New Formulas Involving Fibonacci and Certain Orthogonal Polynomials

Abstract

In this paper, new formulas for the Fibonacci polynomials, including high-order derivatives and repeated integrals of them, are derived in terms of the polynomials themselves. The results are then used to solve connection problems between the Fibonacci and orthogonal polynomials. The inverse cases are also studied. Finally, new results for the linear products of the Fibonacci and orthogonal polynomials are determined using the earlier result for the moments formula of Fibonacci polynomials.

1. Introduction

Special functions and polynomials play important roles in many branches of science. As an example, the authors in [1] studied applications of some special functions in mathematical physics. Some applications of special functions in numerical analysis can be found in [2]. Many authors have been interested in different sequences of polynomials. For example, the authors of [3] studied Appell sequences of polynomials of Bernoulli and Euler types; Ref. [4] studied some general polynomial sequences. In [5], we can find some properties of Horadam polynomials. In [6], the author developed several characteristics of some polynomial sequences in combinatorial theory.

In this work, we shall be primarily concerned with the sequence of polynomials known as the Fibonacci polynomials, which represent a generalization of the famous Fibonacci numbers [7]. The coefficients of these polynomials can be read off the shallow diagonals of Pascal’s triangle and they appear as Sequence A011973 in the Online Encyclopedia of Integer Sequences [8]. These polynomials can also be expressed in terms of imaginary arguments of Chebyshev polynomials, but as far as the developments in this paper are concerned, it is far more expedient to use their standard real forms. Later in the paper, we shall develop new results concerning connection problems involving both the Fibonacci and Chebyshev polynomials.

The Fibonacci polynomials and their generalizations have been found to be extremely useful in obtaining numerical solutions of standard and fractional differential equations. See for example [9,10,11,12], while other authors have been solely interested in their intrinsic mathematical properties by deriving identities and inequalities together with extensions or generalizations [13,14,15,16,17]. In this paper, we continue the investigation of the properties of the Fibonacci polynomials, especially in connection with orthogonal polynomials.

Over the past few decades, hypergeometric functions have become crucial for solving problems in modern analysis, particularly when special functions are involved. As will be seen here, an important topic in applied analysis is the solution of connection and linear problems involving special polynomials, where determining coefficients that are frequently expressed in terms of hypergeometric functions with specific indices and/or arguments is ultimately required [18,19].

Among the more important formulas for a given set of polynomials are those that express the high-order derivatives and repeated integrals of the polynomials explicitly in terms of their original ones, or by other polynomials. The expressions of the derivatives of certain polynomials are very useful for obtaining spectral and pseudospectral solutions of various kinds of differential equations (see, for example, [20,21]). Furthermore, it is worth noting that the derivatives of a polynomial in terms of another one lead to the solution of the connection problem between the two polynomials. This is an approach to solving the connection problem between two different sets of polynomials. In the literature, the connection problem is crucial in the area of special functions. Several techniques were followed to solve these problems (see, for example, [22,23,24,25]).

The primary aim of this paper is to derive new formulas for the higher-order derivatives and repeated integrals of the Fibonacci polynomials in terms of the polynomials themselves (Section 3), which follow from the basic properties given in Section 2. Interestingly, when studying repeated integrals of the Fibonacci polynomials in Section 3, we shall employ Zeilberger’s algorithm [26,27], which represents an effective computer technique for solving problems in applied analysis. Often, this algorithm is the only means of solving problems involving hypergeometric functions. Another major aim of the paper is to determine connection formulas between the Fibonacci polynomials and various orthogonal polynomials in addition to inverted forms (Section 4). Finally, in Section 5 we consider the linear products of the polynomials with orthogonal polynomials.

2. An Account on Fibonacci and Orthogonal Polynomials

This section provides some important properties of Fibonacci polynomials. In addition, some fundamental characteristics of some celebrated orthogonal polynomials are displayed.

2.1. Properties of Fibonacci Polynomials

The Fibonacci polynomials are obtained from the following second-order recurrence relation [7]:

$$F_{n+2}\left(x\right)=x{F}_{n+1}\left(x\right)+F_n\left(x\right),n\ge 0,$$

(1)

where $F_0\left(x\right)=0,$ and $F_1\left(x\right)=1.$ For $n\ge 1$, they can be written as

$$F_n\left(x\right)=\sum _{j=0}^{⌊\frac{n-1}{2}⌋}\left(\genfrac{}{}{0pt}{}{n-j-1}{j}\right)x^{n-2j-1},$$

