Novel Expressions for Certain Generalized Leonardo Polynomials and Their Associated Numbers

Abstract

This article introduces new polynomials that extend the standard Leonardo numbers, generalizing Fibonacci and Lucas polynomials. A new power form representation is developed for these polynomials, which is crucial for deriving further formulas. This article also presents two connection formulas linking these generalized polynomials to the Fibonacci and Lucas polynomials, as well as several identities involving some generalized and specific Leonardo numbers. Additionally, new product formulas involving the generalized Leonardo polynomials with the Fibonacci and Lucas polynomials are provided, along with computations of definite integrals based on the derived formulas.

1. Introduction

Various sequences of numbers hold significant importance due to their versatility in both academic research and practical applications. They are crucial in mathematical modeling, providing insights into structures and patterns characterized by spatial or temporal invariance. For example, Fibonacci sequences play a key role in modeling biological development patterns, including population dynamics and the arrangement of leaves. They are also employed in financial markets for technical analyses through Fibonacci retracements. Prime numbers form the basis of modern cryptography, enabling secure encrypted communication. Harmonic sequences are utilized in physics to represent oscillatory systems and in computer science to assess algorithmic complexity, notably to evaluate the efficiency of the merge sort algorithm. Catalan numbers are vital in combinatorics for enumerating binary trees, balanced parentheses, and pathways on lattice grids, with additional applications in compiler design and syntax analysis. Across various fields—including biology, computer science, physics, and finance—sequences of numbers bridge abstract mathematical theory and practical applications, underscoring their significant relevance. One can refer to [1,2,3] for specific applications.

Among the important sequences of numbers are the Fibonacci and Lucas numbers and their corresponding polynomials. Many mathematicians are interested in investigating these sequences from both theoretical and practical perspectives. In [4], the authors developed some formulas for the reciprocal sums of Fibonacci and Lucas polynomials. Other formulas regarding the Fibonacci and Lucas polynomials involving the golden ratio were found in [5]. The authors of [6] found formulas for Lucas polynomials and employed them to form a numerical point of view for treating a time-fractional diffusion equation. Many modifications and generalizations of Fibonacci and Lucas polynomials have been introduced and studied by many authors. The authors of [7] employed two generalized classes of polynomials to reduce certain radicals. The authors of [8] introduced a class of shifted Lucas polynomials and employed them to treat the time-fractional FitzHugh–Nagumo differential equation. The authors of [9] investigated certain generalized Fibonacci polynomials. Generalized distance Fibonacci sequences were developed in [10]. In [11], generalized bi-periodic Fibonacci and Lucas polynomial identities were developed. Supersymmetric Fibonacci polynomials were introduced in [12]. Other contributions regarding Fibonacci polynomials and related polynomials can be found in [13,14,15,16,17].

Leonardo numbers and their generalized numbers are of interest. They have connections with well-known sequences, such as Fibonacci and Lucas numbers, and their generalized numbers, such as the generalized Horadam numbers [18]. There are many contributions regarding Leonardo numbers, their modifications, and generalizations. For example, the authors of [19] studied certain properties of these numbers and found expressions of sums and products for them. In [20], Soykan investigated generalized Leonardo sequences and considered some special cases while introducing certain polynomials that satisfied linear third-order recurrence relations, namely the generalized Horadam–Leonardo polynomials, in [21]. In addition, he extracted many particular polynomials from the generalized polynomials introduced. In [22], certain formulas regarding the Gaussian–Leonardo hybrid polynomials were developed. Dual Leonardo numbers were introduced in [23]. Polynomials with generalized Leonardo numbers as coefficients were investigated in [24]. In [25], the authors developed Leonardo and hyper-Leonardo numbers using Riordan arrays. In [26], Leonardo Pisano hybrinomials were introduced. Prasad and Kumari, in [27], introduced certain Leonardo polynomials and developed new formulas for them. In [28], the authors derived determinant formulas for Toeplitz–Hessenberg matrices with generalized Leonardo number entries. Leonardo polynomials were generalized using bivariate and complex polynomials in [29]. In [30], the periodic characteristics of certain Leonardo sequences were presented. In [31], incomplete Leonardo numbers were defined, and some of their properties were developed. In [32], combinatorial identities for generalized Leonardo numbers were given. New families of generalized k-Leonardo and Gaussian–Leonardo numbers were introduced in [33]. In [34], certain generalized Leonardo numbers were defined, and matrix representations were given.

The so-called connection and linearization problems for different polynomials are important problems in special functions. If we have two sets of polynomials ${\left\{{\varphi }_i\left(x\right)\right\}}_{i\ge 0}$ and ${\left\{{\psi }_j\left(x\right)\right\}}_{j\ge 0}$, then to find the two connection formulas between them, we have to find the connection coefficients $C_{k,i}$ and ${\overline{C}}_{k,i}$ such that

$${\varphi }_i\left(x\right)=\sum _{k=0}^i{C}_{k,i}{\psi }_k\left(x\right),$$

(1)

and

$${\psi }_i\left(x\right)=\sum _{k=0}^i{\overline{C}}_{k,i}{\varphi }_k\left(x\right).$$

(2)

Regarding the linearization problem, if we have three sets of polynomials, ${\left\{{\varphi }_i\left(x\right)\right\}}_{i\ge 0}$, ${\left\{{\psi }_j\left(x\right)\right\}}_{j\ge 0}$, and ${\left\{{\theta }_k\left(x\right)\right\}}_{k\ge 0}$, then the problem

$${\varphi }_i\left(x\right){\psi }_j\left(x\right)=\sum _{k=0}^{i+j}{L}_{k,i,j}{\theta }_k\left(x\right),$$

(3)

is called the general linearization problem. Connection and linearization problems for different polynomials have been studied by many authors; see, for example, [35,36,37,38].

The primary objective of this article is to provide novel formulas of certain generalized Leonardo polynomials that generalize both Fibonacci and Lucas polynomials and their combined polynomials, as well as the standard Leonardo numbers. Furthermore, we provide further relationships between these polynomials and Fibonacci and Lucas polynomials. To the best of our knowledge, formulas for these polynomials have not been introduced in the literature.

The rest of this paper is organized as follows: The next section gives some characteristics of the Fibonacci and Lucas sequences. We also give an account of Leonardo numbers and polynomials in this section. New expressions for generalized Leonardo polynomials are given in Section 3. New connection formulas for generalized Leonardo polynomials with Fibonacci and Lucas polynomials are developed in Section 4. Some new identities involving generalized Leonardo numbers and a combining sequence of Fibonacci and Lucas numbers are presented in Section 5. Product formulas with Fibonacci and Lucas polynomials are presented in Section 6. Some closed definite integral formulas are presented in Section 7. Some concluding remarks are reported in Section 8.

2. An Overview of Fibonacci, Lucas, and Leonardo Polynomials

2.1. Fibonacci and Lucas Polynomials

It is well known that the Fibonacci and Lucas polynomials can be generated using the following recurrence formulas [3]:

$$\begin{array}{cc}\hfill F_m\left(x\right)& =x{F}_{m-1}\left(x\right)+F_{m-2}\left(x\right),m\ge 2,F_0\left(x\right)=0,F_1\left(x\right)=1,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill L_m\left(x\right)& =x{L}_{m-1}\left(x\right)+L_{m-2}\left(x\right),m\ge 2,L_0\left(x\right)=2,L_1\left(x\right)=x.\hfill \end{array}$$

(5)

The Binet-type formulas for these polynomials are

$$\begin{array}{cc}\hfill F_m\left(x\right)& ={\displaystyle \frac{{\alpha }^m\left(x\right)-{\beta }^m\left(x\right)}{\sqrt{x^2+4}}},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill L_m\left(x\right)& ={\alpha }^m\left(x\right)+{\beta }^m\left(x\right),\hfill \end{array}$$

(7)

where

$$\begin{array}{c}\hfill \alpha \left(x\right)={\displaystyle \frac{x+\sqrt{x^2+4}}{2}},\end{array}$$

(8)

$$\begin{array}{c}\hfill \beta \left(x\right)={\displaystyle \frac{x-\sqrt{x^2+4}}{2}}.\end{array}$$

(9)

In addition, they can be written in the following explicit expressions:

$$\begin{array}{cc}\hfill F_m\left(x\right)=& \sum _{s=0}^{⌊{\displaystyle \frac{m-1}{2}}⌋}\left({\displaystyle \genfrac{}{}{0pt}{}{m-s-1}{s}}\right)x^{m-2s-1},m\ge 0,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill L_m\left(x\right)=& m\sum _{s=0}^{⌊{\displaystyle \frac{m}{2}}⌋}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{m-s}{s}\right)}{m-s}}{x}^{m-2s},m\ge 1.\hfill \end{array}$$

(11)

The inverse expressions for (10) and (11) have the following forms:

$$\begin{array}{cc}\hfill x^m=& \sum _{s=0}^{⌊{\displaystyle \frac{m}{2}}⌋}{\displaystyle \frac{{(-1)}^s(m-2s+1){(m-s+2)}_{s-1}}{s!}}{F}_{m-2s+1}\left(x\right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill x^m=& \sum _{s=0}^{⌊{\displaystyle \frac{m}{2}}⌋}{\displaystyle \frac{c_{m-2s}{(-1)}^s{(m-s+1)}_s}{s!}}{L}_{m-2s}\left(x\right),\hfill \end{array}$$

(13)

with

$$c_r=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}},\hfill & r=0,\hfill \\ 1,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(14)

Remark 1.

