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Article

On Convolved Fibonacci Polynomials

by
Waleed Mohamed Abd-Elhameed
1,
Omar Mazen Alqubori
2 and
Anna Napoli
3,*
1
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
2
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia
3
Department of Mathematics and Computer Science, University of Calabria, 87036 Rende, Italy
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(1), 22; https://doi.org/10.3390/math13010022
Submission received: 14 November 2024 / Revised: 21 December 2024 / Accepted: 22 December 2024 / Published: 25 December 2024
(This article belongs to the Section Mathematics and Computer Science)
Versions Notes

Abstract

:
This work delves deeply into convolved Fibonacci polynomials (CFPs) that are considered generalizations of the standard Fibonacci polynomials. We present new formulas for these polynomials. An expression for the repeated integrals of the CFPs in terms of their original polynomials is given. A new approach is followed to obtain the higher-order derivatives of these polynomials from the repeated integrals formula. The inversion and moment formulas for these polynomials, which we find, are the keys to developing further formulas for these polynomials. The derivatives of the moments of the CFPs in terms of their original polynomials and different symmetric and non-symmetric polynomials are also derived. New product formulas of these polynomials with some polynomials, including the linearization formulas of these polynomials, are also deduced. Some closed forms for definite and weighted definite integrals involving the CFPs are found as consequences of some of the introduced formulas.

1. Introduction

The use of various polynomials and special functions is fundamental in many fields due to their applications in many fields of applied science; see, for example, [1,2,3]. An extensive number of theoretical publications have explored different polynomial sequences. The authors of [4] have derived some formulas related to Bernoulli polynomials. In [5], the authors derived some other formulas concerning some generalized Bernoulli polynomials. In [6], the authors investigated mixed hypergeometric Bernoulli–Gegenbauer polynomials. In [7], Apostol-type Hermite degenerated polynomials were introduced. In [8], the authors derived some new formulas for general polynomial sequences, while the same authors applied a matrix approach for investigating some Appell polynomials. Generalized polynomials have drawn the attention of several authors because of their theoretical and practical significance. For example, the authors of [9] derived some new formulas for the generalized Hermite polynomials with some applications. In [10], the generalized Bessel polynomials were used to treat multi-order fractional differential equations.
Sequences of Fibonacci and Lucas polynomials and their associated numbers have many applications in various fields, such as combinatorics, algebra, and graph theory. Fibonacci sequences aid in analyzing algebraic characteristics of polynomial rings and can be utilized to solve specific types of polynomial equations. Some applications of Fibonacci polynomials can be found in [11,12]. The standard Fibonacci and Lucas polynomials have been the subject of certain theoretical advancements. The authors of [13,14] presented findings on the Fibonacci and Lucas polynomials and their relationships with various polynomials, such as various orthogonal polynomials. The authors of [15] derived some formulas for Lucas polynomials and employed them to find spectral solutions to the time-fractional diffusion equation. Additional references for these polynomials are available in [16,17,18,19].
Researchers are exploring various modified and generalized Fibonacci and Lucas polynomial sequences beyond the standard ones. The author of [20] developed some formulas using the generalized Fibonacci polynomials via a matrix approach. The authors of [21] developed some formulas related to the generalized bi-periodic Fibonacci and Lucas polynomials. The authors of [22] investigated some distance Fibonacci polynomials. A generalization of the Fibonacci sequence is defined on elliptic curves in [23]. From a numerical perspective, Fibonacci polynomials and their variants are utilized in many applications. For example, the authors of [24] proposed a numerical method for solving a two-dimensional Sobolev equation by employing a combination of Lucas and Fibonacci polynomials. The authors of [25] followed a numerical algorithm for handling some fractional differential equations. The authors of [26] followed a numerical approach for treating advection-diffusion-reaction equations in multi-dimensions using Lucas and Fibonacci polynomials. Vieta–Fibonacci polynomials were employed in [27] to treat a certain fractional model.
Generalized hypergeometric functions play an essential role in mathematical analysis and its applications. They are a basis for practically all significant functions and polynomials. They often appear in the solution of some important problems in special functions such as connection, duplication, and linearization formulas. For example, in the series of papers [28,29,30,31], the authors solved the connection and linearization problems for several orthogonal polynomials. In these articles, the authors found expressions for the connection and linearization coefficients in explicit forms, including generalized hypergeometric functions. Furthermore, the derivative expressions for the different orthogonal polynomials are essential in solving differential equations using spectral methods. Expressions for the derivatives can be obtained in closed forms using these functions. For example, in [32], the authors found these expressions using terminating hypergeometric functions of the type F 3 4 ( 1 ) . For some other contributions regarding the important problems related to special functions, one can refer to [33,34,35,36,37].
One of the generalizations of the Fibonacci sequences is the so-called “convolved ( p , q ) Fibonacci polynomials” that was introduced and considered in [38]. These polynomials generalize some standard sequences, such as Fibonacci and Pell polynomials. In [39], the authors discussed some properties of the convolved generalized Fibonacci and Lucas polynomials. In [40], the authors derived some new formulas for the convolved Pell polynomials. This paper will introduce a class of polynomials, namely, convolved Fibonacci polynomials that generalize the standard Fibonacci polynomials; we will develop new formulas related to them. We can list the main aims of the paper in the following items:
  • Derive a new formula for the repeated integrals of the CFPs based on the structure formula of the polynomials;
  • derive formulas for the derivatives of these polynomials using a new approach using the repeated integrals formula;
  • express the derivatives of the moments of these polynomials in terms of the original polynomials;
  • establish a linearization formula of these polynomials and new product formulas for these polynomials with some polynomials;
  • establish two general theorems for computing the derivatives of the moments of the CFPs in terms of other polynomials;
  • present some closed formulas for the integral formulas as applications to some of the presented formulas.
As far as we know, most of the results for the CFPs in this paper are new. In addition, there are few investigations of this kind of generalized polynomials. We anticipate that the introduced polynomials will be useful in other applications.
The paper follows the following structure: We will discuss some elementary properties of the CFPs in the next section. In addition, an overview of some celebrated polynomials is presented in this section. Section 3 is devoted to developing a new expression for the q times repeated integrals of the CFPs. High-order derivatives of the CFPs as combinations of the CFPs are obtained in Section 4 using a new approach. Derivatives of the moments of the CFPs are developed in Section 5. Some linearization formulas of the CFPs are deduced in Section 6. Further derivative formulas of the CFPs are found in Section 7. Some definite integral formulas are presented in Section 8. A matrix approach is followed in Section 9 to obtain some formulas related to the CFPs. Finally, some conclusions are presented in Section 10.

2. Some Essentials of Fibonacci and Convolved Fibonacci Polynomials

This section introduces some formulas related to Fibonacci polynomials and convolved Fibonacci polynomials. In addition, an account of some different polynomials is provided.

2.1. An Account on Fibonacci Polynomials

The Fibonacci polynomials can be constructed using the recursive formula [11]:
F 0 ( x ) = 0 , F 1 ( x ) = 1 , F m + 2 ( x ) = x F m + 1 ( x ) + F m ( x ) , m ≥ 0 .
These polynomials can be expressed as
F m ( x ) = ∑ r = 0 m − 1 2 m − r − 1 r x m − 2 r − 1 ,
where z is the floor function. The inversion formula for the Fibonacci polynomials is given by
x m = ∑ r = 0 m 2 ( − 1 ) r r ! m − 2 r + 1 m − r + 2 r − 1 F m − 2 r + 1 ( x ) ,
where ( a ) j represents the Pochhammer function defined by:
( a ) j = Γ ( a + j ) Γ ( a ) .
The moment formula for Fibonacci polynomials can be written as
x r F j + 1 ( x ) = ∑ s = 0 r ( − 1 ) s r s F j + r − 2 s + 1 ( x ) , r , j ≥ 0 .