(2)

where $⌊z⌋$ is the floor function or the greatest integer less than or equal to z. On the other hand, the inversion formula for the Fibonacci polynomials can be expressed as

$$x^i=\sum _{j=0}^{⌊\frac{i}{2}⌋}\frac{{(-1)}^j}{j!}\left(i-2j+1\right){\left(i-j+2\right)}_{j-1}{F}_{i-2j+1}\left(x\right),$$

(3)

where ${\left(a\right)}_j$ denotes the Pochhammer notation for $\mathrm{\Gamma }(a+j)/\mathrm{\Gamma }\left(a\right)$, while the structure formula of Fibonacci polynomials is given by

$${\displaystyle F_i\left(x\right)=\frac{1}{i}\left[D{F}_{i-1}\left(x\right)+D{F}_{i+1}\left(x\right)\right],}$$

where $D=d/dx$.

The Fibonacci numbers, given by $F_i=F_i\left(1\right)$, represent a special case of the Fibonacci polynomials.

We denote the special case of $x=1$ for the derivatives of the Fibonacci polynomials by $F_i^{\left(q\right)}$. That is,

$$F_i^{\left(q\right)}=D^q{F}_i\left(x\right){|}_{x=1}.$$

The following theorem presents an expression for the moments of the Fibonacci polynomials, which will become important in later sections of this paper.

Theorem 1.

For all non-negative integers m and n, the following formula, which will be referred to as the moments formula, holds

$${\displaystyle x^m{F}_{n+1}\left(x\right)=\sum _{i=0}^m{(-1)}^i\left(\genfrac{}{}{0pt}{}{m}{i}\right)F_{n+m-2i+1}\left(x\right).}$$

(4)

Proof. 

We shall prove this result by induction. First, we note that for $m=1$ the right-hand side (rhs) yields

$$\sum _{i=0}^1{(-1)}^i\left(\genfrac{}{}{0pt}{}{1}{i}\right)F_{n+2-2i}\left(x\right)=F_{n+2}\left(x\right)-F_n\left(x\right).$$

Based on to (1), this equals $x{F}_{n+1}$. Now we assume that (1) is valid for m. Then we are required to show for $m+1$ that

$${\displaystyle x^{m+1}{F}_{n+1}\left(x\right)=\sum _{i=0}^{m+1}{(-1)}^i\left(\genfrac{}{}{0pt}{}{m+1}{i}\right)F_{n+m-2i+2}\left(x\right).}$$

(5)

If the identity

$$\left(\genfrac{}{}{0pt}{}{m+1}{i}\right)=\left(\genfrac{}{}{0pt}{}{m}{i}\right)+\left(\genfrac{}{}{0pt}{}{m}{i-1}\right),$$

(6)

is inserted into the rhs of (5), we have

$${\displaystyle \mathrm{rhs}=\sum _{i=0}^m{(-1)}^i\left(\genfrac{}{}{0pt}{}{m}{i}\right)F_{n+m-2i+2}\left(x\right)+\sum _{i=1}^{m+1}{(-1)}^i\left(\genfrac{}{}{0pt}{}{m}{i-1}\right)F_{n+m-2i+2}\left(x\right).}$$

The first sum in the above result is simply our assumption. Hence, it equals $x^m{F}_{n+2}\left(x\right)$. In regard to the second sum, we replace $i-1$ by i, which by our assumption yields $-x^m{F}_n\left(x\right)$. Hence the rhs becomes

$$\mathrm{rhs}=x^m\left(F_{n+2}\left(x\right)-F_n\left(x\right)\right).$$

Inserting (1) into the above result, we obtain the lhs of (5).

We now turn our attention to integer values of n. For $n=1$, (4) reduces to

$${\displaystyle x^m{F}_1\left(x\right)=\sum _{i=0}^m{(-1)}^i\left(\genfrac{}{}{0pt}{}{m}{i}\right)F_{m-2i+1}\left(x\right).}$$

Inserting (6) into the rhs, one obtains

$${\displaystyle \mathrm{rhs}=\sum _{i=0}^{m-1}{(-1)}^i\left(\genfrac{}{}{0pt}{}{m-1}{i}\right)\left(F_{m-2i+1}\left(x\right)-F_{m-2i-1}\left(x\right)\right).}$$

From (1), the above equation reduces to

$$\mathrm{rhs}=x\sum _{i=0}^{m-1}{(-1)}^i\left(\genfrac{}{}{0pt}{}{m-1}{i}\right)F_{m-1-2i+1},$$

which according to (4) yields $x^m{F}_1\left(x\right)$. We assume that the $n-1$ case of (4) and preceding values of n are valid.