The symbol ${(.)}_m$ that appears in Formulas (12) and (13) is the Pochhammer symbol. For a non-negative integer m, ${\left(z\right)}_m$ is defined as [39]

$${\displaystyle {\left(z\right)}_m=\frac{\Gamma (z+m)}{\Gamma \left(z\right)}.}$$

(15)

However, ${\left(z\right)}_{-m},m\ge 0$ is defined as

$${\left(z\right)}_{-m}={\displaystyle \frac{1}{{(z-m)}_m}}.$$

(16)

Remark 2.

The Fibonacci and Lucas numbers, denoted, respectively, by $F_m$ and $L_m$, are special numbers for their associated polynomials; that is, we have

$$F_m=F_m\left(1\right),L_m=L_m\left(1\right).$$

2.2. Leonardo Numbers

Leonardo numbers ${\left\{L_m\right\}}_{m\ge 0}$ are generated by the following recurrence relation [27]:

$$L_{m+2}=L_{m+1}+L_m+1,\mathrm{with}{L}_0=L_1=1.$$

(17)

The first few Leonardo numbers are 1, 1, 3, 5, 9, 15, 25, and 41.

2.3. Leonardo Polynomials

Leonardo polynomials are generalizations of Leonardo numbers. The authors of [27] introduced the polynomials generated by the following recurrence relation:

$$U_{m+2}\left(x\right)=x{U}_{m+1}\left(x\right)+U_m\left(x\right)+x;U_0\left(x\right)=1,U_1\left(x\right)=2x-1,$$

which may be written alternatively in the following third-order homogeneous equation:

$$U_{m+3}\left(x\right)=(x+1)U_{m+2}\left(x\right)-(x-1)U_{m+1}\left(x\right)-U_m\left(x\right),$$

where $U_0\left(x\right)=1$, $U_1\left(x\right)=2x-1$, and $U_2\left(x\right)=2x^2+1$. The authors in [27] proved that $U_m\left(x\right)$ can be expressed in terms of $F_m\left(x\right)$.

$$U_m\left(x\right)=2F_m\left(x\right)+1.$$

The above identity enables one to deduce many formulas concerned with $U_m\left(x\right)$.

In this paper, we aim to introduce other new generalized Leonardo polynomials ${\mathcal{GL}}_i^{a,b,T}\left(x\right)$ that involve three parameters and generalize some celebrated sequences of polynomials. These polynomials can be constructed with the aid of the following non-homogeneous recurrence relation:

$${\mathcal{GL}}_i^{a,b,T}\left(x\right)=x{\mathcal{GL}}_{i-1}^{a,b,T}\left(x\right)+{\mathcal{GL}}_{i-2}^{a,b,T}\left(x\right)+T,{\mathcal{GL}}_0^{a,b,T}\left(x\right)=a,{\mathcal{GL}}_1^{a,b,T}\left(x\right)=bx.$$

(18)

The first few Leonardo polynomials are given by

$$\begin{array}{cc}\hfill {\mathcal{GL}}_0\left(x\right)& =a,{\mathcal{GL}}_1\left(x\right)=bx,\hfill \\ \hfill {\mathcal{GL}}_2\left(x\right)& =a+T+b{x}^2,\hfill \\ \hfill {\mathcal{GL}}_3\left(x\right)& =T+ax+bx+Tx+b{x}^3,\hfill \\ \hfill {\mathcal{GL}}_4\left(x\right)& =a+2T+Tx+a{x}^2+2b{x}^2+T{x}^2+b{x}^4,\hfill \\ \hfill {\mathcal{GL}}_5\left(x\right)& =2T+2ax+bx+3Tx+T{x}^2+a{x}^3+3b{x}^3+T{x}^3+b{x}^5.\hfill \end{array}$$

We refer here to the fact that our proposed polynomials generalize the following polynomial sequences:

• The Leonardo polynomials ${\mathcal{L}}_i^{a,b}\left(x\right)$ are defined as ${\mathcal{L}}_i^{a,b}\left(x\right)={\mathcal{GL}}_i^{a,b,1}\left(x\right)$. They can be constructed using the following recurrence relation:

  $${\mathcal{L}}_i^{a,b}\left(x\right)=x{\mathcal{L}}_{i-1}^{a,b}\left(x\right)+{\mathcal{L}}_{i-2}^{a,b}\left(x\right)+1,{\mathcal{L}}_0^{a,b}\left(x\right)=a,{\mathcal{L}}_1^{a,b}\left(x\right)=bx.$$

  (19)
• The combined Fibonacci–Lucas polynomials $F{L}_i^{a,b}\left(x\right)$ that can be obtained by setting $T=0$

  $$F{L}_i^{a,b}\left(x\right)={\mathcal{GL}}_i^{a,b,0}\left(x\right).$$
  The polynomials $F{L}_i^{a,b}\left(x\right)$ satisfy the following recurrence relation:

  $$F{L}_i^{a,b}\left(x\right)=xF{L}_{i-1}^{a,b}\left(x\right)+F{L}_{i-2}^{a,b}\left(x\right),F{L}_0^{a,b}\left(x\right)=a,F{L}_1^{a,b}\left(x\right)=bx.$$

  (20)
• The Fibonacci polynomials $F_{i+1}\left(x\right)$ that can be obtained by setting $a=1,b=1,T=0$

  $$F_{i+1}\left(x\right)={\mathcal{GL}}_i^{1,1,0}\left(x\right).$$
• The Lucas polynomials $L_i\left(x\right)$ that can be obtained by setting $a=2,b=1,T=0$

  $$L_i\left(x\right)={\mathcal{GL}}_i^{2,1,0}\left(x\right).$$

Remark 3.

The sequence of generalized Leonardo numbers can be obtained from their corresponding polynomials; that is,

$${\mathcal{GL}}_i^{a,b,T}={\mathcal{GL}}_i^{a,b,T}\left(1\right).$$

Remark 4.

The sequence of generalized Leonardo numbers involves four celebrated sequences of numbers, namely Leonardo numbers, combined Fibonacci–Lucas numbers, Fibonacci numbers, and Lucas numbers. The following table displays the recurrence relations for these sequences of numbers and a few numbers for each sequence.

2.4. Comparison with Some Related Polynomials

In this part, aiming to illustrate the novelty of our approach and the presented results, we compare our introduced generalized Leonardo polynomials with some other related polynomials that were investigated in other papers. We compare between our sequence and the following two sequences here:

• This sequence was proposed by Prasad and Kumari [27]. They studied the Leonardo polynomials defined by

  $$U_{m+2}\left(x\right)=x{U}_{m+1}\left(x\right)+U_m\left(x\right)+x,U_0\left(x\right)=1,U_1\left(x\right)=2x-1.$$

  (21)
• Our proposed generalized Leonardo polynomials are generated by

  $${\mathcal{GL}}_i^{a,b,T}\left(x\right)=x{\mathcal{GL}}_{i-1}^{a,b,T}\left(x\right)+{\mathcal{GL}}_{i-2}^{a,b,T}\left(x\right)+T,{\mathcal{GL}}_0^{a,b,T}\left(x\right)=a,{\mathcal{GL}}_1^{a,b,T}\left(x\right)=bx,$$

  (22)

  which can also be written as

  $${\mathcal{GL}}_{m+3}\left(x\right)=(x+1){\mathcal{GL}}_{m+2}\left(x\right)+(1-x){\mathcal{GL}}_{m+1}\left(x\right)-{\mathcal{GL}}_m\left(x\right),$$

  (23)

  with ${\mathcal{GL}}_0\left(x\right)=a,{\mathcal{GL}}_1\left(x\right)=bx,{\mathcal{GL}}_2\left(x\right)=b{x}^2+a+T.$
• The sequence proposed by Sokyan [21] is generated by the following recurrence relation:

  $$W_{n+3}\left(x\right)=(r\left(x\right)+1)W_{n+2}\left(x\right)+\left(s\left(x\right)-r\left(x\right)\right)W_{n+1}\left(x\right)-s\left(x\right)W_n\left(x\right),$$

  (24)

  with the following initial values:

  $$W_0\left(x\right)=c_0\left(x\right),W_1\left(x\right)=c_1\left(x\right),W_2\left(x\right)=c_2\left(x\right).$$

We refer to the following important points here:

• The polynomials investigated by Prasad and Kumari in [27] are generalizations of the Leonardo numbers. These numbers can be obtained by setting $x=1$. Also, our proposed polynomials generated by (18) generalize the Leonardo numbers.
• The polynomials in (21) do not generalize Fibonacci and Lucas numbers and their associated polynomials; however, our polynomials generalize the following sequences of polynomials:

  (a)

  Leonardo polynomials and their associated numbers;

  (b)

  Fibonacci polynomials and their associated numbers;

  (c)

  Lucas polynomials and their associated numbers;

  (d)

  The combined Fibonacci and Lucas polynomials that were investigated in [40].