2.2. An Account on the Generalized Convolved Fibonacci Polynomials

In [38], the authors constructed the generalized convolved Fibonacci polynomials (or ( p , q ) convolved Fibonacci polynomials) CF j p , q , μ ( x ) , where p = p ( x ) and q = q ( x ) are two polynomials with real coefficients, and μ is a complex number. According to [38], these polynomials may be generated using the following generating function:
1 − p ( x ) t − q ( x ) t 2 − μ = ∑ j = 0 ∞ CF j p , q , μ ( x ) t j .
In addition, they have the following series representation:
CF j p , q , μ ( x ) = ∑ r = 0 j 2 μ + r − 1 r μ + j − r − 1 j − 2 r p j − 2 r ( x ) q r ( x ) .
The recursive formula satisfied by CF j p , q , μ ( x ) is given by:
j CF j p , q , μ ( x ) − ( μ + j − 1 ) p ( x ) CF j − 1 p , q , μ ( x ) − ( 2 μ + j − 2 ) q ( x ) CF j − 2 p , q , μ ( x ) = 0 , j ≥ 2 ,
with the initial values:
CF 0 p , q , μ ( x ) = 1 , CF 1 p , q , μ ( x ) = μ p ( x ) .
The CFPs denoted by F j μ ( x ) are particular polynomials of the generalized polynomials CF j p , q , μ ( x ) corresponding to the choices:
p ( x ) = x , q ( x ) = 1 ,
and μ is restricted to be a positive real number.
Remark 1. 
It is worth mentioning here that F j μ ( x ) are also particular cases of the convolved h ( x ) -Fibonacci polynomials introduced by Ramírez in [39].
Remark 2. 
The polynomials F j μ ( x ) were employed previously in [41] to solve the Fitzhugh–Nagumo non-linear equation. Here, we will concentrate on developing new theoretical results regarding these polynomials.
Remark 3. 
The standard Fibonacci polynomials are particular ones of the CFPs in the sense that
F j + 1 ( x ) = F j 1 ( x ) .
The analytic and inversion formulas for these polynomials can be written, respectively, as [41]:
F j μ ( x ) = ∑ s = 0 j 2 ( μ ) j − s s ! ( j − 2 s ) ! x j − 2 s ,
x j = j ! Γ ( μ ) ∑ s = 0 j 2 ( − 1 ) s ( j + μ − 2 s ) Γ ( 1 + j + μ − s ) s ! F j − 2 s μ ( x ) .

2.3. An Overview on Some Polynomials

We will present an overview of orthogonal and non-orthogonal polynomials. It is useful to consider two classes of symmetric and non-symmetric polynomials. Let us denote them as ϕ s ( x ) and ψ s ( x ) , respectively. They have the following analytic forms:
ϕ s ( x ) = ∑ r = 0 s 2 G r , s x s − 2 r ,
ψ s ( x ) = ∑ r = 0 s M r , s x s − r ,
where G r , s and M r , s are known coefficients.
Furthermore, suppose that the inversion formulas of (11) and (12) are as follows:
x s = ∑ r = 0 s 2 G ¯ r , s ϕ s − 2 r ( x ) ,
x s = ∑ r = 0 s M ¯ r , s ψ s − r ( x ) ,
where G ¯ r , s and M ¯ r , s are known coefficients.
We give some well-known polynomials expressed in (11) and (12). The normalized shifted Jacobi polynomials are represented by [42]:
R ˜ ℓ ( ν , θ ) ( x ) = ∑ r = 0 ℓ ( − 1 ) r ℓ ! Γ ( ν + 1 ) ( θ + 1 ) ℓ ( ν + θ + 1 ) 2 ℓ − r r ! ( ℓ − r ) ! Γ ( ℓ + ν + 1 ) ( ν + θ + 1 ) ℓ ( θ + 1 ) ℓ − r x ℓ − r ,
where R ˜ ℓ ( ν , θ ) ( x ) are orthogonal on [ 0 , 1 ] in the sense that:
∫ 0 1 ( 1 − x ) ν x θ R ˜ ℓ ( ν , θ ) ( x ) R ˜ k ( ν , θ ) ( x ) d x = 0 , k ≠ ℓ , h ˜ ℓ ν , θ , k = ℓ ,
where
h ˜ ℓ ν , θ = k ! Γ ( ν + 1 ) 2 Γ ( k + θ + 1 ) ( 2 k + ν + θ + 1 ) Γ ( k + ν + 1 ) Γ ( k + ν + θ + 1 ) .
Also, the Bernoulli polynomials can be written as:
B s ( x ) = ∑ r = 0 ℓ ℓ r B r x ℓ − r ,
where B r are the Bernoulli numbers.
Among the important orthogonal polynomials that can be represented as in (11) are the Hermite and ultraspherical polynomials [43]. In addition, other important symmetric polynomials that can be represented as in (11) are the following two sequences: G F r R , T ( x ) and G L r A , B ( x ) that generalize, respectively, the standard Fibonacci and Lucas polynomials. They can be generated with the aid of the following recursive formulas:
G F r R , T ( x ) = R x G F r − 1 R , T ( x ) + T G F r − 2 R , T ( x ) , G F 0 R , T ( x ) = 1 , G F 1 R , T ( x ) = R x , r ≥ 2 ,
G L r A , B ( x ) = A x G L r − 1 A , B ( x ) + B G L r − 2 A , B ( x ) , G L 0 A , B ( x ) = 2 , G L 1 A , B ( x ) = A x , r ≥ 2 .
The series representations of G F r R , T ( x ) , and G L r A , B ( x ) are as follows [42]:
G F r R , T ( x ) = ∑ ℓ = 0 r 2 r − ℓ ℓ T ℓ R r − 2 ℓ x r − 2 ℓ , r ≥ 0 ,
G L r A , B ( x ) = r ∑ ℓ = 0 r 2 B ℓ A r − 2 ℓ r − ℓ ℓ r − ℓ x r − 2 ℓ , r ≥ 1 .
One can consult [43,44,45] for a survey on some special polynomials.

3. The q Times Repeated Integration of the CFPs

The main purpose of this section is to state and prove a new formula expressing explicitly the q times repeated integrals of the CFPs in terms of their original polynomials. This formula is based on the following structure formula for these polynomials.
Lemma 1. 
The following is the structure formula of the F j μ ( x )
F j μ ( x ) = 1 j + μ d d x F j + 1 μ ( x ) + d d x F j − 1 μ ( x ) , j ≥ 0 .
Proof. 
Formula (23) can be proved easily by applying the analytic formula of the CFPs in (9). □
Theorem 1. 
Let I j ( q ) ( x ) = ∫ ∫ … ∫ ︷ q t i m e s F j μ ( x ) d x 1 d x 2 … d x q denote the q times repeated integration of F j μ ( x ) . The following formula is valid:
I j ( q ) ( x ) = Γ ( j + μ ) ∑ ℓ = 0 q ( j − 2 ℓ + q + μ ) ( 1 − ℓ + q ) ℓ ℓ ! Γ ( j − ℓ + q + μ + 1 ) ( j + μ − ℓ ) ℓ F j + q − 2 ℓ μ + π q − 1 ( x ) , q ≥ 1 ,
where π q − 1 ( x ) is a polynomial of degree at most ( q − 1 ) .
Proof. 
We will prove Formula (24) by induction on q. At first, the structure Formula (23) leads to the following relation:
∫ F j μ ( x ) d x = 1 j + μ F j + 1 μ ( x ) + F j − 1 μ ( x ) + c ,
for arbitrary constant c. Thus, the theorem is correct for q = 1 . Now, assume the validity of relation (24). We have to prove the following relation:
I j ( q + 1 ) ( x ) = Γ ( j + μ ) ∑ ℓ = 0 q + 1 ( 1 + j − 2 ℓ + q + μ ) ( 2 − ℓ + q ) ℓ ℓ ! Γ ( 2 + j − ℓ + q + μ ) ( j − ℓ + μ ) ℓ F j + q − 2 ℓ + 1 μ + π ¯ q ( x ) ,
where π ¯ q ( x ) is a polynomial of degree at most q.
Based on the application of the induction hypothesis together with (25), the following formula can be obtained:
I j ( q + 1 ) ( x ) = ∫ I j ( q ) ( x ) d x = Γ ( j + μ ) ∑ ℓ = 0 q ( 1 − ℓ + q ) ℓ ℓ ! Γ ( j − ℓ + μ + q + 1 ) ( j + μ − ℓ ) ℓ × F j − 2 ℓ + q + 1 μ ( x ) + F j − 2 ℓ + q − 1 μ ( x ) + π ¯ q ( x ) ,
where π ¯ q ( x ) is a polynomial of degree at most q.
Some algebraic computations lead to the following formula:
I j ( q + 1 ) ( x ) = Γ ( j + μ ) Γ ( j + μ + q + 1 ) F j + q + 1 μ ( x ) + 1 ( j + μ ) ( j − q + μ ) q F j + q − 1 μ ( x ) + ∑ ℓ = 1 q Γ ( j + μ ) ( 1 − ℓ + q ) ℓ ℓ ! ( j − ℓ + μ + q ) ! ( j + μ − ℓ ) ℓ + Γ ( j + μ ) ( 2 − ℓ + q ) ℓ − 1 ( ℓ − 1 ) ! Γ ( 2 + j − ℓ + q + μ ) ( 1 + j − ℓ + μ ) ℓ − 1 F j + q + 1 − 2 ℓ μ ( x ) + π ¯ q ( x ) .
It is not difficult to show the identity
Γ ( j + μ ) ( − ℓ + q + 1 ) ℓ ℓ ! Γ ( j + μ − ℓ + 1 ) ℓ Γ ( j − ℓ + q + μ + 1 ) + Γ ( j + μ ) ( − ℓ + q + 2 ) ℓ − 1 ( ℓ − 1 ) ! Γ ( j + μ − ℓ + 2 ) ℓ − 1 Γ ( j − ℓ + q + μ + 2 ) = Γ ( j + μ ) ( − ℓ + q + 2 ) ℓ ( j − 2 ℓ + q + μ + 1 ) ℓ ! Γ ( j + μ − ℓ + 1 ) ℓ Γ ( j − ℓ + q + μ + 2 ) .
Consequently, the following relation is obtained:
I j ( q + 1 ) ( x ) = ∑ ℓ = 0 q + 1 ( 1 + j − 2 ℓ + q + μ ) Γ ( j + μ ) ( 2 − ℓ + q ) ℓ ℓ ! Γ ( j − ℓ + q + μ + 2 ) ( j − ℓ + μ ) ℓ F j + q − 2 ℓ + 1 μ + π ¯ q ( x ) .
This ends the proof of Theorem 1. □