According to (1), the lhs of (5) can be expressed as

$$x^m{F}_{n+1}\left(x\right)=x^{m+1}{F}_n\left(x\right)+x^m{F}_{n-1}\left(x\right).$$

By the assumption, the above formula becomes

$${\displaystyle x^m{F}_{n+1}\left(x\right)=\sum _{i=0}^{m+1}{(-1)}^i\left(\genfrac{}{}{0pt}{}{m+1}{i}\right)F_{n+m-2i+1}\left(x\right)+\sum _{i=0}^m{(-1)}^i\left(\genfrac{}{}{0pt}{}{m}{i}\right)F_{n+m-2i-1}\left(x\right).}$$

From (6), we find that

$$\begin{array}{cc}\hfill x^m{F}_{n+1}\left(x\right)& {\displaystyle =\sum _{i=0}^m{(-1)}^i\left(\genfrac{}{}{0pt}{}{m}{i}\right)F_{n+m-2i+1}\left(x\right)+\sum _{i=1}^{m+1}{(-1)}^i\left(\genfrac{}{}{0pt}{}{m}{i-1}\right)F_{n+m-2i+1}\left(x\right)}\hfill \\ & +\sum _{i=0}^m{(-1)}^i\left(\genfrac{}{}{0pt}{}{m}{i}\right)F_{n+m-2i-1}\left(x\right).\hfill \end{array}$$

Replacing $i-1$ by i in the second sum on the rhs, we obtain

$${\displaystyle x^m{F}_{n+1}\left(x\right)=\sum _{i=0}^m{(-1)}^i\left(\genfrac{}{}{0pt}{}{m}{i}\right)F_{n+m-2i+1}\left(x\right),}$$

which is just (4). □

2.2. Properties of Orthogonal Polynomials

All orthogonal sequences used in this paper are families of the classical and symmetric polynomials (with the exception of Chebyshev polynomials of the fifth and sixth kind).

Let ${\varphi }_k\left(x\right)$, $k\ge 0$ be a symmetric orthogonal polynomial of degree k and ${\psi }_k\left(x\right)$, $k\ge 0$ be a non-symmetric orthogonal polynomial of degree k:

$$\begin{array}{cc}\hfill {\varphi }_k\left(x\right)=& \sum _{m=0}^{⌊\frac{k}{2}⌋}{A}_{m,k}{x}^{k-2m},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\psi }_k\left(x\right)=& \sum _{m=0}^k{B}_{m,k}{x}^{k-m}.\hfill \end{array}$$

(8)

The inversion formulas, whereby the powers of x in (7) and (8) are expressed in terms of orthogonal polynomials, are given by [28]

$$\begin{array}{cc}\hfill x^k=& \sum _{m=0}^{⌊\frac{k}{2}⌋}{\overline{A}}_{m,k}{\varphi }_{k-2m},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill x^k=& \sum _{m=0}^k{\overline{B}}_{m,k}{\psi }_{k-m}.\hfill \end{array}$$

(10)

It should be emphasised here that the coefficients $A_{m,k},B_{m,k},{\overline{A}}_{m,k}$ and ${\overline{B}}_{m,k}$ are crucial for deriving formulas involving sequences of orthogonal polynomials.

Among the most important orthogonal polynomials are the Chebyshev polynomials, of which there are six kinds. The first two kinds are the most famous and appear in software packages, e.g., Maple (version 17) and Mathematica (version 12). The third and fourth kinds have a history going back at least to 1993 [29]. As will be seen later, they can be expressed in terms of the Chebyshev polynomials of the second kind. The last two kinds are comparatively recent [30]. In addition, the first, second, fifth, and sixth kinds are also symmetric. On the other hand, the third and fourth kinds are expressible in terms of the second kind and are non-symmetric. All six kinds of Chebyshev polynomials can be represented by trigonometric functions [30,31,32], which are

$$\begin{array}{cc}\hfill & {\displaystyle T_j(cos\theta )=cos\left(j\theta \right),U_j(cos\theta )=\frac{sin\left((j+1)\theta \right)}{sin\theta },}\hfill \\ \hfill & {\displaystyle V_j(cos\theta )=\frac{cos\left(\left(j+\frac{1}{2}\right)\theta \right)}{cos\left(\frac{\theta }{2}\right)},W_j(cos\theta )=\frac{sin\left(\left(j+\frac{1}{2}\right)\theta \right)}{sin\left(\frac{\theta }{2}\right)},}\hfill \end{array}$$

$$X_j(cos\theta )=\left\{\begin{array}{cc}{\displaystyle \frac{cos\left(\right(j+1\left)\theta \right)}{2^{j}cos\theta },}\hfill & j\mathrm{even},\hfill \\ {\displaystyle \frac{(jcos\theta cos\left((j+1)\theta \right)+cos\left(j\theta \right)){sec}^2\theta }{2^{j}j},}\hfill & j\mathrm{odd}.\hfill \end{array}\right.$$