• Prasad and Kumari’s polynomials in [27] are Fibonacci polynomials multiplied by a factor and with a factor added; that is, they are given explicitly the following formula [27]:

  $$U_m\left(x\right)=2F_m\left(x\right)+1.$$
  Thus, it is evident that they are not generalizations of any celebrated polynomials. They are generalizations of the standard Leonardo numbers, not the generalized Leonardo numbers given in the first row of

  Table 1. Some sequences of numbers.

  Sequence Name	Recurrence Relation	Initials	First Few Terms
  Generalized Leonardo Numbers	${\mathcal{GL}}_n={\mathcal{GL}}_{n-1}+{\mathcal{GL}}_{n-2}+T$	a, b	$a,b,a+b+T,a+2b+2T,2a+3b+4T,3a+5b+7T$
  Leonardo Numbers	${\mathcal{L}}_n={\mathcal{L}}_{n-1}+{\mathcal{L}}_{n-2}+1$	a, b	$a,b,a+b+1,a+2b+2,2a+3b+4,3a+5b+7$
  Combined Fibonacci–Lucas Numbers	$F{L}_n^{a,b}=F{L}_{n-1}^{a,b}+F{L}_{n-2}^{a,b}$	a, b	$a,b,a+b,a+2b,2a+3b,3a+5b$
  Fibonacci Numbers	$F_n=F_{n-1}+F_{n-2}$	0, 1	$0,1,1,2,3,5,8,13$
  Lucas Numbers	$L_n=L_{n-1}+L_{n-2}$	2, 1	$2,1,3,4,7,11,18$

  .
• Our polynomials involve three free parameters: $a,b$, and T. This means that they involve infinite sequences based on the selection of $a,b$, and T.
• The polynomials investigated by Sokyan in [21], which are generated using the recurrence relation in (24), are considered generalizations of our generalized polynomials by taking the choices $r\left(x\right)=x,s\left(x\right)=1$.
• Although our proposed polynomials are special versions of those from [21], the results and approaches followed in this paper are different. The results obtained in [21] concentrate on using the generating function and Binet’s form. Our approach in this paper depends on developing an explicit power form representation of the generalized Leonardo polynomial. This formula will be the key to other essential problems in special functions, such as connection and linearization formulas. This approach differs from that of Prasad and Kumari in [27] and that of Sokyan in [21].
• From a partial point of view, the explicit formula for the Leonardo polynomials that will be given in the next section enables these polynomials to be selected as basis functions, which can be utilized in the scope of the numerical solutions for differential equations.

3. A New Expression for the Generalized Leonardo Polynomials

This section develops a new expression for the generalized Leonardo polynomials ${\mathcal{GL}}_i^{a,b,T}\left(x\right)$ as powers of x. This expression will be pivotal in establishing several formulas for these polynomials.

Theorem 1.

Let i be any positive integer. The following power form representation holds for ${\mathcal{GL}}_i^{a,b,T}\left(x\right)$:

$$\begin{array}{cc}\hfill {\mathcal{GL}}_i^{a,b,T}\left(x\right)=& \sum _{r=0}^{⌊{\displaystyle \frac{i}{2}}⌋}{\displaystyle \frac{((1+i-2r)(b(i-2r)+ar)+(i-r)rT){(2+i-2r)}_{r-2}}{r!}}{x}^{i-2r}\hfill \\ \hfill & +T\sum _{r=0}^{⌊{\displaystyle \frac{i-1}{2}}⌋}{\displaystyle \frac{{(i-2r+1)}_{r-1}}{(r-1)!}}{x}^{i-2r-1},\hfill \end{array}$$

(25)

where $⌊z⌋$ represents the floor function.

Proof. 

We will prove that the following expression holds:

$$\begin{array}{cc}\hfill {\mathcal{GL}}_i^{a,b,T}\left(x\right)=& \sum _{r=0}^{⌊{\displaystyle \frac{i}{2}}⌋}{F}_{r,i}{x}^{i-2r}+\sum _{r=0}^{⌊{\displaystyle \frac{i-1}{2}}⌋}{G}_{r,i}{x}^{i-2r-1},\hfill \end{array}$$

(26)

where

$$\begin{array}{cc}\hfill F_{r,i}=& {\displaystyle \frac{((1+i-2r)(b(i-2r)+ar)+(i-r)rT){(2+i-2r)}_{r-2}}{r!}},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill G_{r,i}=& {\displaystyle \frac{T{(i-2r+1)}_{r-1}}{(r-1)!}}.\hfill \end{array}$$

(28)

Now, consider the polynomials ${\eta }_i\left(x\right)$ given by

$$\begin{array}{cc}\hfill {\eta }_i\left(x\right)=& \sum _{r=0}^{⌊{\displaystyle \frac{i}{2}}⌋}{F}_{r,i}{x}^{i-2r}+\sum _{r=0}^{⌊{\displaystyle \frac{i-1}{2}}⌋}{G}_{r,i}{x}^{i-2r-1},\hfill \end{array}$$

(29)

where $F_{r,i},$ and $G_{r,i}$ are given by (27) and (28), respectively.

First, it is evident that

$${\eta }_1\left(x\right)={\mathcal{GL}}_1^{a,b,T}\left(x\right)=bx,$$

and then, to show that ${\mathcal{GL}}_k^{a,b,T}\left(x\right)={\eta }_k\left(x\right)\forall k\ge 2,$ we show that ${\eta }_i\left(x\right)$ satisfies the same recurrence relation for ${\mathcal{GL}}_i^{a,b,T}\left(x\right)$ in (18); that is, we will show that

$${\varphi }_k\left(x\right)={\eta }_{k+2}\left(x\right)-x{\eta }_{k+1}\left(x\right)-{\eta }_k\left(x\right)-T=0.$$

(30)

Now, Formula (29) enables ${\varphi }_k\left(x\right)$ to be expressed in the following form:

$$\begin{array}{cc}\hfill {\varphi }_i\left(x\right)=& \sum _{r=0}^{⌊{\displaystyle \frac{k}{2}}⌋+1}{F}_{r,k+2}{x}^{k-2r+2}+\sum _{r=0}^{⌊{\displaystyle \frac{k+1}{2}}⌋}{G}_{r,k+2}{x}^{k-2r+1}\hfill \\ \hfill & -x\left(\sum _{r=0}^{⌊{\displaystyle \frac{k+1}{2}}⌋}{F}_{r,k+1}{x}^{k-2r+1}+\sum _{r=0}^{⌊{\displaystyle \frac{k}{2}}⌋}{G}_{r,k+1}{x}^{k-2r}\right)-\left(\sum _{r=0}^{⌊{\displaystyle \frac{k}{2}}⌋}{F}_{r,k}{x}^{k-2r}+\sum _{r=0}^{⌊{\displaystyle \frac{k-1}{2}}⌋}{G}_{r,k}{x}^{k-2r-1}\right)\hfill \\ \hfill & -T.\hfill \end{array}$$

(31)

Now, we consider two cases corresponding to k being even and k being odd.

First, we prove (30), but by replacing k with $2k$, we prove that ${\varphi }_{2k}\left(x\right)=0$.

Now, we can write

$$\begin{array}{cc}\hfill {\varphi }_{2k}\left(x\right)=& \sum _{r=0}^{k+1}{F}_{r,2k+2}{x}^{2k+2-2r}+\sum _{r=0}^k{G}_{r,2k+2}{x}^{2k-2r+1}\hfill \\ \hfill & -\left(\sum _{r=0}^k{F}_{r,2k+1}{x}^{2k-2r+2}+\sum _{r=0}^k{G}_{r,2k+1}{x}^{2k-2r+1}\right)\hfill \\ \hfill & -\left(\sum _{r=0}^k{F}_{r,2k}{x}^{2k-2r}+\sum _{r=0}^{k-1}{G}_{r,2k}{x}^{2k-2r-1}\right)-T.\hfill \end{array}$$

(32)

The last formula can be written as

$${\varphi }_{2k}\left(x\right)=\sum _1+\sum _2,$$

(33)

where

$$\begin{array}{cc}\hfill \sum _1=& \sum _{r=0}^k{G}_{r,2k+2}{x}^{2k-2r+1}-\sum _{r=0}^k{G}_{r,2k+1}{x}^{2k-2r+1}-\sum _{r=0}^{k-1}{G}_{r,2k}{x}^{2k-2r-1},\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill \sum _2=& \sum _{r=0}^{k+1}{F}_{r,2k+2}{x}^{2k+2-2r}-\sum _{r=0}^k{F}_{r,2k+1}{x}^{2k-2r+2}-\sum _{r=0}^k{F}_{r,2k}{x}^{2k-2r}-T.\hfill \end{array}$$

(35)

Regarding ${\displaystyle \sum _1}$ in (34), it can be written as

$$\begin{array}{c}\hfill \sum _1=\sum _{r=0}^k(G_{r,2k+2}-G_{r,2k+1})x^{2k-2r+1}-\sum _{r=1}^k{G}_{r-1,2k}{x}^{2k-2r+1}.\end{array}$$

Form (28), we note that $G_{-1,2k}=0$, and therefore, ${\displaystyle \sum _1}$ takes the form

$$\sum _1=\sum _{r=0}^k\left(G_{r,2k+2}-G_{r,2k+1}-G_{r-1,2k}\right)x^{2k-2r+1}.$$

(36)