4. A New Approach for Developing the Derivatives of the CFPs

This section follows a new approach based on the repeated integrals formula for deriving the expression of the higher-order derivatives of the CFPs.
Theorem 2. 
For every non-negative r and every integer r ≥ q ≥ 1 , the following formula holds:
D q F r μ ( x ) = ∑ m = 0 r − q 2 ( − 1 ) m m + q − 1 m ( − 2 m + μ − q + r ) ( − m + μ − q + r + 1 ) q − 1 × F r − q − 2 m μ ( x ) .
Proof. 
Our strategy to prove Theorem 2 is built on making use of Theorem 1. In other words, we will use the q times repeated integration of CFPs to prove the formula for the q-th derivative of CFPs.
First, consider the sum
G n ( x ) = ∑ r = 0 n a r F r μ ( x ) .
Integrating both sides of (30) q times, we obtain
J n + q ( x ) = ∫ ∫ … ∫ ︷ q t i m e s G n ( x ) d x 1 d x 2 … d x q = ∑ r = 0 n a r I r ( q ) ( x ) ,
where I j ( q ) ( x ) is given by (24). Therefore, we have
J n + q ( x ) = ∑ r = 0 n a r ∑ k = 0 q ( r + μ − 1 ) ! ( − k + q + 1 ) k ( − 2 k + q + r + μ ) k ! ( r + μ − k ) k Γ ( − k + q + r + μ + 1 ) F r + q − 2 k μ ( x ) + θ q − 1 ( x ) ,
where θ q − 1 ( x ) a polynomial of degree not exceeding ( q − 1 ) .
Rewriting the expansion in the right-hand side of (32), we obtain
J n + q ( x ) = ∑ r = q n + q ∑ i = 0 q ( r + μ ) ( − i + q + 1 ) i Γ ( 2 i − q + r + μ ) i ! Γ ( i + r + μ + 1 ) ( i − q + r + μ ) i a r − q + 2 i F r μ ( x ) + θ ¯ q − 1 ( x ) ,
where θ ¯ q − 1 ( x ) is a polynomial of degree less than or equal to ( q − 1 ) .
Now, if we assume that
J n + q ( x ) = ∑ r = 0 n + q A r F r μ ( x ) ,
then relations (33) and (34) yield the following recurrence relation:
A r = ∑ i = 0 q ( r + μ ) ( − i + q + 1 ) i Γ ( 2 i − q + r + μ ) i ! Γ ( i + r + μ + 1 ) ( i − q + r + μ ) i a r − q + 2 i , r = n + q , n + q − 1 , … , q + 1 , q , a n + 1 = a n + 2 = … = a n + q = 0 .
The last recurrence relation can be solved exactly to give
a r = ∑ k = r + q ( k + r + q ) even n + q ( − 1 ) k − q − r 2 ( r + μ ) k + q − r − 2 2 k − q − r 2 k − q + r + 2 2 + μ q − 1 A k .
Now, if we differentiate (34) with respect to x, then
G n ( x ) = ∑ r = 0 n + q A r D q F r μ ( x ) ,
and accordingly, we have
∑ r = 0 n a r F r μ ( x ) = ∑ r = 0 n + q A r D q F r μ ( x ) .
In view of (36), the last relation is converted into
∑ r = 0 n ∑ k = r + q ( k + r + q ) even n + q ( − 1 ) k − q − r 2 ( r + μ ) k + q − r − 2 2 k − q − r 2 k − q + r + 2 2 + μ q − 1 A k F r μ ( x ) = ∑ r = 0 n + q A r D q F r μ ( x ) .
If the left-hand side of the last equation is expanded and rearranged, then, after some lengthy manipulations, the following relation is obtained
∑ r = 0 n + q A r ∑ m = 0 r − q 2 ( − 1 ) m m + q − 1 m ( − 2 m − q + r + μ ) ( − m − q + r + μ + 1 ) q − 1 F r − q − 2 m μ ( x ) = ∑ r = 0 n + q A r D q F r μ ( x ) ,
which immediately yields the relation
D q F r μ ( x ) = ∑ m = 0 r − q 2 ( − 1 ) m m + q − 1 m ( − 2 m + μ − q + r ) ( − m + μ − q + r + 1 ) q − 1 × F r − q − 2 m μ ( x ) .
Theorem 2 is now proved. □
Remark 4. 
If we set μ = q = 1 , then the following first derivative formula for the Fibonacci polynomials can be obtained as a special case of (29):
D F r 1 ( x ) = D F r + 1 ( x ) = ∑ m = 0 r − 1 2 ( − 1 ) m ( r − 2 m ) F r − 2 m ( x ) .
Remark 5. 
The last formula coincides with the result of Proposition 16 in [46].

