, Florez et al., 2022, Coletti et al., 30 Sep 2025).

A frequently used special case is the recurrence

$k$2

where $k$3 and $k$4 are positive integers such that $k$5 (Tsuno, 2021). Its generating function is

$k$6

The associated Lucas-type sequence satisfies $k$7, $k$8, with generating function

$k$9

(Tsuno, 2021).

Another standard parameterization uses real or integer parameters $r$0, defining

$r$1

with corresponding generalized Lucas polynomials $r$2 defined by the same recurrence and initial conditions $r$3, $r$4 (Amdeberhan et al., 2013). This family specializes to classical Fibonacci numbers when $r$5, to ordinary integers when $r$6, and to $r$7-integers when $r$8 (Amdeberhan et al., 2013).

Broader generalizations are also standard. The generalized Fibonacci polynomials $r$9 are defined by

$\{G_n(x)\}$0

where $\{G_n(x)\}$1 are real-coefficient polynomials with $\{G_n(x)\}$2, $\{G_n(x)\}$3, and $\{G_n(x)\}$4 (Uçar et al., 2017). At higher order, the $\{G_n(x)\}$5-Fibonacci polynomial in variables $\{G_n(x)\}$6 is defined by

$\{G_n(x)\}$7

and $\{G_n(x)\}$8 recovers the classical Fibonacci polynomials (Park et al., 2023).

2. Closed forms, generating functions, and structural identities

The second-order GFP framework is governed by the characteristic equation

$\{G_n(x)\}$9

If $$G_0(x)=p_0(x),\quad G_1(x)=p_1(x),\quad G_n(x)=d(x)G_{n-1}(x)+g(x)G_{n-2}(x)\quad \text{for } n\ge 2,$$0 and $$G_0(x)=p_0(x),\quad G_1(x)=p_1(x),\quad G_n(x)=d(x)G_{n-1}(x)+g(x)G_{n-2}(x)\quad \text{for } n\ge 2,$$1 are its roots, then the Binet-like formulas are

$$G_0(x)=p_0(x),\quad G_1(x)=p_1(x),\quad G_n(x)=d(x)G_{n-1}(x)+g(x)G_{n-2}(x)\quad \text{for } n\ge 2,$$2

for Fibonacci type and

$$G_0(x)=p_0(x),\quad G_1(x)=p_1(x),\quad G_n(x)=d(x)G_{n-1}(x)+g(x)G_{n-2}(x)\quad \text{for } n\ge 2,$$3

for Lucas type, where $$G_0(x)=p_0(x),\quad G_1(x)=p_1(x),\quad G_n(x)=d(x)G_{n-1}(x)+g(x)G_{n-2}(x)\quad \text{for } n\ge 2,$$4 in the Lucas-type normalization (Flórez et al., 2017, Flórez et al., 2018). In the $$G_0(x)=p_0(x),\quad G_1(x)=p_1(x),\quad G_n(x)=d(x)G_{n-1}(x)+g(x)G_{n-2}(x)\quad \text{for } n\ge 2,$$5 notation, if $$G_0(x)=p_0(x),\quad G_1(x)=p_1(x),\quad G_n(x)=d(x)G_{n-1}(x)+g(x)G_{n-2}(x)\quad \text{for } n\ge 2,$$6 are the roots of $$G_0(x)=p_0(x),\quad G_1(x)=p_1(x),\quad G_n(x)=d(x)G_{n-1}(x)+g(x)G_{n-2}(x)\quad \text{for } n\ge 2,$$7, then

$$G_0(x)=p_0(x),\quad G_1(x)=p_1(x),\quad G_n(x)=d(x)G_{n-1}(x)+g(x)G_{n-2}(x)\quad \text{for } n\ge 2,$$8

(Uçar et al., 2017).

Generating functions are equally basic. For the two-parameter family $$G_0(x)=p_0(x),\quad G_1(x)=p_1(x),\quad G_n(x)=d(x)G_{n-1}(x)+g(x)G_{n-2}(x)\quad \text{for } n\ge 2,$$9,

$p_0(x)$0

and for $p_0(x)$1 one obtains

$p_0(x)$2

(Amdeberhan et al., 2013, Uçar et al., 2017). For $p_0(x)$3-Fibonacci polynomials,

$p_0(x)$4

for $p_0(x)$5 (Park et al., 2023).