Regarding ${\displaystyle \sum _2}$ in (35), it can be written as

$$\begin{array}{c}\hfill \sum _2=\sum _{r=0}^{k+1}{F}_{r,2k+2}{x}^{2k+2-2r}-\sum _{r=0}^k{F}_{r,2k+1}{x}^{2k-2r+2}-\sum _{r=1}^{k+1}{F}_{r-1,2k}{x}^{2k-2r+2}-T.\end{array}$$

From (27), it can be noted that the following two identities hold:

$$F_{k+1,2k+1}=T,F_{-1,2k}=0,$$

and thus, ${\displaystyle \sum _2}$ can be written as

$$\sum _2=\sum _{r=0}^{k+1}(F_{r,2k+2}-F_{r,2k+1}-F_{r-1,2k})x^{2k-2r+2}.$$

(37)

Now, from the two sums in (36) and (37), Formula (32) can be written in the following form:

$$\begin{array}{cc}\hfill {\varphi }_{2k}\left(x\right)=& \sum _{r=0}^k(G_{r,2k+2}-G_{r-1,2k}-G_{r,2k+1})x^{2k-2r+1}\hfill \\ \hfill & +\sum _{r=0}^{k+1}(F_{r,2k+2}-F_{r,2k+1}-F_{r-1,2k})x^{2k-2r+2}.\hfill \end{array}$$

(38)

From the expressions of $F_{r,k},G_{r,k}$ in (27) and (28), some algebraic computations lead to

$$\begin{array}{c}\hfill F_{r,2k+2}-F_{r,2k+1}-F_{r-1,2k}=0,\end{array}$$

(39)

$$\begin{array}{c}\hfill G_{r,2k+2}-G_{r-1,2k}-G_{r,2k+1}=0,\end{array}$$

(40)

and therefore, ${\varphi }_{2k}\left(x\right)=0$. Similarly, we can show that ${\varphi }_{2k+1}\left(x\right)=0$. This ends the proof. □

As direct consequences of Theorem 1, the expressions for the combined Fibonacci–Lucas polynomials that are generated by the recurrence relation (20) can be deduced. The following corollary presents this result.

Corollary 1.

Let i be any positive integer. The following power form representation holds for $F_i^{a,b}\left(x\right)$:

$$F_i^{a,b}\left(x\right)=\sum _{r=0}^{⌊{\displaystyle \frac{i}{2}}⌋}{\displaystyle \frac{((1+i-2r)(b(i-2r)+ar)+(i-r)rT){(2+i-2r)}_{r-2}}{r!}}{x}^{i-2r}.$$

(41)

Proof. 

Formula (41) can easily be obtained from Formula (25) by setting $T=0$. □

4. New Connection Formulas with Fibonacci and Lucas Polynomials

This section is devoted to establishing two new connection formulas for generalized Leonardo polynomials with both Fibonacci and Lucas polynomials. These formulas are presented in the following two theorems.

4.1. Connection with Fibonacci Polynomials

Theorem 2.

Consider any positive integer j. The following generalized Leonardo–Fibonacci connection formula holds:

$${\mathcal{GL}}_j^{a,b,T}\left(x\right)=b{F}_{j+1}\left(x\right)+(a-b)F_{j-1}\left(x\right)+T\sum _{p=0}^{j-2}{F}_{p+1}\left(x\right).$$

(42)

Proof. 

We begin with the power form representation in (25), and after that, we apply the inversion formula for Fibonacci polynomials in (12) to obtain

$$\begin{array}{cc}\hfill {\mathcal{GL}}_j^{a,b,T}\left(x\right)=& \sum _{r=0}^{⌊{\displaystyle \frac{j}{2}}⌋}{\displaystyle \frac{((1+j-2r)(b(j-2r)+ar)+(j-r)rT){(2+j-2r)}_{r-2}}{r!}}\times \hfill \\ \hfill & \sum _{s=0}^{⌊{\displaystyle \frac{j}{2}}⌋-r}{\displaystyle \frac{{(-1)}^s(1+j-2r-2s){(2+j-2r-s)}_{s-1}}{s!}}{F}_{j-2r-2s+1}\left(x\right)\hfill \\ \hfill & +T\sum _{r=0}^{⌊{\displaystyle \frac{j-1}{2}}⌋}{\displaystyle \frac{{(1+j-2r)}_{r-1}}{(r-1)!}}\times \hfill \\ \hfill & \sum _{s=0}^{⌊{\displaystyle \frac{j-1}{2}}⌋-r}{\displaystyle \frac{{(-1)}^s(j-2(r+s)){(1+j-2r-s)}_{s-1}}{s!}}{F}_{j-2r-2s}\left(x\right).\hfill \end{array}$$

(43)

Performing some algebraic calculations on the last formula, the following formula can be obtained:

$$\begin{array}{cc}\hfill {\mathcal{GL}}_j^{a,b,T}\left(x\right)=& \sum _{p=0}^{⌊{\displaystyle \frac{j}{2}}⌋}(1+j-2p)\times \hfill \\ \hfill & \sum _{r=0}^p{\displaystyle \frac{{(-1)}^{p-r}((1+j-2r)(b(j-2r)+ar)+(j-r)rT)(j-r-1)!}{(1+j-2r)(p-r)!r!(j-p-r+1)!}}{F}_{j-2p+1}\left(x\right)\hfill \\ \hfill & +T\sum _{p=0}^{⌊{\displaystyle \frac{j-1}{2}}⌋}(j-2p)\sum _{r=0}^p{\displaystyle \frac{{(-1)}^{p-r}(j-r-1)!}{(j-2r)(r-1)!(j-p-r)!(p-r)!}}{F}_{j-2p}\left(x\right).\hfill \end{array}$$

(44)

Now, to obtain a reduced formula for ${\mathcal{GL}}_j^{a,b,T}\left(x\right)$, we need to simplify the last formula. This can be accomplished using symbolic algebra. First, consider the following two sums:

$$\begin{array}{cc}\hfill M_{p,j}=& \sum _{r=0}^p{\displaystyle \frac{{(-1)}^{p-r}((1+j-2r)(b(j-2r)+ar)+(j-r)rT)(j-r-1)!}{(1+j-2r)(p-r)!r!(j-p-r+1)!}},\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill {\overline{M}}_{p,j}=& \sum _{r=0}^p{\displaystyle \frac{{(-1)}^{p-r}(j-r-1)!}{(j-2r)(r-1)!(j-p-r)!(p-r)!}}.\hfill \end{array}$$

(46)

Regarding $M_{p,j}$, we note that

$$M_{0,j}={\displaystyle \frac{b}{j+1}},M_{1,j}={\displaystyle \frac{a-b+T}{j-1}}.$$

In addition, it can be shown using Zeilberger’s algorithm [41] that $M_{p,j}$, for $p\ge 2$, satisfies the following recurrence relation:

$$(j-2p-1)M_{p+1,j}-(j-2p+1)M_{p,j}=0,M_{2,j}={\displaystyle \frac{T}{j-3}}.$$

Thus, it is easy to see that

$$M_{p,j}=\left\{\begin{array}{cc}{\displaystyle \frac{b}{j+1},}\hfill & p=0,\hfill \\ {\displaystyle \frac{a-b+T}{j-1},}\hfill & p=1,\hfill \\ {\displaystyle \frac{T}{j-2p+1},}\hfill & p\ge 2.\hfill \end{array}\right.$$

(47)

Regarding ${\overline{M}}_{p,j}$, we note that

$${\overline{M}}_{0,j}=0.$$

In addition, it can be shown using Zeilberger’s algorithm [41] that ${\overline{M}}_{p,j}$, for $p\ge 1$, satisfies the following recurrence relation:

$$(j-2p-2){\overline{M}}_{p+1,j}-(j-2p){\overline{M}}_{p,j}=0,{\overline{M}}_{1,j}={\displaystyle \frac{1}{j-2}}.$$

Thus, it is easy to see that

$${\overline{M}}_{p,j}=\left\{\begin{array}{cc}0,\hfill & p=0,\hfill \\ {\displaystyle \frac{1}{j-2p},}\hfill & p\ge 1.\hfill \end{array}\right.$$

(48)

Inserting the two reduction Formulas (47) and (48) into (44) leads to the following simplified connection formula:

$${\mathcal{GL}}_i^{a,b,T}\left(x\right)=b{F}_{j+1}\left(x\right)+(a-b+T)F_{j-1}\left(x\right)+T\sum _{p=2}^{⌊{\displaystyle \frac{j}{2}}⌋}{F}_{j-2p+1}\left(x\right)+T\sum _{p=1}^{⌊{\displaystyle \frac{j-1}{2}}⌋}{F}_{j-2p}\left(x\right),$$

(49)

which can be written again in the following form:

$${\mathcal{GL}}_j^{a,b,T}\left(x\right)=b{F}_{j+1}\left(x\right)+(a-b)F_{j-1}\left(x\right)+T\sum _{p=0}^{j-2}{F}_{p+1}\left(x\right).$$

This completes the proof. □

The following corollary presents the connection formula between the combined polynomials $F_i^{a,b}\left(x\right)$ that are defined in (20) and the Fibonacci polynomials.

Corollary 2.