5. Derivatives of the Moments Between Two CFPs

In this section, we will establish the derivatives of the moments of the polynomials F j μ ( x ) in terms of other CFPs: F j δ ( x ) . The core for deriving these formulas is built on the series representation in (9) and its inverse Formula (10).
Theorem 3. 
Let r , j and q be non-negative integers with r + j ≥ q , and let μ , δ be two positive real numbers. The following formula holds:
D q x r F j μ ( x ) = Γ ( δ ) Γ ( j + μ ) ( j + r ) ! Γ ( μ ) j ! ∑ s = 0 1 2 ( j + r − q ) ( − 1 ) s ( j − q + r − 2 s + δ ) s ! Γ ( 1 + j − q + r − s + δ ) × F 3 4 − s , 1 2 − j 2 , − j 2 , − j + q − r + s − δ − j 2 − r 2 , 1 2 − j 2 − r 2 , 1 − j − μ | 1 F j + r − q − 2 s δ ( x ) .
Note that F 3 4 ( 1 ) is a specific instance of the generalized hypergeometric function F q p ( z ) for p = 4 , q = 3 at z = 1 , see [43].
Proof. 
The series form in (9) allows one to write the following expression:
D q x r F j μ ( x ) = ∑ ℓ = 0 1 2 ( j + r − q ) ( 1 + j − 2 ℓ − q + r ) q ( μ ) j − ℓ ( j − 2 ℓ ) ! ℓ ! x j + r − 2 ℓ − q .
The application of Formula (10), transforms Formula (39) into
D q x r F j μ ( x ) = Γ ( δ ) Γ ( μ ) ∑ ℓ = 0 1 2 ( j + r − q ) ( j − 2 ℓ + r ) ! Γ ( j − ℓ + μ ) ( j − 2 ℓ ) ! ℓ ! × ∑ t = 0 1 2 ( j + r − q ) − ℓ ( − 1 ) t ( j − 2 ℓ − q + r − 2 t + δ ) t ! Γ ( 1 + j − 2 ℓ − q + r − t + δ ) F j + r − 2 ℓ − q − 2 t δ ( x ) .
Some algebraic computations turn the last formula into
D q x r F j μ ( x ) = Γ ( δ ) Γ ( μ ) ∑ s = 0 1 2 ( j + r − q ) ( − j + q − r + 2 s − δ ) × ∑ ℓ = 0 s ( − 1 ) 1 − ℓ + s ( j − 2 ℓ + r ) ! Γ ( j − ℓ + μ ) ( j − 2 ℓ ) ! ℓ ! ( s − ℓ ) ! Γ ( 1 + j − ℓ − q + r − s + δ ) F j + r − q − 2 s δ ( x ) ,
which can be written in the form of (38). □
Remark 6. 
It is worth noting that Formula (38) produces two essential formulas as special cases. The following two corollaries exhibit these formulas.
Corollary 1. 
Let j ≥ q ≥ 1 . D q F j μ ( x ) can be expressed as combinations of F j δ ( x ) as follows:
D q F j μ ( x ) = Γ ( δ ) ( μ ) j ∑ s = 0 j − q 2 ( − 1 ) s ( j − q − 2 s + δ ) ( q − δ + μ ) s s ! Γ ( 1 + j − q − s + δ ) ( j − s + μ ) s F j − q − 2 s δ ( x ) .
Proof. 
Setting r = 0 in (38) yields
D q F j μ ( x ) = Γ ( δ ) ( μ ) j ∑ s = 0 j − q 2 ( − 1 ) s ( j − q − 2 s + δ ) s ! Γ ( 1 + j − q − s + δ ) F 1 2 − s , − j + q + s − δ 1 − j − μ | 1 F j − q − 2 s δ ( x ) ,
where F 1 2 a , b c | z is the well-known Gaussian hypergeometric function; see [43].
The F 1 2 ( 1 ) in (43) can be summed using the Chu-Vandermonde identity to give
F 1 2 − s , − j + q + s − δ 1 − j − μ | 1 = ( q − δ + μ ) s ( j − s + μ ) s ,
and hence (43) reduces to the following formula:
D q F j μ ( x ) = Γ ( δ ) ( μ ) j ∑ s = 0 j − q 2 ( − 1 ) s ( j − q − 2 s + δ ) ( q − δ + μ ) s s ! Γ ( 1 + j − q − s + δ ) ( j − s + μ ) s F j − q − 2 s δ ( x ) ,
and this finalizes the proof. □
Corollary 2. 
The following is the moment formula of F j μ ( x ) in terms of F j δ ( x ) for all non-negative integers j and r.
x r F j μ ( x ) = Γ ( δ ) ( μ ) j ( j + r ) ! j ! ∑ s = 0 1 2 ( j + r ) ( − 1 ) s ( j + r − 2 s + δ ) s ! Γ ( 1 + j + r − s + δ ) × F 3 4 − s , 1 2 − j 2 , − j 2 , − j − r + s − δ − j 2 − r 2 , 1 2 − j 2 − r 2 , 1 − j − μ | 1 F j + r − 2 s δ ( x ) .
Proof. 
It is a consequence of (38) by setting q = 0 . □
Corollary 3. 
The following is the moment formula of F j μ ( x ) in terms of their original ones for all non-negative integers j and r.
x r F j μ ( x ) = Γ ( j + μ ) ( j + r ) ! j ! ∑ s = 0 1 2 ( j + r ) ( − 1 ) s ( j + r − 2 s + μ ) s ! Γ ( 1 + j + r − s + μ ) × F 3 4 − s , 1 2 − j 2 , − j 2 , − j − r + s − μ − j 2 − r 2 , 1 2 − j 2 − r 2 , 1 − j − μ | 1 F j + r − 2 s μ ( x ) .
Proof. 
Direct from (45) setting δ = μ . □
Remark 7. 
The F 3 4 ( 1 ) appears in (45) can not be summed in general; however, it can be summed for special choices of μ. In the following, we give one of these cases. More precisely, we will give a reduced formula for the moments of the standard Fibonacci polynomials corresponding to the case μ = 1 .
Corollary 4. 
The moment’s formula for the standard Fibonacci polynomials can be expressed as
x r F j + 1 ( x ) = ∑ s = 0 r ( − 1 ) s r s F j + r − 2 s + 1 ( x ) .
Proof. 
Setting μ = 1 in (46) yields the following formula:
x r F j + 1 ( x ) = ( j + r ) ! ∑ s = 0 j + r 2 ( − 1 ) s ( 1 + j + r − 2 s ) ( j + r − s + 1 ) ! s ! F 3 4 − s , 1 2 − j 2 , − j 2 , − 1 − j − r + s − j , − j 2 − r 2 , 1 2 − j 2 − r 2 | 1 × F j + r − 2 s + 1 ( x ) .
The last F 3 4 ( 1 ) is balanced. We can sum it up using suitable symbolic algebra tools. We use Zeilberger’s algorithm [47] for such reduction. Set
H s , j , r = F 3 4 − s , 1 2 − j 2 , − j 2 , − 1 − j − r + s − j , − j 2 − r 2 , 1 2 − j 2 − r 2 | 1 ,
The applications of Zeilberger’s algorithm [47] to show the validity of the following recurrence relation:
− ( 1 + j + r − 2 s ) ( r − s ) ( s + 1 ) H s , j , r − ( − 1 + j + r − 2 s ) × j ( − 2 + r − 2 s ) − 2 ( r − s ) ( s + 1 ) H s + 1 , j , r + ( − 3 + j + r − 2 s ) ( − 1 + j − s ) ( j + r − s ) × H s + 2 , j , r = 0 ,
with the following initial conditions:
H 0 , j , r = 1 , H 1 , j , r = r r + j − 1 .
The explicit solution of (48) is given by
H s , j , r = ( r − s + 1 ) s ( j + r − 2 s + 1 ) ( j + r − s + 2 ) s − 1 ,
and therefore, Formula (47) reduces to the following formula:
x r F j + 1 ( x ) = ∑ s = 0 r ( − 1 ) s r s F j + r − 2 s + 1 ( x ) .
This ends the proof of Corollary 4. □
Remark 8. 
This formula coincides with the well-known moment formula of Fibonacci polynomials (see [13] (Theorem 1)).

6. Some Linearization Formulas of the CFPs

This section is devoted to presenting some product formulas of the CFPs. First, we will derive a new linearization formula for the CFPs. In addition, we will derive some linearization formulas of the CFPs with some celebrated polynomials.
Theorem 4. 
Let i and j be non-negative integers. The following linearization formula holds:
F i μ ( x ) F j μ ( x ) = 1 Γ ( μ ) ∑ s = 0 min ( i , j ) ( − 1 ) s ( i + j − 2 s + μ ) ( i + j − 2 s ) ! Γ ( i − s + μ ) Γ ( j − s + μ ) ( μ ) s ( i − s ) ! ( j − s ) ! s ! Γ ( 1 + i + j − s + μ ) × ( i + j − 2 s + 2 μ ) s F i + j − 2 s μ ( x ) .
Proof. 
Consider the case j ≥ i . We start from the series representation of the F i μ ( x ) to write
F i μ ( x ) F j μ ( x ) = ∑ ℓ = 0 i 2 ( μ ) i − ℓ ℓ ! ( i − 2 ℓ ) ! x i − 2 ℓ F j μ ( x ) ,
which gives after making use of the moment Formula (46)
F i μ ( x ) F j μ ( x ) = Γ ( j + μ ) j ! ∑ ℓ = 0 i 2 ( i + j − 2 ℓ ) ! ( μ ) i − ℓ ℓ ! ( i − 2 ℓ ) ! ∑ s = 0 i + j 2 − ℓ ( − 1 ) s ( i + j − 2 ( ℓ + s ) + μ ) s ! Γ ( 1 + i + j − 2 ℓ − s + μ ) × F 3 4 − s , 1 2 − j 2 , − j 2 , − i − j + 2 ℓ + s − μ − i 2 − j 2 + ℓ , 1 2 − i 2 − j 2 + ℓ , 1 − j − μ | 1 F j + i − 2 ℓ − 2 s μ ( x ) .
The last formula can be written after some algebraic computations in the form
F i μ ( x ) F j μ ( x ) = ( μ ) j j ! ∑ s = 0 i ( i + j + μ − 2 s ) ∑ ℓ = 0 s ( − 1 ) s − ℓ ( i + j − 2 ℓ ) ! Γ ( i − ℓ + μ ) ( i − 2 ℓ ) ! ℓ ! ( s − ℓ ) ! Γ ( 1 + i + j − ℓ − s + μ ) × F 3 4 ℓ − s , 1 2 − j 2 , − j 2 , − i − j + ℓ − μ + s − i 2 − j 2 + ℓ , 1 2 − i 2 − j 2 + ℓ , 1 − j − μ | 1 F i + j − 2 s μ ( x ) .
Now, to obtain a reduced form for the last formula, we resort to computational algebra for such a purpose. The Zeilberger’s algorithm [47] may be used. In this regard, let
M s , μ , i , j = ∑ ℓ = 0 s ( − 1 ) s − ℓ ( i + j − 2 ℓ ) ! Γ ( i − ℓ + μ ) ( i − 2 ℓ ) ! ℓ ! ( s − ℓ ) ! Γ ( 1 + i + j − ℓ − s + μ ) × F 3 4 ℓ − s , 1 2 − j 2 , − j 2 , − i − j + ℓ − μ + s − i 2 − j 2 + ℓ , 1 2 − i 2 − j 2 + ℓ , 1 − j − μ | 1 ,
it can be shown that the following recursive formula is satisfied by M s , μ , i , j :
( s + 1 ) ( − 1 + i + j − 2 s ) ( i + j − 2 s ) ( − 1 + i + μ − s ) ( − 1 + j + μ − s ) × ( − 1 + i + j + 2 μ − s ) M s + 1 , μ , i , j − ( − 2 + i + j + 2 μ − 2 s ) ( − 1 + i + j + 2 μ − 2 s ) × ( j − s ) ( i + j + μ − s ) ( − i + s ) ( μ + s ) M s , μ , i , j = 0 ,
with
M 0 , μ , i , j = ( i + j ) ! Γ ( i + μ ) i ! Γ ( 1 + i + j + μ ) ,
whose solution can be easily obtained to be
M s , μ , i , j = ( − 1 ) s j ! ( i + j − 2 s ) ! Γ ( i − s + μ ) Γ ( j − s + μ ) ( μ ) s ( i + j − 2 s + 2 μ ) s s ! ( i − s ) ! ( j − s ) ! Γ ( j + μ ) Γ ( 1 + i + j − s + μ ) .
The above reduction turns Formula (52) into the following simplified formula:
F i μ ( x ) F j μ ( x ) = 1 Γ ( μ ) ∑ s = 0 i ( − 1 ) s ( i + j − 2 s + μ ) ( i + j − 2 s ) ! Γ ( i − s + μ ) Γ ( j − s + μ ) ( μ ) s ( i − s ) ! ( j − s ) ! s ! Γ ( 1 + i + j − s + μ ) × ( i + j − 2 s + 2 μ ) s F i + j − 2 s μ ( x ) .
This completes the proof for the case j ≥ i . The proof for the case j < i is similar. This ends the proof of Theorem 4. □
Theorem 5. 
Let i , j be two non-negative integers. The product formula of the CFPs and the generalized Fibonacci polynomials that are expressed as in (21) is given by
G F i R , T ( x ) F j μ ( x ) = R − j ( μ ) j ∑ s = 0 j ( − μ ) s s ! ( j − s ) ! F 1 2 − s , s − j 1 − j − μ | R 2 T G F i + j − 2 s R , T ( x ) .
Proof. 
Making use of the power form representation of G F i R , T ( x ) in (21) along with the moment formula of Fibonacci polynomials in (4) leads to
G F i R , T ( x ) F j μ ( x ) = ∑ m = 0 j 2 ( μ ) j − m ( j − 2 m ) ! m ! ∑ r = 0 j − 2 m R − j + 2 m ( − T ) r j − 2 m r G F i + j − 2 m − 2 r R , T ( x ) ,
which can be written as
G F i R , T ( x ) F j μ ( x ) = ∑ s = 0 j ∑ ℓ = 0 s R − j + 2 ℓ ( − T ) s − ℓ j − 2 ℓ s − ℓ ( μ ) j − ℓ ( j − 2 ℓ ) ! ℓ ! G F i + j − 2 s R , T ( x ) ,
which is equivalent to
G F i R , T ( x ) F j μ ( x ) = R − j ( μ ) j ∑ s = 0 j ( − T ) s s ! ( j − s ) ! F 1 2 − s , s − j 1 − j − μ | R 2 T G F i + j − 2 s R , T ( x ) .
This ends the proof. □
Theorem 6. 
Let i and j be two non-negative integers. The product formula of the CFPs and the generalized Lucas polynomials that are defined in (22) is given by
G L i A , B ( x ) F j μ ( x ) = A − j ( μ ) j ∑ s = 0 j ( − B ) s s ! ( j − s ) ! F 1 2 − s , s − j 1 − j − μ | A 2 B G L i + j − 2 s A , B ( x ) .
Proof. 
The proof is similar to the proof of Theorem 6 based on the power form representation of G L i A , B ( x ) in (22) along with the moment formula of Fibonacci polynomials in (4). □