The literature extends classical identities systematically. For Fibonacci-type GFP,

$p_0(x)$6

giving the generalized Cassini identity (Flórez et al., 2017). In the $p_0(x)$7-matrix formalism, for recurrence $p_0(x)$8, one has

$p_0(x)$9

and therefore

$p_1(x),d(x),g(x)\in \mathbb{Z}[x]$0

for Fibonacci-type initial value $p_1(x),d(x),g(x)\in \mathbb{Z}[x]$1 (Chen et al., 2020). The same formalism yields the Honsberger addition formula

$p_1(x),d(x),g(x)\in \mathbb{Z}[x]$2

and the generalized d’Ocagne identity

$p_1(x),d(x),g(x)\in \mathbb{Z}[x]$3

(Chen et al., 2020).

For generalized Fibonacci numbers in the broad recurrence $p_1(x),d(x),g(x)\in \mathbb{Z}[x]$4, the generalized Catalan identity states that for two second-order recurrences $p_1(x),d(x),g(x)\in \mathbb{Z}[x]$5 and $p_1(x),d(x),g(x)\in \mathbb{Z}[x]$6,

$p_1(x),d(x),g(x)\in \mathbb{Z}[x]$7

where $p_1(x),d(x),g(x)\in \mathbb{Z}[x]$8 is the solution of the associated homogeneous recurrence (Tangboonduangjit et al., 2015). This identity underlies determinant formulas and many GFP specializations.

3. Divisibility, gcd structure, irreducibility, and integer-valued substitutions

A defining arithmetic question is when GFP retain Fibonacci-like divisibility. The strong divisibility property is

$p_1(x),d(x),g(x)\in \mathbb{Z}[x]$9

The complete characterization states that a GFP satisfies the strong divisibility property if and only if it is of Fibonacci type (Flórez et al., 2017). For Lucas-type polynomials, the gcd depends on the 2-adic valuation $\gcd(d(x),g(x))=1$0: if $\gcd(d(x),g(x))=1$1, then

$\gcd(d(x),g(x))=1$2

whereas otherwise the gcd is governed by a reduction algorithm and is often $\gcd(d(x),g(x))=1$3 in classical cases (Flórez et al., 2017).

The two-parameter family $\gcd(d(x),g(x))=1$4 also exhibits a Fibonacci-style gcd law: $\gcd(d(x),g(x))=1$5 (Amdeberhan et al., 2013). This family supports Fibonomial coefficients

$\gcd(d(x),g(x))=1$6

which specialize to ordinary binomial coefficients when $\gcd(d(x),g(x))=1$7, to classical Fibonomials when $\gcd(d(x),g(x))=1$8, and to $\gcd(d(x),g(x))=1$9-binomial coefficients when $p_0(x)=0$0 (Amdeberhan et al., 2013).

Irreducibility theory is highly developed for second-order GFP. For Fibonacci-type sequences, under certain conditions the polynomials are irreducible if and only if $p_0(x)=0$1 is a prime number (Florez et al., 2022). The paper records that the Fibonacci polynomials, Pell polynomials, Fermat polynomials, Lucas polynomials, Pell-Lucas polynomials, and Fermat-Lucas polynomials are irreducible when $p_0(x)=0$2 is a prime number, whereas Chebyshev polynomials of the second kind, Morgan-Voyce polynomials of Fibonacci type, and Vieta polynomials are reducible when $p_0(x)=0$3 is a prime number (Florez et al., 2022). For multivariate $p_0(x)=0$4-Fibonacci polynomials, the irreducibility statement is stronger: for $p_0(x)=0$5, $p_0(x)=0$6 is irreducible over $p_0(x)=0$7 (Park et al., 2023).