(11)

$$Y_j(cos\theta )=\left\{\begin{array}{cc}{\displaystyle \frac{sin\left(\right(2+j\left)\theta \right)}{2^{j}sin\left(2\theta \right)},}\hfill & j\mathrm{even},\hfill \\ {\displaystyle \frac{sin\left(\right(1+j\left)\theta \right)+(1+j)cos\theta sin\left(\right(2+j\left)\theta \right)}{2^{j+1}(1+j){cos}^2\theta sin\theta },}\hfill & j\mathrm{odd}.\hfill \end{array}\right.$$

(12)

Because of their importance in the derivation of new formulas in this paper, the coefficients, $A_{m,k},B_{m,k}$, ${\overline{A}}_{m,k}$ and ${\overline{B}}_{m,k}$ for several of the more well-known symmetric and non-symmetric polynomials are displayed in Table 1. As a result of the trigonometric forms given above, it should be noted that the Chebyshev polynomials of the third and fourth kind can be expressed in terms of the Chebyshev polynomials of the second kind:

$$\begin{array}{c}\hfill V_j(cos\theta )=U_j(cos\theta )-U_{j-1}(cos\theta ),W_j(cos\theta )=U_j(cos\theta )+U_{j-1}(cos\theta ).\end{array}$$

From the above equations, we immediately observe that

$$U_j\left(x\right)=\frac{1}{2}\left[V_j\left(x\right)+W_j\left(x\right)\right].$$

(13)

Table 1. Coefficients of various orthogonal polynomials and their inverted power forms.