Let j be any non-negative integer. The following connection formula is valid:

$$F_j^{a,b}\left(x\right)=b{F}_{j+1}\left(x\right)+(a-b)F_{j-1}\left(x\right).$$

(50)

Proof. 

Formula (50) is a particular formula of (42) for a case corresponding to $T=0$. □

Remark 5.

We can define the combined Fibonacci–Lucas polynomials with negative indices from the connection Formula (50). The following corollary exhibits this expression.

Corollary 3.

Let j be any non-negative integer. The following expression is valid:

$$F_{-j}^{a,b}\left(x\right)=b{(-1)}^j{F}_{j-1}\left(x\right)+{(-1)}^j(a-b)F_{j+1}\left(x\right).$$

(51)

Proof. 

Based on the connection Formula (50), we can write

$$F_{-j}^{a,b}\left(x\right)=b{F}_{-j+1}\left(x\right)+(a-b)F_{-j-1}\left(x\right).$$

(52)

Based on the well-known formula [3]

$$F_{-j}\left(x\right)={(-1)}^{j+1}{F}_j\left(x\right),$$

Formula (52) turns into (51). □

Remark 6.

From Formula (51), the combined Fibonacci–Lucas numbers with negative indices can be explicitly expressed. The following corollary exhibits this expression.

Corollary 4.

The combined Fibonacci–Lucas numbers for negative indices can be explicitly computed using the following formula:

$$F_{-j}^{a,b}=b{(-1)}^j{F}_{j-1}+{(-1)}^j(a-b)F_{j+1}.$$

(53)

Proof. 

Formula (53) is a direct special case of (51) by setting $x=1$. □

Remark 7.

The first few polynomials $F_{-j}^{a,b}\left(x\right),j\ge 0$ are

$$\begin{array}{cc}\hfill F_0^{a,b}\left(x\right)& =a,F_1^{a,b}\left(x\right)=(b-a)x,F_2^{a,b}\left(x\right)=(a-b)x^2+a,\hfill \\ \hfill F_3^{a,b}\left(x\right)& =(b-a)x^3+(b-2a)x,F_4^{a,b}\left(x\right)=(a-b)x^4+(3a-2b)x^2+a,\hfill \end{array}$$

and thus, the first few combined numbers $F_{-j}^{a,b},j\ge 0$ are

$$\begin{array}{cc}\hfill F_0^{a,b}& =a,F_1^{a,b}=b-a,F_2^{a,b}=2a-b,\hfill \\ \hfill F_3^{a,b}& =2b-3a,F_4^{a,b}=5a-3b.\hfill \end{array}$$

4.2. Connection with Lucas Polynomials

In this part, we give a new connection formula between Leonardo and Lucas polynomials. First, the following two lemmas are required.

Lemma 1.

For every non-negative integer p, one has

$$\begin{array}{cc}\hfill & \sum _{r=0}^p{\displaystyle \frac{{(-1)}^{p-r}((1+j-2r)(b(j-2r)+ar)+(j-r)rT)(j-r-1)!}{(1+j-2r)(p-r)!r!(j-p-r)!}}\hfill \\ \hfill & =\left\{\begin{array}{cc}b,\hfill & \mathit{if}p=0,\hfill \\ {\displaystyle \frac{1}{2}}\left({(-1)}^{p+1}(2a-4b+T)+T\right),\hfill & \mathit{if}p>0.\hfill \end{array}\right.\hfill \\ \hfill & =\left\{\begin{array}{cc}b,\hfill & \mathit{if}p=0,\hfill \\ 2b-a,\hfill & \mathit{if}p>0\mathit{and}p\mathit{is}\mathit{even},\hfill \\ a-2b+T,\hfill & \mathit{if}p\mathit{is}\mathit{odd}.\hfill \end{array}\right.\hfill \end{array}$$

(54)

Proof. 

Let

$$H_{p,j}=\sum _{r=0}^p{\displaystyle \frac{{(-1)}^{p-r}((1+j-2r)(b(j-2r)+ar)+(j-r)rT)(j-r-1)!}{(1+j-2r)(p-r)!r!(j-p-r)!}}.$$

It is evident that $H_{0,j}=b$. In addition, it can be shown using Zeilberger’s algorithm [41] that $H_{p,j}$ satisfies the following recurrence relation for all $p\ge 1$:

$$H_{p+2,j}-H_{p,j}=0,H_{1,j}=a-2b+T,H_{2,j}=2b-a.$$

(55)

Therefore, $H_{p,j}$ has the following explicit form:

$$H_{p,j}=\left\{\begin{array}{cc}b,\hfill & \mathit{if}p=0,\hfill \\ 2b-a,\hfill & \mathit{if}p>0\mathit{and}p\mathit{is}\mathit{even},\hfill \\ a-2b+T,\hfill & \mathit{if}p\mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

(56)

This proves Lemma 1. □

Lemma 2.

For every integer with $p\ge 1$, one has

$$\sum _{r=1}^p{\displaystyle \frac{{(-1)}^{p-r}(j-r-1)!}{(j-2r)(j-p-r-1)!(p-r)!(r-1)!}}={\displaystyle \frac{1}{2}}\left({(-1)}^{p+1}+1\right).$$

(57)

Proof. 

This is similar to the proof for Lemma 1. □

Theorem 3.

Consider any positive integer j. The following generalized Leonardo–Lucas connection formula holds:

$$\begin{array}{cc}\hfill {\mathcal{GL}}_j^{a,b,T}\left(x\right)=& b{L}_j\left(x\right)+{\displaystyle \frac{1}{2}}\sum _{p=1}^{⌊{\displaystyle \frac{j}{2}}⌋}{c}_{j-2p}\left(T+{(-1)}^{p+1}(2a-4b+T)\right)L_{j-2p}\left(x\right)\hfill \\ \hfill & +T\sum _{p=0}^{⌊{\displaystyle \frac{j-3}{4}}⌋}{c}_{j-4p-3}{L}_{j-4p-3}\left(x\right),\hfill \end{array}$$

(58)

with

$$c_j=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}},\hfill & j=0,\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

(59)

Proof. 

Starting from the representation in (25) and applying the inversion formula for Lucas polynomials in (13) yields

$$\begin{array}{cc}\hfill {\mathcal{GL}}_j^{a,b,T}\left(x\right)=& \sum _{r=0}^{⌊{\displaystyle \frac{j}{2}}⌋}{\displaystyle \frac{((1+j-2r)(b(j-2r)+ar)+(j-r)rT){(2+j-2r)}_{r-2}}{r!}}\times \hfill \\ \hfill & \sum _{s=0}^{⌊{\displaystyle \frac{j}{2}}⌋-r}{\displaystyle \frac{{(-1)}^s{c}_{j-2(r+s)}{(1+j-2r-s)}_s}{s!}}{L}_{j-2r-2s}\left(x\right)\hfill \\ \hfill & +T\sum _{r=1}^{⌊{\displaystyle \frac{j}{2}}⌋}{\displaystyle \frac{{(1+j-2r)}_{r-1}}{(r-1)!}}\sum _{s=0}^{⌊{\displaystyle \frac{j-1}{2}}⌋-r}{\displaystyle \frac{{(-1)}^s{c}_{j-2r-2s-1}{(j-2r-s)}_s}{s!}}{L}_{j-2r-2s-1}\left(x\right).\hfill \end{array}$$

(60)

The above formula can be written as

$$\begin{array}{cc}\hfill {\mathcal{GL}}_j^{a,b,T}\left(x\right)=& \sum _{p=0}^{⌊{\displaystyle \frac{j}{2}}⌋}{c}_{j-2p}\sum _{r=0}^p{\displaystyle \frac{{(-1)}^{p-r}((1+j-2r)(b(j-2r)+ar)+(j-r)rT)(j-r-1)!}{(1+j-2r)(p-r)!r!(j-p-r)!}}\times \hfill \\ \hfill & L_{j-2p}\left(x\right)+T\sum _{p=0}^{⌊{\displaystyle \frac{j-1}{2}}⌋}{c}_{j-2p-1}\sum _{r=1}^p{\displaystyle \frac{{(-1)}^{p-r}(j-r-1)!}{(j-2r)(j-p-r-1)!(p-r)!(r-1)!}}{L}_{j-2p-1}\left(x\right).\hfill \end{array}$$

(61)

Now, based on the application of Lemmas 1 and 2, the last formula turns into the following form:

$$\begin{array}{cc}\hfill {\mathcal{GL}}_j^{a,b,T}\left(x\right)=& b{L}_j\left(x\right)+{\displaystyle \frac{1}{2}}\sum _{p=1}^{⌊{\displaystyle \frac{j}{2}}⌋}{c}_{j-2p}\left(T+{(-1)}^{p+1}(2a-4b+T)\right)L_{j-2p}\left(x\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\sum _{p=0}^{⌊{\displaystyle \frac{j-1}{2}}⌋}{c}_{j-2p-1}\left(\left(-1+{(-1)}^p\right)T\right)L_{j-2p-1}\left(x\right),\hfill \end{array}$$

(62)

which can be written as

$$\begin{array}{cc}\hfill {\mathcal{GL}}_j^{a,b,T}\left(x\right)=& b{L}_j\left(x\right)+{\displaystyle \frac{1}{2}}\sum _{p=1}^{⌊{\displaystyle \frac{j}{2}}⌋}{c}_{j-2p}\left(T+{(-1)}^{p+1}(2a-4b+T)\right)L_{j-2p}\left(x\right)\hfill \\ \hfill & +T\sum _{p=0}^{⌊{\displaystyle \frac{j-3}{4}}⌋}{c}_{j-4p-3}{L}_{j-4p-3}\left(x\right).\hfill \end{array}$$

(63)

This proves the theorem. □

5. New Identities Involving Generalized Leonardo Numbers

This section presents new identities involving the two sequences of numbers: the generalized Leonardo numbers ${\mathcal{GL}}_i^{c,d,T}\left(x\right)$ and the combined Fibonacci–Lucas numbers $F{L}_i^{a,b}\left(x\right)$. Some of the introduced identities generalize some published existing identities in the literature. First, the following lemma is needed.