Resultants and discriminants encode further algebraic structure. For Fibonacci-type GFP with notation from the source, one has

$p_0(x)=0$8

and, in the case $p_0(x)=0$9, $p_1(x)$0 constant, and $p_1(x)$1 constant,

$p_1(x)$2

For Lucas-type polynomials,

$p_1(x)$3

(Flórez et al., 2018). The same paper gives derivative formulas, including

$p_1(x)$4

when $p_1(x)$5 is constant (Flórez et al., 2018).

A distinct arithmetic classification concerns integer values of generating functions after rational substitution. For

$p_1(x)$6

if $p_1(x)$7 and $p_1(x)$8, then

$p_1(x)$9

if and only if

$k$00

A crucial step in the proof is the polynomial Pell-type equation

$k$01

(Tsuno, 2021).

4. Determinants, matrices, and convolution constructions

Matrix realizations are a major organizing principle in GFP theory. For generalized bivariate Fibonacci $k$02-polynomials $k$03, lower Hessenberg matrices give exact determinant and permanent formulas. If $k$04 is the $k$05 Hessenberg matrix with diagonal $k$06, superdiagonal $k$07, and $k$08 on the $k$09-th subdiagonal, then

$k$10

An alternative Hessenberg matrix $k$11 with superdiagonal $k$12 gives the same determinant, and analogous matrices $k$13 and $k$14 satisfy

$k$15

(Kaygisiz et al., 2011).

For generalized order-$k$16 Fibonacci polynomials $k$17, determinant and permanent formulas again use structured Hessenberg matrices. The paper gives matrices $k$18, $k$19, $k$20, and $k$21 such that

$k$22

(Sahin et al., 2011). These constructions generalize determinantal and permanental representations of generalized order-$k$23 Fibonacci and Pell numbers.

Determinantal identities also extend to matrices whose entries are powers or products of generalized Fibonacci numbers. For the matrix

$k$24

Theorem 5 in the source gives a closed formula for $k$25 in terms of $k$26, $k$27, binomial coefficients, and companion sequence terms $k$28 (Tangboonduangjit et al., 2015). The proofs use determinant calculus, the generalized Catalan identity, and a factorization lemma of the form

$k$29

(Tangboonduangjit et al., 2015).

A different matrix-theoretic direction concerns right circulant matrices. For $k$30, the determinant is given explicitly in Theorem 3.1 of the source, and the paper derives eigenvalues and determinants for right circulant matrices with entries $k$31 (Uçar et al., 2017). These formulas are then used to construct coding and decoding algorithms based on invertible right circulant matrices (Uçar et al., 2017).

The convolution viewpoint links GFP to symmetric function theory and arithmetic functions. Under convolution product, the rational $k$32-th roots $k$33 of GFP admit determinant and permanent representations by lower Hessenberg/Stirling matrices. The general expansion is

$k$34

where the Stirling operators of the first kind are $k$35 (Conci et al., 2014). The paper states that this yields matrix representations of multiplicative arithmetic functions under the Dirichlet product into its divisible closure (Conci et al., 2014).

5. Higher-order, multivariate, finite-field, and negative-index extensions

The GFP literature extends far beyond second-order univariate recurrences. The $k$36-Fibonacci polynomials furnish a multivariate theory with explicit Binet-type formula, Cassini-like determinant identity, irreducibility for $k$37, and an explicit partition sum

$k$38

This formula connects the polynomials directly to integer partitions (Park et al., 2023). The same work identifies

$k$39

so the $k$40-Fibonacci polynomials coincide with complete ordinary Bell polynomials in the appropriate truncation, and it expresses Fubini numbers via a specialization involving $k$41 (Park et al., 2023).

For $k$42-generalized Fibonacci polynomials,

$k$43

the extension to negative indices reveals a phenomenon absent from the classical $k$44 case: there are exactly $k$45 indices at which the polynomials vanish identically (Mane, 15 Jul 2025). The vanishing indices occur in blocks, with the $k$46-th block at

$k$47

for $k$48 (Mane, 15 Jul 2025). The positive-index generating function is

$k$49

and the negative-index theory requires a separate generating function (Mane, 15 Jul 2025).