7. Further Derivative Formulas of the CFPs

In this section, we give some other formulas for the CFPs. We will find expressions for the derivatives of the CFPs in terms of several polynomials.

7.1. Expressions for the Derivatives of the CFPs in Terms of Symmetric Polynomials

This section presents a general theorem for computing the derivatives of the CFPs in terms of any family of symmetric polynomials expressed as in (11).
Theorem 7. 
Let m and q be two non-negative integers with m ≥ q ≥ 1 . Consider any symmetric polynomial ϕ s ( x ) that can be represented as in (11). In terms of ϕ m ( x ) , D q F m μ ( x ) can be expressed as
D q F m μ ( x ) = 1 Γ ( μ ) ∑ s = 0 m − q 2 ∑ r = 0 s Γ ( m + μ − r ) G ¯ s − r , m − 2 r − q ( m − q − 2 r ) ! r ! ϕ m − q − 2 s ( x ) ,
where G ¯ r , s are the inversion coefficients of ϕ s ( x ) that appear in (13).
Proof. 
Making use of the power form representation of the CFPs given in (9) leads to the following formula:
D q F m μ ( x ) = 1 Γ ( μ ) ∑ r = 0 m − q 2 Γ ( m + μ − r ) ( m − q − 2 r ) ! r ! x m − 2 r − q .
Based on the inversion formula of ϕ i ( x ) that expressed as in (13), the last formula can be converted into
D q F m μ ( x ) = 1 Γ ( μ ) ∑ r = 0 m − q 2 Γ ( m + μ − r ) ( m − q − 2 r ) ! r ! ∑ t = 0 m − q 2 − r G ¯ t , m − 2 r − q ϕ m − 2 r − q − 2 t ( x ) .
Some algebraic manipulations convert the last formula into
D q F m μ ( x ) = 1 Γ ( μ ) ∑ s = 0 m − q 2 ∑ r = 0 s Γ ( m + μ − r ) ( m − q − 2 r ) ! r ! G ¯ s − r , m − 2 r − q ϕ m − q − 2 s ( x ) ,
which finalizes the proof. □
Remark 9. 
It is evident from Theorem 7 that we can obtain an expression of the derivatives of the CFPs in terms of any symmetric polynomial once the inversion coefficients G ¯ r , s are substituted. The following corollary exhibits expressions for some derivatives.
Corollary 5. 
For m ≥ q ≥ 1 , the derivatives D q F m μ ( x ) can be written as combinations of the ultraspherical polynomials U i ( λ ) ( x ) , Hermite polynomials H i ( x ) , generalized Fibonacci polynomials G F i R , T ( x ) , and generalized Lucas polynomials G L i A , B ( x ) as follows:
D q F m μ ( x ) = 2 − m + q − 2 λ π ( m + 2 ) ! Γ 1 2 + λ ∑ s = 0 m − q 2 ( m − 2 s − q + λ ) Γ ( m − 2 s − q + 2 λ ) s ! ( m − 2 s − q ) ! Γ ( 1 + m − s − q + λ ) × F 1 2 − s , − m + s + q − λ − 2 − m | − 4 U m − q − 2 s ( λ ) ( x ) ,
D q F m μ ( x ) = 2 − 1 − m + q ( m + 2 ) ! ∑ s = 0 m − q 2 1 s ! ( m − 2 s − q ) ! × F 1 1 ( − s ; − 2 − m ; 4 ) H m − q − 2 s ( x ) ,
D q F m μ ( x ) = R − m + q ( μ ) m ∑ s = 0 m − q 2 ( − 1 ) s + 1 T s ( − 1 − m + 2 s + q ) s ! ( m − s − q + 1 ) ! × F 1 2 − s , − 1 − m + s + q 1 − m − μ | R 2 T G F m − q − 2 s R , T ( x ) ,
D q F m μ ( x ) = A − m + q ( μ ) m ∑ s = 0 m − q 2 c m − q − 2 s ( − B ) s s ! ( m − s − q ) ! × F 1 2 − s , − m + s + q 1 − m − μ | A 2 B G L m − q − 2 s A , B ( x ) ,
and
c m = 1 2 , m = 0 , 1 , m ≥ 1 .
Proof. 
The above formulas can be deduced by applying Theorem 7, considering the inversion coefficients of the different symmetric polynomials. □