Lemma 3.

For any positive integer k and with $a^2+ab-b^2\ne 0$, the following recursive formula holds:

$$\begin{array}{cc}\hfill {\mathcal{GL}}_{k+1}^{c,d,T}=& F{L}_k^{a,b}+{\displaystyle \frac{a^2-b(-c+b+d)+a(c+b+T)}{a^2+ab-b^2}}F{L}_{k+1}^{a,b}\hfill \\ \hfill & +{\displaystyle \frac{-a^2+b(-c+b-T)+a(-b+d+T)}{a^2+ab-b^2}}F{L}_{k+2}^{a,b}-T.\hfill \end{array}$$

(64)

Proof. 

We are going to show the validity of the following identity:

$$\begin{array}{c}\hfill {\xi }_k^{a,b,c,d,T}={\mathcal{GL}}_{k+1}^{c,d,T}-F{L}_k^{a,b}-{\displaystyle \frac{a^2-b(-c+b+d)+a(c+b+T)}{a^2+ab-b^2}}F{L}_{k+1}^{a,b}\\ \hfill -{\displaystyle \frac{-a^2+b(-c+b-T)+a(-b+d+T)}{a^2+ab-b^2}}F{L}_{k+2}^{a,b}+T=0.\end{array}$$

(65)

The two power form representations of the generalized Leonardo polynomials and the combined Fibonacci polynomials that are given, respectively, in (25), and (41) enable their associated numbers to be expressed in the following forms:

$$\begin{array}{cc}\hfill F{L}_i^{a,b}=& \sum _{r=0}^{⌊{\displaystyle \frac{i}{2}}⌋}{R}_{r,i},\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill {\mathcal{GL}}_i^{c,d,T}=& \sum _{r=0}^{⌊{\displaystyle \frac{i}{2}}⌋}{H}_{r,i}+\sum _{r=0}^{⌊{\displaystyle \frac{i-1}{2}}⌋}{\overline{H}}_{r,i},\hfill \end{array}$$

(67)

with

$$\begin{array}{cc}\hfill R_{r,i}=& {\displaystyle \frac{\left((1+i-2r)(b(i-2r)+ar)\right){(2+i-2r)}_{-2+r}}{r!}},\hfill \\ \hfill H_{r,i}=& {\displaystyle \frac{((1+i-2r)(d(i-2r)+cr)+(i-r)rT){(2+i-2r)}_{-2+r}}{r!}},\hfill \\ \hfill {\overline{H}}_{r,i}=& {\displaystyle \frac{T{(i-2r+1)}_{r-1}}{(r-1)!}}.\hfill \end{array}$$

Therefore, we can write ${\xi }_k^{a,b,c,d,T}$ as

$$\begin{array}{c}\hfill {\xi }_k^{a,b,c,d,T}=\sum _{r=0}^{⌊{\displaystyle \frac{k+1}{2}}⌋}{H}_{r,k+1}+\sum _{r=0}^{⌊{\displaystyle \frac{k}{2}}⌋}{\overline{H}}_{r,k+1}-\sum _{r=0}^{⌊{\displaystyle \frac{k}{2}}⌋}{R}_{r,k}+{\gamma }_1\sum _{r=0}^{⌊{\displaystyle \frac{k+1}{2}}⌋}{R}_{r,k+1}+{\gamma }_2\sum _{r=0}^{⌊{\displaystyle \frac{k+2}{2}}⌋}{R}_{r,k+2}+T,\end{array}$$

(68)

where

$$\begin{array}{cc}\hfill {\gamma }_1& =-{\displaystyle \frac{a^2-b(-c+b+d)+a(c+b+T)}{a^2+ab-b^2}},\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill {\gamma }_2& =-{\displaystyle \frac{-a^2+b(-c+b-T)+a(-b+d+T)}{a^2+ab-b^2}}.\hfill \end{array}$$

(70)

We will prove that ${\xi }_{2k}^{a,b,c,d,T}=0$ and ${\xi }_{2k+1}^{a,b,c,d,T}=0$.

Now, we have

$${\xi }_{2k}^{a,b,c,d,T}=\sum _{r=0}^k{H}_{r,2k+1}+\sum _{r=0}^k{\overline{H}}_{r,2k+1}-\sum _{r=0}^k{R}_{r,2k}+{\gamma }_1\sum _{r=0}^k{R}_{r,2k+1}+{\gamma }_2\sum _{r=0}^{k+1}{R}_{r,2k+2}+T,$$

which can be written as

$${\xi }_{2k}^{a,b,c,d,T}=\sum _{r=0}^k{Z}_k+{\gamma }_2{R}_{k+1,2k+2}+T,$$

where $Z_k$ is given by

$$Z_k=H_{r,2k+1}+{\overline{H}}_{r,2k+1}-R_{r,2k}+{\gamma }_1{R}_{r,2k+1}+{\gamma }_2{R}_{r,2k+2}.$$

To find a closed formula for ${\displaystyle \sum _{r=0}^k{Z}_k,k\ge 1}$, set

$${\displaystyle M_k=\sum _{r=0}^k{Z}_k,}$$

it can be shown using Zeilberger’s algorithm that

$$M_{k+1}-M_k=0,M_1={\displaystyle \frac{-a\left(a^2+b(-b+c)+a(b-d)\right)+b(-2a+b)T}{a^2+ab-b^2}},$$

and therefore, it is clear that

$$M_k={\displaystyle \frac{-a\left(a^2+b(-b+c)+a(b-d)\right)+b(-2a+b)T}{a^2+ab-b^2}}.$$

Noting that

$${\gamma }_2{R}_{k+1,2k+2}+T=-M_k,$$

hence ${\xi }_{2k}^{a,b,c,d,T}=0$. Similarly, it can be shown that ${\xi }_{2k+1}^{a,b,c,d,T}=0$. This completes the proof. □

Theorem 4.

Let m be any positive integer. The following identity holds for any real numbers $a,b$ with $a^2+ab-b^2\ne 0$:

$$\begin{array}{c}\hfill y^{m+1}{\mathcal{GL}}_{m+1}^{c,d,T}=c+\sum _{i=0}^m{y}^i\left({\displaystyle \frac{y(-b(c+T)+a(d+T\left)\right)}{a^2+ab-b^2}}F{L}_{i+1}^{a,b}+(y-1){\mathcal{GL}}_i^{c,d,T}\right.\\ \hfill \left.+{\displaystyle \frac{y\left(a\right(c-d)+b(2c-d+T\left)\right)}{a^2+ab-b^2}}F{L}_{i-1}^{a,b}\right).\end{array}$$

(71)

Proof. 

We will provide the proof through induction on m. For $m=0$, the theorem holds since each side equals $dy$. Assume now that (71) holds, and we will prove the following identity:

$$\begin{array}{c}\hfill y^{m+2}{\mathcal{GL}}_{m+2}^{c,d,T}=c+\sum _{i=0}^{m+1}{y}^i\left({\displaystyle \frac{y(-b(c+T)+a(d+T\left)\right)}{a^2+ab-b^2}}F{L}_{i+1}^{a,b}+(y-1){\mathcal{GL}}_i^{c,d,T}\right.\\ \hfill \left.+{\displaystyle \frac{y\left(a\right(c-d)+b(2c-d+T\left)\right)}{a^2+ab-b^2}}F{L}_{i-1}^{a,b}\right).\end{array}$$

(72)

Now, to prove (72), we assume the following polynomial:

$$\begin{array}{c}\hfill G\left(y\right)=c+\sum _{i=0}^{m+1}{y}^i\left({\displaystyle \frac{y(-b(c+T)+a(d+T\left)\right)}{a^2+ab-b^2}}F{L}_{i+1}^{a,b}+(y-1){\mathcal{GL}}_i^{c,d,T}\right.\\ \hfill \left.+{\displaystyle \frac{y\left(a\right(c-d)+b(2c-d+T\left)\right)}{a^2+ab-b^2}}F{L}_{i-1}^{a,b}\right),\end{array}$$