Over finite fields, one studies the generalized Fibonacci sequence of polynomials

$k$50

modulo an arbitrary polynomial $k$51, with $k$52 (Chen et al., 2023). The sequence modulo $k$53 is periodic. The rank of apparition $k$54 is the least positive integer $k$55 such that $k$56, and the period $k$57 is the least positive integer $k$58 such that

$k$59

If $k$60, then

$k$61

(Chen et al., 2023). The same paper emphasizes that the polynomial case is much more complicated than the integer case, and that the quotient $k$62 need not stabilize for prime powers.

A separate higher-order perspective appears in generalized Fibonacci polynomial sequences of the first and second kind. For order $k$63, the first-kind recurrence is

$k$64

whereas the second-kind recurrence is

$k$65

(Laugier et al., 2012). Their generating functions are rational; for example,

$k$66

(Laugier et al., 2012).

6. Geometric, combinatorial, analytic, and applied directions

GFP have a rich combinatorial and geometric ecology. In the Hosoya-like polynomial triangle, one forms entries

$k$67

where $k$68 is a GFP sequence (Florez et al., 2017). The triangle satisfies double recurrences

$k$69

(Florez et al., 2017). Its main structural theorem is the polynomial Star of David property: if $k$70 and $k$71 are the alternating vertices of the corresponding hexagon, then

$k$72

and under broad conditions

$k$73

(Florez et al., 2017). The same framework gives geometric interpretations of Cassini’s and Catalan’s identities and extends to the gibonomial triangle (Florez et al., 2017).

The two-parameter family $k$74 has a tiling interpretation: $k$75 is the generating function for linear tilings of $k$76 squares using monominos of weight $k$77 and dominos of weight $k$78 (Amdeberhan et al., 2013). Its explicit monomial expansion is

$k$79

(Amdeberhan et al., 2013). The same source proves a generalized recurrence

$k$80

and gives a binomial-theorem analogue

$k$81

(Amdeberhan et al., 2013).

Analytic and figurate-number extensions also appear. Simplicial $k$82-polytopic numbers defined on GFP are introduced by

$k$83

with GFP factorial $k$84 (López, 20 Jan 2025). The paper studies generating functions, $k$85-identities, reciprocal sums, and introduces the generalized Fibonacci Zeta function

$k$86

(López, 20 Jan 2025).

Orthogonality introduces a different classification. In second-order GFP with Binet formula similar to Fibonacci and Lucas numbers, the paper on zeros and orthogonality states that familiar orthogonal polynomials include the Fermat, Fermat-Lucas, both types of Chebyshev polynomials, both types of Morgan-Voyce polynomials, and Vieta and Vieta-Lucas polynomials, whereas the Fibonacci, Lucas, Pell, and Pell-Lucas sequences are not orthogonal (Coletti et al., 30 Sep 2025). The same work gives a root-finding technique: for Fibonacci-type GFP, if $k$87 and

$k$88

then $k$89 is a root of $k$90 (Coletti et al., 30 Sep 2025). It also identifies sufficient conditions under which an orthogonal GFP family induces a birth-and-death Markov chain and highlights Chebyshev polynomials of the first kind and Fermat-Lucas as examples (Coletti et al., 30 Sep 2025).

Applications include block coding and cryptography. Right circulant matrices with entries from generalized Fibonacci and Lucas polynomials are used to formulate coding and decoding algorithms; the paper emphasizes message blocking, multiplication by circulant encoding matrices, and decoding via inverses whose existence is controlled by determinant formulas (Uçar et al., 2017). Another application concerns polynomial inputs such as $k$91: for square-free natural number $k$92, the recurrence

$k$93

has generating function

$k$94

and the paper verifies that the integer-valued substitution classification is of the same form as the polynomial case for $k$95, but not generally for higher square-free $k$96 (Tsuno, 2021).

Taken together, these developments show that GFP are not a single sequence family but a broad research program on polynomial recurrences. The unifying core is the persistence of Fibonacci–Lucas phenomena—Binet formulas, rational generating functions, determinant models, divisibility patterns, and combinatorial expansions—under systematic generalization to polynomial coefficients, multivariate recurrences, finite fields, negative indices, and application-specific algebraic structures (Flórez et al., 2017, Flórez et al., 2018, Chen et al., 2023).