7.2. Derivatives of the CFPs in Terms of Non-Symmetric Polynomials

This section presents a general theorem for computing the derivatives of the CFPs in terms of any family of non-symmetric polynomials expressed as in (12). Some specific formulas will also be presented.
Theorem 8. 
Let ψ i ( x ) be any non-symmetric polynomial that can be expressed as in (12). In terms of ψ i ( x ) , the following derivative expression holds:
D q F m μ ( x ) = 1 Γ ( μ ) ∑ s = 0 m − q 2 ∑ r = 0 s Γ ( m + μ − r ) r ! ( m − q − 2 r ) ! M ¯ 2 s − 2 r , m − 2 r − q ψ m − q − 2 s ( x ) + 1 Γ ( μ ) ∑ s = 0 1 2 ( m − q − 1 ) ∑ r = 0 s Γ ( m + μ − r ) r ! ( m − q − 2 r ) ! M ¯ 2 s − 2 r + 1 , m − 2 r − q ψ m − q − 2 s − 1 ( x ) ,
where the coefficients M ¯ r , s are the inversion coefficients of ψ i ( x ) in (14).
Proof. 
Starting from Formula (12) and making use of the inversion Formula (14), we can write
D q F m μ ( x ) = 1 Γ ( μ ) ∑ r = 0 m − q 2 Γ ( m + μ − r ) r ! ( m − q − 2 r ) ! ∑ t = 0 m − 2 r − q M ¯ t , m − 2 r − q ψ m − 2 r − q − t ( x ) ,
which can be transformed after some lengthy manipulations into (68). □
Remark 10. 
It is evident from Theorem 8 that we can obtain an expression of the derivatives in terms of any symmetric polynomial once the coefficients M ¯ r , s are known. The following corollary exhibits some derivative expressions.
Corollary 6. 
For m ≥ q ≥ 1 , the derivatives D q F m μ ( x ) can be written as combinations of Bernoulli polynomials B i ( x ) , the generalized Laguerre polynomials L i α ( x ) , and shifted Jacobi polynomials R ˜ ℓ ( ν , θ ) ( x ) as follows:
D q F m μ ( x ) = ( μ ) m ∑ s = 0 m − q 2 1 ( 2 s + 1 ) ! ( m − 2 s − q ) ! F 1 2 − s , − 1 2 − s 1 − m − μ | − 4 B m − q − 2 s ( x ) + 1 Γ ( μ ) ∑ s = 0 1 2 ( m − q − 1 ) Γ ( − 1 + m + μ − s ) ( 2 s + 2 ) ! + Γ ( m + μ ) ( s + 1 ) ! F 1 2 − 1 − s , − 1 2 − s 1 − m − μ | − 4 ( s + 1 ) ! ( 2 s + 2 ) ! ( m − 2 s − q − 1 ) ! × B m − q − 2 s − 1 ( x ) ,
D q F m μ ( x ) = ( − 1 ) m − q ( μ ) m Γ ( 1 + m − q + α ) × ∑ s = 0 m − q 2 1 ( 2 s ) ! Γ ( 1 + m − 2 s − q + α ) F 3 2 − s , 1 2 − s 1 − m − μ , − m 2 + q 2 − α 2 , 1 2 − m 2 + q 2 − α 2 | − 1 L m − q − 2 s α ( x ) − ∑ s = 0 1 2 ( m − q − 1 ) 1 ( 2 s + 1 ) ! Γ ( m − 2 s − q + α ) F 3 2 − s , − 1 2 − s 1 − m − μ , − m 2 + q 2 − α 2 , 1 2 − m 2 + q 2 − α 2 | − 1 L m − q − 2 s − 1 α ( x ) ,
D q F m μ ( x ) = ( μ ) m Γ ( 1 + m − q + β ) Γ ( 1 + α ) × ( ∑ s = 0 m − q 2 ( 1 + 2 m − 4 s − 2 q + α + β ) Γ ( 1 + m − 2 s − q + α ) Γ ( 1 + m − 2 s − q + α + β ) ( 2 s ) ! ( m − 2 s − q ) ! Γ ( 1 + m − 2 s − q + β ) Γ ( 2 + 2 m − 2 s − 2 q + α + β ) × F 3 4 − s , 1 2 − s , − 1 2 − m + s + q − α 2 − β 2 , − m + s + q − α 2 − β 2 1 − m − μ , − m 2 + q 2 − β 2 , 1 2 − m 2 + q 2 − β 2 | − 4 R ˜ m − q − 2 s ( α , β ) ( x ) + ∑ s = 0 1 2 ( m − q − 1 ) ( − 1 + 2 m − 4 s − 2 q + α + β ) Γ ( m − 2 s − q + α ) Γ ( m − 2 s − q + α + β ) ( 2 s + 1 ) ! ( m − 2 s − q − 1 ) ! Γ ( m − 2 s − q + β ) Γ ( 1 + 2 m − 2 s − 2 q + α + β ) × F 3 4 − s , − 1 2 − s , − m + s + q − α 2 − β 2 , 1 2 − m + s + q − α 2 − β 2 1 − m − μ , − m 2 + q 2 − β 2 , 1 2 − m 2 + q 2 − β 2 | − 4 R ˜ m − q − 2 s − 1 ( α , β ) ( x ) ) .
Proof. 
The above formulas can be deduced by applying Theorem 8, considering the inversion coefficients of the different non-symmetric polynomials [13]. □

8. Some New Definite and Weighted Definite Integral Formulas Involving the CFPs

Some definite and weighted definite integrals will be deduced as an application to some of the derived formulas. More precisely, some connection and linearization formulas involving the CFPs will be the keys to deriving such integral formulas.

8.1. Some Definite Integrals

Corollary 7. 
For a non-negative integer r, one has
∫ 0 1 F r μ ( x ) d x = W r ,
with
W r = ( μ ) r F 1 2 − 1 2 ( r + 1 ) , − r 2 1 − r − μ | − 4 ( r + 1 ) ! , i f r e v e n , − r + 1 ! Γ 1 2 − 1 + r + μ + r + 1 2 ! Γ ( r + μ ) F 1 2 − 1 2 ( r + 1 ) , − r 2 1 − r − μ | − 4 ( r + 1 ) ! r + 1 2 ! Γ ( μ ) , i f r o d d .
Proof. 
From the derivative formula in (69), and setting q = 0 , we get the following connection formula between the CFPs and Bernoulli polynomials:
F r μ ( x ) = ( μ ) r ∑ m = 0 r 2 1 ( 2 m + 1 ) ! ( r − 2 m ) ! F 1 2 − m , − 1 2 − m 1 − r − μ | − 4 B r − 2 m ( x ) + 1 Γ ( μ ) ∑ m = 0 r − 1 2 − Γ ( − 1 + r + μ − m ) ( 2 m + 2 ) ! + Γ ( r + μ ) ( m + 1 ) ! ( m + 1 ) ! ( 2 m + 2 ) ! ( r − 2 m − 1 ) ! × F 1 2 − 1 − m , − 1 2 − m 1 − r − μ | − 4 B r − 2 m − 1 ( x ) .
Based on the identity [4]:
∫ 0 1 B m ( x ) d x = 1 , m = 0 , 0 , m > 0 ,
it is easy to conclude Formula (72). □
Corollary 8. 
Consider two non-negative integers r and ℓ. The following integral formula holds:
∫ 0 1 F r μ ( x ) F ℓ μ ( x ) d x = ∫ 0 1 F r μ ( x ) F ℓ μ ( x ) d x = ∑ s = 0 min ( r , ℓ ) U s , r , ℓ ,
where
U s , r , ℓ = ( − 1 ) s + 1 ( − r − ℓ + 2 s − μ ) ( r + ℓ − 2 s ) ! Γ ( r − s + μ ) Γ ( ℓ − s + μ ) ( μ ) s ( r + ℓ − 2 s + 2 μ ) s ( 1 + r + ℓ − 2 s ) ! ( r − s ) ! ( ℓ − s ) ! s ! Γ 1 2 ( 3 + r + ℓ − 2 s ) Γ ( μ ) 2 Γ ( 1 + r + ℓ − s + μ ) × Γ 1 2 ( 3 + r + ℓ − 2 s ) Γ ( r + ℓ − 2 s + μ ) F 1 2 − 1 2 ( 1 + r + ℓ ) + s , − r 2 − ℓ 2 + s 1 − r − ℓ + 2 s − μ | − 4 , ( r + ℓ ) e v e n , 1 2 ( 1 + r + ℓ − 2 s ) ! Γ ( r + ℓ − 2 s + μ ) F 1 2 − 1 2 ( 1 + r + ℓ ) + s , − r 2 − ℓ 2 + s 1 − r − ℓ + 2 s − μ | − 4 − ( 1 + r + ℓ − 2 s ) ! Γ 1 2 ( − 1 + r + ℓ − 2 s ) + μ , ( r + ℓ ) o d d .
Proof. 
The product Formula (49) enables one to write
∫ 0 1 F r μ ( x ) F ℓ μ ( x ) d x = ∑ s = 0 min ( r , ℓ ) Z s , r , ℓ ∫ 0 1 F r + ℓ − 2 s μ ( x ) d x ,
with
Z s , r , ℓ = ( − 1 ) s ( r + ℓ − 2 s + μ ) ( r + ℓ − 2 s ) ! Γ ( r − s + μ ) Γ ( ℓ − s + μ ) ( μ ) s ( r + ℓ − 2 s + 2 μ ) s Γ ( μ ) ( r − s ) ! ( ℓ − s ) ! s ! Γ ( 1 + r + ℓ − s + μ ) .
Now Formula (72) leads to (74). □