(73)

and we have to show that

$$G\left(y\right)=y^{m+2}{\mathcal{GL}}_{m+2}^{c,d,T}.$$

Now, we can write

$$\begin{array}{cc}\hfill G\left(y\right)=& c+\sum _{i=0}^m{y}^i\left({\displaystyle \frac{y(-b(c+T)+a(d+T\left)\right)}{a^2+ab-b^2}}F{L}_{i+1}^{a,b}+(y-1){\mathcal{GL}}_i^{c,d,T}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{y\left(a\right(c-d)+b(2c-d+T\left)\right)}{a^2+ab-b^2}}F{L}_{i-1}^{a,b}\right)\hfill \\ \hfill & +y^{m+1}\left({\displaystyle \frac{a(c-d)+b(2c-d+T)}{a^2+ab-b^2}}yF{L}_m^{a,b}+{\displaystyle \frac{-b(c+T)+a(d+T)}{a^2+ab-b^2}}yF{L}_{m+2}^{a,b}\right.\hfill \\ \hfill & \left.+(y-1){\mathcal{GL}}_{m+1}^{c,d,T}\right).\hfill \end{array}$$

(74)

Applying the inductive step, the last formula can be written as

$$\begin{array}{cc}\hfill G\left(y\right)=& y^{m+2}{\mathcal{GL}}_{m+1}^{c,d,T}+y^{m+1}\left({\displaystyle \frac{a(c-d)+b(2c-d+T)}{a^2+ab-b^2}}F{L}_m^{a,b}\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{-b(c+T)+a(d+T)}{a^2+ab-b^2}}yF{L}_{m+2}^{a,b}\right).\hfill \end{array}$$

(75)

Applying the recurrence relation (64) to (75), we can write

$$\begin{array}{cc}\hfill G\left(y\right)=& y^{m+2}\left(-T+F{L}_m^{a,b}+{\displaystyle \frac{a^2-b(b-c+d)+a(b+c+T)}{a^2+ab-b^2}}F{L}_{m+1}^{a,b}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{-a^2+b(b-c-T)+a(-b+d+T)}{a^2+ab-b^2}}F{L}_{m+2}^{a,b}\right)\hfill \\ \hfill & +y^{m+1}\left({\displaystyle \frac{a(c-d)+b(2c-d+T)}{a^2+ab-b^2}}yF{L}_m^{a,b}+{\displaystyle \frac{-b(c+T)+a(d+T)}{a^2+ab-b^2}}yF{L}_{m+2}^{a,b}\right).\hfill \end{array}$$

(76)

The last formula can be written as

$$G\left(y\right)=y^{2+m}\left(R_m-T\right),$$

(77)

where

$$R_m={\xi }_{1}F{L}_m^{a,b}+{\xi }_2{F}_{m+1}^{a,b}+{\xi }_3{F}_{m+2}^{a,b},$$

with

$$\begin{array}{cc}\hfill {\xi }_1=& 1+{\displaystyle \frac{a(c-d)+b(2c-d+T)}{a^2+ab-b^2}},\hfill \\ \hfill {\xi }_1=& {\displaystyle \frac{a^2-b(b-c+d)+a(b+c+T)}{a^2+ab-b^2}},\hfill \\ \hfill {\xi }_3=& {\displaystyle \frac{-a^2-ab+b^2-2b(c+T)+2a(d+T)}{a^2+ab-b^2}}.\hfill \end{array}$$

Using the mixed recurrence relation again (64) yields the following formula:

$${\mathcal{GL}}_{m+2}^{c,d,T}=-T+{\xi }_{1}F{L}_m^{a,b}+{\xi }_{2}F{L}_{m+1}^{a,b}+{\xi }_{3}F{L}_{m+2}^{a,b},$$

and therefore, we obtain

$$G\left(y\right)=y^{m+2}{\mathcal{GL}}_{m+2}^{c,d,T}.$$

This ends the proof. □

Remark 8.

Taking into consideration the special numbers of the generalized Leonardo numbers given in Table 1, some specific formulas for (71) can be deduced. The following corollaries exhibit these results.

Corollary 5.

Let m be any positive integer. The following identity holds:

$$\begin{array}{c}\hfill y^{m+1}{\mathcal{GL}}_{m+1}^{c,d,T}=c+\sum _{i=0}^m{y}^i\left(y(c+T)F_{i+1}+(y-1){\mathcal{GL}}_i^{c,d,T}+\left(y(-2c+d-T)\right)F_{i-1}\right).\end{array}$$

(78)

Proof. 

We obtain a special case for (71) by setting $a=0,b=1$. □

Corollary 6.

Let m be any positive integer. The following identity holds:

$$\begin{array}{cc}\hfill y^{m+1}{\mathcal{GL}}_{m+1}^{c,d,T}=& c+\sum _{i=0}^m{y}^i\left({\displaystyle \frac{1}{5}}(4c-3d+T)y{L}_{i-1}+{\displaystyle \frac{1}{5}}(-c+2d+T)y{L}_{i+1}\right.\hfill \\ \hfill & \left.+(y-1){\mathcal{GL}}_i^{c,d,T}\right).\hfill \end{array}$$

(79)

Proof. 

We obtain a special case for (71) by setting $a=2,b=1$. □

Corollary 7.

Let m be any positive integer with $a^2+ab-b^2\ne 0$. The following identity holds:

$$\begin{array}{c}\hfill y^{m+1}{F}_{m+1}=\sum _{i=0}^m{y}^i\left(-{\displaystyle \frac{(a+b)y}{a^2+ab-b^2}}F{L}_{i-1}^{a,b}+{\displaystyle \frac{ay}{a^2+ab-b^2}}F{L}_{i+1}^{a,b}+(y-1)F_i\right).\end{array}$$

(80)

Proof. 

We obtain a special case for (71) by setting $c=0,d=1,T=0$. □

Corollary 8.

Sury’s identity [42,43]: For every positive integer m, one has

$$2^{m+1}{F}_{m+1}=\sum _{i=0}^m{2}^i{L}_i.$$

(81)

Proof. 

Substituting $a=0,$$b=1$, and $y=2$ into (80) yields the following formula:

$$2^{m+1}{F}_{m+1}=\sum _{i=0}^m{2}^i\left(F_i+2F_{i-1}\right).$$

(82)

Based on the following two identities [3]

$$\begin{array}{cc}\hfill F_i+F_{i-1}=& F_{i+1},\hfill \\ \hfill F_{i+1}+F_{i-1}=& L_i,\hfill \end{array}$$

Formula (82) is converted into the following identity:

$$2^{m+1}{F}_{m+1}=\sum _{i=0}^m{2}^i{L}_i.$$

(83)

which coincides with Sury’s identity that was developed in [42,43]. □

Corollary 9.

Let m be any positive integer with $a^2+ab-b^2\ne 0$: The following identity holds:

$$\begin{array}{c}\hfill y^{m+1}{L}_{m+1}=2+\sum _{i=0}^m{y}^i\left({\displaystyle \frac{(a+3b)y}{a^2+ab-b^2}}F{L}_{i-1}^{a,b}+{\displaystyle \frac{(a-2b)y}{a^2+ab-b^2}}F{L}_{i+1}^{a,b}+(y-1)L_i\right).\end{array}$$

(84)

Proof. 

We obtain a special case for (71) by setting $c=2,d=1,T=0$. □

6. Product Formulas with Fibonacci and Lucas Polynomials

We give two new formulas for the products of Leonardo polynomials with Fibonacci and Lucas polynomials.

Theorem 5.

Consider the two non-negative integers r and s. The following product formula holds:

$$\begin{array}{cc}\hfill F_r\left(x\right){\mathcal{GL}}_s^{a,b,T}\left(x\right)=& b\sum _{m=0}^{min(r-1,s)}{(-1)}^m{F}_{r+s-2m}\left(x\right)+(a-b)\sum _{m=0}^{min(r-1,s-2)}{(-1)}^m{F}_{r+s-2m-2}\left(x\right)\hfill \\ \hfill & +T\sum _{p=0}^{s-2}\sum _{m=0}^{min(r-1,p)}{(-1)}^m{F}_{r+p-2m}\left(x\right).\hfill \end{array}$$

(85)

Proof. 

From (42), the generalized Leonardo polynomials can be expressed as combinations of Fibonacci polynomials as

$${\mathcal{GL}}_s^{a,b,T}\left(x\right)=b{F}_{s+1}\left(x\right)+(a-b)F_{s-1}\left(x\right)+T\sum _{p=0}^{s-2}{F}_{p+1}\left(x\right).$$

(86)

If we multiply both sides of the last formula by $F_r\left(x\right)$, then we obtain

$$\begin{array}{c}\hfill F_r\left(x\right){\mathcal{GL}}_s^{a,b,T}\left(x\right)=b{F}_r\left(x\right)F_{s+1}\left(x\right)+(a-b)F_r\left(x\right)F_{s-1}\left(x\right)+T\sum _{p=0}^{s-2}{F}_r\left(x\right)F_{p+1}\left(x\right).\end{array}$$

(87)

If we make use of the following linearization formula of Fibonacci polynomials [44]

$$F_r\left(x\right)F_s\left(x\right)=\sum _{m=0}^{min(r-1,s-1)}{(-1)}^m{F}_{r+s-2m-1}\left(x\right),$$

(88)

then the following product formula is obtained:

$$\begin{array}{cc}\hfill F_r\left(x\right){\mathcal{GL}}_s^{a,b,T}\left(x\right)=& b\sum _{m=0}^{min(r-1,s)}{(-1)}^m{F}_{r+s-2m}\left(x\right)+(a-b)\sum _{m=0}^{min(r-1,s-2)}{(-1)}^m{F}_{r+s-2m-2}\left(x\right)\hfill \\ \hfill & +T\sum _{p=0}^{s-2}\sum _{m=0}^{min(r-1,p)}{(-1)}^m{F}_{r+p-2m}\left(x\right).\hfill \end{array}$$

(89)

This ends the proof. □

Theorem 6.