8.2. An Example of Weighted Integrals

Some weighted integral formulas can be obtained by using the derivatives and connection formulas of the CFPs with different orthogonal polynomials. The following corollary displays one of these formulas.
Corollary 9. 
For all non-negative integers m and ℓ, the following integral formula holds for α , β > − 1 :
∫ 0 1 x β ( 1 − x ) α F m μ ( x ) R ˜ ℓ ( α , β ) ( x ) d x = A m − ℓ 2 , m h ℓ , if ( m − ℓ ) e v e n , A ¯ m − ℓ − 1 2 , m h ℓ , if ( m − ℓ ) o d d ,
where
A s , m = ( 1 + 2 m − 4 s + α + β ) Γ ( 1 + m − 2 s + α ) Γ ( 1 + m + β ) Γ ( 1 + m − 2 s + α + β ) ( μ ) m ( m − 2 s ) ! ( 2 s ) ! Γ ( 1 + α ) Γ ( 1 + m − 2 s + β ) Γ ( 2 + 2 m − 2 s + α + β ) × F 3 4 − s , 1 2 − s , − 1 2 − m + s − α 2 − β 2 , − m + s − α 2 − β 2 − m 2 − β 2 , 1 2 − m 2 − β 2 , 1 − m − μ | − 4 ,
A ¯ s , m = ( − 1 + 2 m − 4 s + α + β ) Γ ( m − 2 s + α ) Γ ( 1 + m + β ) Γ ( m − 2 s + α + β ) ( μ ) m ( m − 2 s − 1 ) ! ( 2 s + 1 ) ! Γ ( 1 + α ) Γ ( m − 2 s + β ) Γ ( 1 + 2 m − 2 s + α + β ) × F 3 4 − s , − 1 2 − s , − m + s − α 2 − β 2 , 1 2 − m + s − α 2 − β 2 − m 2 − β 2 , 1 2 − m 2 − β 2 , 1 − m − μ | − 4 ,
and h ℓ is given by
h ℓ = ℓ ! Γ ( ℓ + β + 1 ) Γ ( α + 1 ) 2 ( 2 ℓ + λ ) Γ ( ℓ + λ ) Γ ( k + α + 1 ) .
Proof. 
From the derivative Formula (71), and setting q = 0 , we obtain the following connection formula:
F m μ ( x ) = ∑ s = 0 m 2 A s , m R ˜ m − 2 s ( α , β ) ( x ) + ∑ s = 0 m − 1 2 A ¯ s , m R ˜ m − 2 s − 1 ( α , β ) ( x ) ,
where A s , m and A ¯ s , m are as given in (77), and (78), respectively. Multiplying both sides of (80) by w ( x ) = x β ( 1 − x ) α and integrating from 0 to 1, we obtain
∫ 0 1 w ( x ) F μ ( x ) R ˜ ℓ ( α , β ) ( x ) d x = ∑ s = 0 m 2 A s , m ∫ 0 1 w ( x ) R ˜ ℓ ( α , β ) ( x ) R ˜ m − 2 s ( α , β ) ( x ) d x + ∑ s = 0 m − 1 2 A ¯ s , m ∫ 0 1 w ( x ) R ˜ ℓ ( α , β ) ( x ) R ˜ m − 2 s − 1 ( α , β ) ( x ) d x .
The application of the orthogonality relation in (16) yields
∫ 0 1 w ( x ) F m μ ( x ) R ˜ ℓ ( α , β ) ( x ) d x = A m − ℓ 2 , m h ℓ , if ( m − ℓ ) e v e n , A ¯ m − ℓ − 1 2 , m h ℓ , if ( m − ℓ ) o d d ,
where h ℓ is given by (79). This proves the result. □

9. A Matrix Approach to the CFPs

This section is devoted to following another approach to derive some characteristics and formulas related to the CFPs.

9.1. Coefficients of the CFPs

In general, the convolved Fibonacci polynomials of degree i can be written in the following monomial form
F j μ ( x ) = ∑ k = 0 j t j , k μ x k , t j , k μ ∈ IR , j ≥ 0 ,
where
t j , k μ = ( μ ) ⌊ j + k 2 ⌋ j − k 2 ! k ! , ( j − k ) even , 0 , ( j − k ) odd .
This form includes the odd polynomials
F j o , μ ( x ) = ∑ k = 0 j 2 t j , k o , μ x j − 2 k , t j , k o , μ = ( μ ) 2 j + 1 − k 2 j + 1 − 2 k ! k ! , k ≤ j , 0 , otherwise ,
and the even convolved Fibonacci polynomials
F j e , μ ( x ) = ∑ k = 0 j 2 t j , k e , μ x j − 2 k , t j , k e , μ = ( μ ) 2 j − k 2 j − 2 k ! k ! , k ≤ j , 0 , otherwise .
From (6), it is clear that the coefficients t j , k μ satisfy the recurrence relation
t j , k μ = μ + j − 1 j t j − 1 , k − 1 μ + 2 μ + j − 2 j t j − 2 , k μ , j ≥ k ≥ 2 ,
with the initial values
t 0 , j μ = δ j , 0 , t j , 0 μ = ( μ ) j 2 j 2 ! , j even , 0 , j odd , t 1 , k μ = ( μ ) , k = 1 , 0 , k ≠ 1 ,
where δ j , i is Kronecker’s delta symbol.
Since the coefficients are all positive, we understand that the convolved Fibonacci polynomials F j μ ( x ) are positive functions if j is even. If j is odd, then F j μ ( x ) ≥ 0 for x ≥ 0 and F j μ ( x ) < 0 for x < 0 .
Moreover, from (9) we have F j μ ( − x ) = ( − 1 ) i F j μ ( x ) . Therefore F j μ ( x ) are even functions if j is even and are odd functions if j is odd.

9.2. Vector Representation of the CFPs

Let F i μ i ∈ IN be the polynomial sequence whose elements are the convolved Fibonacci polynomials. If we set T j μ = t j , 0 μ , t j , 1 μ , … , t j , j μ and X j = 1 , x , x 2 , … , x j T , then, each element F j μ x of the sequence can be written in the following matrix form
F j μ x = T j μ X j .
Now, let F μ x = F 0 μ ( x ) , F 1 μ ( x ) , … , F i μ ( x ) , … T and If X = [ 1 , x , x 2 , ⋯ ] , we have
F μ x = T μ X ,
where T μ = t i , j μ i , j ≥ 0 is a lower triangular infinite matrix. The matrix T μ (related to F μ x ) can be factorized [48] as
T μ = D 1 B μ D 1 − 1 ,
where D 1 = d i a g { i ! i ≥ 0 and B μ = b i , j μ i , j ≥ 0 is an infinite lower triangular matrix with entries
b i , j μ = j ! i ! t i , j μ , j ≤ i , 0 j > i ,
T μ = 1 0 0 0 0 … … … … 0 μ 0 0 0 μ 0 μ ( 1 + μ ) 2 0 0 ⋱ 0 μ ( 1 + μ ) 0 μ ( 1 + μ ) ( 2 + μ ) 6 0 ⋱ μ ( 1 + μ ) 2 0 μ ( 1 + μ ) ( 2 + μ ) 2 0 μ ( 1 + μ ) ( 2 + μ ) ( 3 + μ ) 24 0 ⋱ 0 μ ( 1 + μ ) ( 2 + μ ) 2 0 μ ( 1 + μ ) ( 2 + μ ) ( 3 + μ ) 6 0 ⋱ ⋱ ⋮ ⋮ ⋮ ⋮ ⋱ ⋱ ⋱ .
For μ = 1 the matrix T μ coincides with the matrix B mentioned in [46].
T μ is an invertible matrix since its diagonal elements are t k , k μ = μ k k ! , ∀ k , which are clearly non-zero. Let S μ = s i , j μ i , j ≥ 0 be the inverse of T μ . The matrix S μ is also an infinite lower triangular matrix and can be written as
S μ = D 1 B μ − 1 D 1 − 1 ,
where B μ − 1 can be easily computed (see for example [49]).
Alternatively, the entries of S μ can be directly calculated by the following algorithm. Let T k , m be the square block of order k − m + 1 of T μ with the lower-left element t k , m μ , the upper-left element t m , m μ and the lower-right element t k , k μ . Similarly, we define the blocks S k , m of S μ and the blocks I k , m of the identity matrix I. Then we have
T k , m S k , m = I k , m , k ≥ m .
Relation (81) is a system in the unknowns s k , m μ . The solution of the system gives s k , k μ = 1 t k , k μ = k ! μ k and, for k > m ,
s k , m μ = ( − 1 ) k − m ∏ i = m k t i , i μ t m + 1 , m μ t m + 1 , m + 1 μ 0 … … 0 t m + 2 , m μ t m + 2 , m + 1 μ t m + 2 , m + 2 μ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋱ 0 ⋮ ⋱ t k − 1 , k − 1 μ t k , m μ ⋯ ⋯ ⋯ t k , k − 2 μ t k , k − 1 μ .
We observe that the above determinant is a Hessenberg determinant. It can be calculated recursively [50] or by Gauss elimination.