Consider the two non-negative integers r and s. The following product formula holds:

$$\begin{array}{cc}\hfill & L_r\left(x\right){\mathcal{GL}}_s^{a,b,T}\left(x\right)=b\left(L_{r+s}\left(x\right)+{(-1)}^r{L}_{s-r}\left(x\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sum _{m=1}^{⌊{\displaystyle \frac{s}{2}}⌋}{c}_{s-2m}\left(T+{(-1)}^{m+1}(2a-4b+T)\right)\left(L_{r+s-2m}\left(x\right)+{(-1)}^r{L}_{s-2m-r}\left(x\right)\right)\hfill \\ \hfill & +T\sum _{m=0}^{⌊{\displaystyle \frac{s-3}{4}}⌋}{c}_{s-4m-3}\left(L_{r+s-4m-3}\left(x\right)+{(-1)}^r{L}_{s-4m-3-r}\left(x\right)\right),\hfill \end{array}$$

(90)

and the constants $c_r$ are defined in (14).

Proof. 

We begin with the connection formula in (58)

$$\begin{array}{cc}\hfill {\mathcal{GL}}_s^{a,b,T}\left(x\right)=& b{L}_s\left(x\right)+{\displaystyle \frac{1}{2}}\sum _{m=1}^{⌊{\displaystyle \frac{s}{2}}⌋}{c}_{s-2m}\left(T+{(-1)}^{m+1}(2a-4b+T)\right)L_{s-2m}\left(x\right)\hfill \\ \hfill & +T\sum _{m=0}^{⌊{\displaystyle \frac{s-3}{4}}⌋}{c}_{s-4m-3}{L}_{s-4m-3}\left(x\right),\hfill \end{array}$$

(91)

and accordingly, we have

$$\begin{array}{cc}\hfill L_r\left(x\right){\mathcal{GL}}_s^{a,b,T}\left(x\right)=& b{L}_s\left(x\right)L_r\left(x\right)+{\displaystyle \frac{1}{2}}\sum _{m=1}^{⌊{\displaystyle \frac{s}{2}}⌋}{c}_{s-2m}\left(T+{(-1)}^{m+1}(2a-4b+T)\right)L_r\left(x\right)L_{s-2m}\left(x\right)\hfill \\ \hfill & +T\sum _{m=0}^{⌊{\displaystyle \frac{s-3}{4}}⌋}{c}_{s-4m-3}{L}_r\left(x\right)L_{s-4m-3}\left(x\right).\hfill \end{array}$$

(92)

Based on the linearization formula of Lucas polynomials given by

$$L_r\left(x\right)L_s\left(x\right)=L_{r+s}\left(x\right)+{(-1)}^r{L}_{s-r}\left(x\right),$$

(93)

the following formula can be obtained:

$$\begin{array}{cc}\hfill & L_r\left(x\right){\mathcal{GL}}_s^{a,b,T}\left(x\right)=b\left(L_{r+s}\left(x\right)+{(-1)}^r{L}_{s-r}\left(x\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sum _{m=1}^{⌊{\displaystyle \frac{s}{2}}⌋}{c}_{s-2m}\left(T+{(-1)}^{m+1}(2a-4b+T)\right)\left(L_{r+s-2m}\left(x\right)+{(-1)}^r{L}_{s-2m-r}\left(x\right)\right)\hfill \\ \hfill & +T\sum _{m=0}^{⌊{\displaystyle \frac{s-3}{4}}⌋}{c}_{s-4m-3}\left(L_{r+s-4m-3}\left(x\right)+{(-1)}^r{L}_{s-4m-3-r}\left(x\right)\right).\hfill \end{array}$$

(94)

The proof is now complete. □

Theorem 7.

Consider the two non-negative integers r and s. The following product formula holds:

$$\begin{array}{cc}\hfill & L_r\left(x\right){\mathcal{GL}}_s^{a,b,T}\left(x\right)=b\left(F_{r+s+1}\left(x\right)+F_{r+s-1}\left(x\right)+{(-1)}^r(F_{s-r+1}\left(x\right)+F_{s-r-1}\left(x\right))\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sum _{m=1}^{⌊{\displaystyle \frac{s}{2}}⌋}{c}_{s-2m}\left(T+{(-1)}^{m+1}(2a-4b+T)\right)\times \hfill \\ \hfill & \left({(-1)}^r(F_{-2m-r+s-1}\left(x\right)+F_{-2m-r+s+1}\left(x\right))+F_{-2m+r+s-1}\left(x\right)+F_{-2m+r+s+1}\left(x\right)\right)+\hfill \\ \hfill & T\sum _{m=0}^{⌊{\displaystyle \frac{s-3}{4}}⌋}{c}_{s-4m-3}\left({(-1)}^r(F_{-4m-r+s-4}\left(x\right)+F_{-4m-r+s-2}\left(x\right))+F_{-4m+r+s-4}\left(x\right)+F_{-4m+r+s-2}\left(x\right)\right).\hfill \end{array}$$

(95)

Proof. 

Starting from the linearization Formula (90), along with the following formula [44],

$$\begin{array}{cc}\hfill L_r\left(x\right)=& F_{r+1}\left(x\right)+F_{r-1}\left(x\right),\hfill \end{array}$$

Formula (95) can be obtained. □

7. Some Definite Integrals

We give some definite integrals based on some of the formulas in the previous sections.

Corollary 10.

For any positive integer r, the following integral formula is valid:

$${\int }_0^1{\mathcal{GL}}_r\left(x\right)dx=b{A}_{r+1}+(a-b)A_{r-1}+T\sum _{p=0}^{r-2}{A}_{p+1},$$

(96)

where

$$A_r=\left\{\begin{array}{cc}{\displaystyle \frac{{}_{2}F_1\left(\begin{array}{c}\frac{-r}{2},\frac{1-r}{2}\\ 1-r\end{array}|-4\right)}{r}},\hfill & \mathit{if}r\mathit{is}\mathit{odd},\hfill \\ -{\displaystyle \frac{2}{r}}+{\displaystyle \frac{{}_{2}F_1\left(\begin{array}{c}\frac{-r}{2},\frac{1-r}{2}\\ 1-r\end{array}|-4\right)}{r}},\hfill & \mathit{if}r\mathit{is}\mathit{even}.\hfill \end{array}\right.$$

Proof. 

Based on the connection Formula (42), one can write

$${\int }_0^1{\mathcal{GL}}_r\left(x\right)dx=b{\int }_0^1{F}_{r+1}\left(x\right)dx+(a-b){\int }_0^1{F}_{r-1}\left(x\right)dx+T\sum _{p=0}^{r-2}{\int }_0^1{F}_{p+1}\left(x\right)dx.$$

(97)

If we make use of the following definite integral [45]

$${\int }_0^1{F}_r\left(x\right)dx=\left\{\begin{array}{cc}{\displaystyle \frac{{}_{2}F_1\left(\begin{array}{c}\frac{-r}{2},\frac{1-r}{2}\\ 1-r\end{array}|-4\right)}{r}},\hfill & \mathrm{if}r\mathrm{is}\mathrm{odd},\hfill \\ -{\displaystyle \frac{2}{r}}+{\displaystyle \frac{{}_{2}F_1\left(\begin{array}{c}\frac{-r}{2},\frac{1-r}{2}\\ 1-r\end{array}|-4\right)}{r}},\hfill & \mathrm{if}r\mathrm{is}\mathrm{even},\hfill \end{array}\right.$$

(98)

then (96) can be obtained. □

Corollary 11.

For any positive integer r, the following integral formula is valid:

$$\begin{array}{cc}\hfill {\int }_0^1{F}_r\left(x\right){\mathcal{GL}}_s\left(x\right)dx& =b\sum _{m=0}^{min(r-1,s)}{(-1)}^m{A}_{r+s-2m}+(a-b)\sum _{m=0}^{min(r-1,s-2)}{(-1)}^m{A}_{r+s-2m-2}\hfill \\ \hfill & +T\sum _{p=0}^{s-2}\sum _{m=0}^{min(r-1,p)}{(-1)}^m{A}_{r+p-2m}.\hfill \end{array}$$

(99)

Proof. 

Formula (99) is a direct consequence of Formula (85), along with (98). □

8. Concluding Remarks

In this paper, we introduced a new class of generalized Leonardo polynomials which serve as an extension of the Fibonacci and Lucas polynomials and their combined polynomials. We introduced new connection formulas that link these polynomials with the Fibonacci and Lucas polynomials. Additionally, we provided novel product formulas involving Fibonacci and Lucas polynomials. The explicit representation of Leonardo polynomials can also be used to acquire other connection formulas with other orthogonal and non-orthogonal polynomials. In addition, we think that the derived results may have potential applications in numerical analyses in general and in the region of solving differential equations in particular. These findings open several avenues for future research. One promising direction is the extension of these polynomials to more generalized polynomials, potentially revealing new algebraic and combinatorial properties. We plan to utilize the introduced polynomials and related ones as basis functions to solve certain differential equations using spectral methods.