9.3. The Conjugate Sequence

Let F ¯ i μ i ∈ IN be the conjugate sequence of F i μ i ∈ IN , that is [48], the polynomial sequence such that F i μ ∘ F ¯ i μ = i d i , where " ∘ " denotes the umbral composition and i d i i ∈ IN , with i d i x = x i , the polynomial sequence related to the identity matrix I.
The sequence F ¯ i μ i ∈ IN is related to the matrix S μ in the sense that
F ¯ i μ ( x ) = ∑ k = 0 i s i , k μ x k , s i , k μ ∈ IR , i ≥ 0 .
Let us denote by F ¯ μ x the infinite vector F ¯ μ x = F ¯ 0 μ x , F ¯ 1 μ x , … , F ¯ i μ x , … T . It is easy to prove the following relations:
F μ x = T μ X = T μ 2 F ¯ μ x , and F ¯ μ x = S μ X = S μ 2 F μ x .
For every i ∈ IN , let T μ , i and S μ , i be the principal sub-matrices of order i of T μ and S μ , respectively. Moreover, let
F μ , i x = F 0 μ x , F 1 μ x , … , F i μ x T , and F ¯ μ , i x = F ¯ 0 μ x , F ¯ 1 μ x , … , F ¯ i μ x T .
Then
F μ , i x = T μ , i X i = T μ , i 2 F ¯ μ , i x ,
and
F ¯ μ , i x = S μ , i X i = S μ , i 2 F μ , i x .
Remark 11. 
From the first equality in (82), we obtain
x j = ∑ k = 0 j s j , k μ F k μ ( x ) , j = 0 , ⋯ , i .
Comparing (10) and (83), we obtain the following expression for the elements of the matrix S μ :
s j , k μ = j ! Γ ( μ ) ( − 1 ) α j k ( j + μ − 2 α j k ) Γ ( 1 + j + μ − α j k ) α j k ! , ( j − k ) e v e n , 0 , ( j − k ) o d d ,
with α i k = ⌊ j − k 2 ⌋ .

9.4. Recurrence Relations and Determinant Forms

From (83), F i μ x can be written as:
F i μ x = 1 s i , i μ x i − ∑ k = 1 i − 1 s i , k μ F k μ ( x ) = Γ ( i + μ + 1 ) i ! ( i + μ ) Γ ( μ ) x i − ∑ k = 1 i − 1 s i , k μ F k μ ( x ) .
Furthermore, for i ≥ 0 , relations (83) can be thought of as an infinite linear system in the unknowns F k μ ( x ) , k ≥ 0 . Solving the first ( i + 1 ) equations in the unknowns F 0 μ ( x ) , ⋯ , F i μ ( x ) by Cramer’s rule yields the following determinant form
F i μ x = ( − 1 ) i ∏ k = 0 i Γ ( k + μ + 1 ) k ! ( k + μ ) Γ ( μ ) 1 x x 2 x 3 ⋯ x i − 1 x i s 0 , 0 μ 0 s 2 , 0 μ 0 ⋯ s i − 1 , 0 μ s i , 0 μ 0 s 1 , 1 μ 0 s 3 , 1 μ ⋯ s i − 1 , 1 μ s i , 1 μ ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ 0 ⋯ ⋯ ⋯ 0 s i − 1 , i − 1 μ 0 , i ≥ 1 .
Similarly, from (82), ∀ i ∈ IN , we get x i = ∑ k = 0 i t i , k μ F ¯ k μ ( x ) . Therefore, the conjugate polynomials can be expressed as
F ¯ i μ x = 1 t i , i μ x i − ∑ k = 1 i − 1 t i , k μ F ¯ k μ ( x ) ,
and we obtain
F ¯ i μ x = − 1 i ∏ k = 0 i k ! ( μ ) k 1 x x 2 x 3 ⋯ x i − 1 x i t 0 , 0 μ 0 t 2 , 0 μ 0 ⋯ t i − 1 , 0 μ t i , 0 μ 0 t 1 , 1 μ 0 t 3 , 1 μ ⋯ t i − 1 , 1 μ t i , 1 μ ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ 0 ⋯ ⋯ ⋯ 0 t i − 1 , i − 1 μ 0 , i ≥ 1 .

9.5. Derivation Matrix

Derivation matrices are used to solve many engineering and physical problems and are employed in several areas of numerical analysis, such as differential equations, integral equations, etc.
Let F μ , i x ′ = F 0 μ ′ ( x ) , F 1 μ ′ ( x ) , … , F i μ ′ ( x ) T be the vector whose elements are the first derivative of the polynomials F k μ ( x ) , k = 0 , ⋯ , i . The derivation matrix of the polynomial sequence F i μ i ∈ IN is the matrix D ˜ = d k , j k , j ≥ 0 such that
F μ , i x ′ = D ˜ F μ , i x .
The entries of the derivation matrix of a polynomial sequence related to matrix T = t i , j i , j ≥ 0 are given by [51]
d k , j = ∑ i = j k i + 1 t k , i + 1 s i , j , k > j , 0 , otherwise .
Therefore, the elements of D ˜ in (84) are
d i , j = ( − 1 ) i − j + 1 2 ( μ + j ) , i + j odd , 0 , i + j even i > j .
From (84), for higher-order derivatives we obtain
F μ , i x ( q ) = D ˜ q F μ , i x , q ≥ 1 .
Particularly, for q = 1 ,
F i μ ′ ( x ) = D F i μ ( x ) = ∑ s = 0 ⌊ i − 1 2 ⌋ ( − 1 ) s ( i + μ − 1 − 2 s ) F μ , i − 2 s − 1 ( x ) .
The same formulas can be obtained by (29).
Observe that, from (85), the derivation matrix D is a nilpotent matrix. In fact, it is an upper triangular matrix whose main diagonal elements are zeros. Therefore, D s = 0 for s > i .

9.6. Basis

Observe that each of the two sets F 0 μ x , ⋯ , F i μ x and F ¯ 0 μ x , ⋯ , F ¯ i μ x forms a basis for P i , the set of polynomials of degree ≤ i .
Hence, if q i ( x ) = ∑ k = 0 i q i , k x k ∈ P i , then
q i ( x ) = ∑ k = 0 i c i , k F k μ ( x ) = ∑ k = 0 i d i , k F ¯ k μ ( x ) ,
with c i , k = ∑ j = 0 ⌊ i − k 2 ⌋ q i , i − k − 2 j s k + 2 j , k μ , d i , k = ∑ j = 0 ⌊ i − k 2 ⌋ q i , i − k − 2 j t k + 2 j , k μ , k = 0 , 1 , … , i .

10. Conclusions

The paper has established and developed new formulas for the CFPs, representing notable progress in the research of generalized polynomials. We have developed important formulas of repeated integrals and the higher-order derivatives of the CFPs. Other generalizing formulas are developed based on some essential formulas of these polynomials. The hypergeometric functions have important roles since most derived expressions involve these functions. These hypergeometric functions may be summed in closed forms using standard formulas or computer algebra algorithms. In such cases, these formulas turn into simplified forms. Some integral and weighted definite integral formulae are given based on some derived formulas. A matrix approach for these polynomials was also followed. The formulas in this paper may be helpful in many applications, particularly in numerical analysis. We hope to employ these formulas to solve differential equations in future work. Another target for us is to develop new formulas for other generalizations of Fibonacci and other polynomials.

Author Contributions

Conceptualization, W.M.A.-E. and A.N.; Methodology, W.M.A.-E., O.M.A. and A.N.; Software, W.M.A.-E. and A.N.; Formal analysis, W.M.A.-E., O.M.A. and A.N.; Validation, W.M.A.-E., O.M.A. and A.N.; Writing—original draft, W.M.A.-E. and A.N.; Writing—review & editing, W.M.A.-E., O.M.A. and A.N. All authors have read and agreed to the published version of the manuscript.

Funding

No funding received for this paper.

Data Availability Statement

Data are contained within the article.

Acknowledgments

The authors would like to thank the editorial office of the mathematics journal for its support and the referees for their constructive comments, which helped us enhance the paper in its present form.

Conflicts of Interest

The authors declare no conflict of interest.

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Abd-Elhameed, W.M.; Alqubori, O.M.; Napoli, A. On Convolved Fibonacci Polynomials. Mathematics 2025, 13, 22. https://doi.org/10.3390/math13010022

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Abd-Elhameed WM, Alqubori OM, Napoli A. On Convolved Fibonacci Polynomials. Mathematics. 2025; 13(1):22. https://doi.org/10.3390/math13010022

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Abd-Elhameed, Waleed Mohamed, Omar Mazen Alqubori, and Anna Napoli. 2025. "On Convolved Fibonacci Polynomials" Mathematics 13, no. 1: 22. https://doi.org/10.3390/math13010022

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Abd-Elhameed, W. M., Alqubori, O. M., & Napoli, A. (2025). On Convolved Fibonacci Polynomials. Mathematics, 13(1), 22. https://doi.org/10.3390/math13010022

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