${\mathit{\varphi }}_{\mathit{k}}\left(\mathit{x}\right)$ (${\mathit{\psi }}_{\mathit{k}}\left(\mathit{x}\right)$)	${\mathit{A}}_{\mathit{m},\mathit{k}}\left({\mathit{B}}_{\mathit{m},\mathit{k}}\right)$	${\overline{\mathit{A}}}_{\mathit{m},\mathit{k}}\left({\overline{\mathit{B}}}_{\mathit{m},\mathit{k}}\right)$
First kind Chebyshev polynomial ($T_k\left(x\right)$)	${\displaystyle \frac{{(-1)}^m{2}^{-1+k-2m}k(k-m-1)!}{m!(k-2m)!}}$	${\xi }_{k-2m}{2}^{1-k}\left(\genfrac{}{}{0pt}{}{k}{m}\right)$
Second kind Chebyshev polynomial ($U_k\left(x\right)$)	${(-1)}^m\left(\genfrac{}{}{0pt}{}{k-m}{m}\right)2^{k-2m}$	$\frac{2^{-k}(1+k-2m)k!}{(k-m+1)!m!}$
Legendre polynomial ($P_k\left(x\right)$)	$\frac{{(-1)}^m{2}^{k-2m}\mathrm{\Gamma }\left(\frac{1}{2}+k-m\right)}{\sqrt{\pi }(k-2m)!m!}$	$\frac{2^{-k}\sqrt{\pi }\left(\frac{1}{2}+k-2m\right)k!}{\mathrm{\Gamma }\left(\frac{3}{2}+k-m\right)m!}$
Ultraspherical polynomial ($C_k^{\left(\lambda \right)}\left(x\right)$)	$\frac{{(-1)}^m{2}^{-1+k-2m}k!\mathrm{\Gamma }(k-m+\lambda )\mathrm{\Gamma }(1+2\lambda )}{(k-2m)!m!\mathrm{\Gamma }(1+\lambda )\mathrm{\Gamma }(k+2\lambda )}$	$\frac{2^{-k+1}(k-2m+\lambda )k!\mathrm{\Gamma }(\lambda +1)\mathrm{\Gamma }(k-2m+2\lambda )}{(k-2m)!m!\mathrm{\Gamma }(2\lambda +1)\mathrm{\Gamma }(1+k-m+\lambda )}$
Fifth kind Chebyshev polynomial ($X_k\left(x\right)$)	$\frac{{\left(-\frac{1}{4}\right)}^m(k-m)!}{m!}\left\{\begin{array}{cc}\frac{k+1}{(k-2m+1)!},\hfill & k\mathrm{even},\hfill \\ \frac{k+2}{(k-2m+2)(k-2m)!},\hfill & k\mathrm{odd}.\hfill \end{array}\right.$	$\frac{4^{-m}}{m!(k-m+1)!}\left\{\begin{array}{cc}(k+1)!,\hfill & k\mathrm{even},\hfill \\ \frac{(1+k-2m)(k+2)!}{(1+k)(2+k-2m)}\hfill & k\mathrm{odd}.\hfill \end{array}\right.$
Sixth kind Chebyshev polynomial ($Y_k\left(x\right)$)	$\frac{{\left(-\frac{1}{4}\right)}^m(k-m+1)!}{m!}\left\{\begin{array}{cc}\frac{1}{{\left((k-2m+1)!\right)}^2},\hfill & k\mathrm{even},\hfill \\ \frac{(2+k)(1+k-2m)}{(1+k)(k-2m+2)!},\hfill & k\mathrm{odd}.\hfill \end{array}\right.$	$\frac{4^{-m}}{m!(k-m+2)!}\left\{\begin{array}{cc}(2+k-2m)(k+1)!,\hfill & k\mathrm{even},\hfill \\ (2+k)(1+k-2m)k!,\hfill & k\mathrm{odd}.\hfill \end{array}\right.$
Hermite polynomial ($H_k\left(x\right)$)	${\displaystyle \frac{{(-1)}^{m}k!2^{k-2m}}{m!(k-2m)!}}$	${\displaystyle \frac{k!}{2^{k}m!(k-2m)!}}$
Third kind Chebyshev polynomial ($V_k\left(x\right)$)	$\frac{2^{k-m}}{(k-m)!}\left\{\begin{array}{cc}\frac{{(-1)}^{\frac{m}{2}}\left(k-\frac{m}{2}\right)!}{\frac{m}{2}!},\hfill & m\mathrm{even},\hfill \\ \frac{{(-1)}^{\frac{1+m}{2}}\left(k-\left(\frac{m+1}{2}\right)\right)!}{\frac{m-1}{2}!},\hfill & m\mathrm{odd}.\hfill \end{array}\right.$	$\frac{1}{2^k}\left\{\begin{array}{cc}\left(\genfrac{}{}{0pt}{}{k}{\frac{m}{2}}\right),\hfill & m\mathrm{even},\hfill \\ \left(\genfrac{}{}{0pt}{}{k}{\frac{m-1}{2}}\right),\hfill & m\mathrm{odd}.\hfill \end{array}\right.$
Fourth kind Chebyshev polynomial ($W_k\left(x\right)$)	$\frac{2^{k-m}}{(k-m)!}\left\{\begin{array}{cc}\frac{{(-1)}^{\frac{m}{2}}\left(k-\frac{m}{2}\right)!}{\frac{m}{2}!},\hfill & m\mathrm{even},\hfill \\ \frac{{(-1)}^{\frac{-1+m}{2}}\left(k-\left(\frac{m+1}{2}\right)\right)!}{\frac{m-1}{2}!},\hfill & m,\mathrm{odd}.\hfill \end{array}\right.$	$\frac{1}{2^k}\left\{\begin{array}{cc}\left(\genfrac{}{}{0pt}{}{k}{\frac{m}{2}}\right),\hfill & m\mathrm{even},\hfill \\ -\left(\genfrac{}{}{0pt}{}{k}{\frac{m-1}{2}}\right),\hfill & m\mathrm{odd}.\hfill \end{array}\right.$

Table 1. Coefficients of various orthogonal polynomials and their inverted power forms.

${\mathit{\varphi }}_{\mathit{k}}\left(\mathit{x}\right)$ (${\mathit{\psi }}_{\mathit{k}}\left(\mathit{x}\right)$)	${\mathit{A}}_{\mathit{m},\mathit{k}}\left({\mathit{B}}_{\mathit{m},\mathit{k}}\right)$	${\overline{\mathit{A}}}_{\mathit{m},\mathit{k}}\left({\overline{\mathit{B}}}_{\mathit{m},\mathit{k}}\right)$
First kind Chebyshev polynomial ($T_k\left(x\right)$)	${\displaystyle \frac{{(-1)}^m{2}^{-1+k-2m}k(k-m-1)!}{m!(k-2m)!}}$	${\xi }_{k-2m}{2}^{1-k}\left(\genfrac{}{}{0pt}{}{k}{m}\right)$
Second kind Chebyshev polynomial ($U_k\left(x\right)$)	${(-1)}^m\left(\genfrac{}{}{0pt}{}{k-m}{m}\right)2^{k-2m}$	$\frac{2^{-k}(1+k-2m)k!}{(k-m+1)!m!}$
Legendre polynomial ($P_k\left(x\right)$)	$\frac{{(-1)}^m{2}^{k-2m}\mathrm{\Gamma }\left(\frac{1}{2}+k-m\right)}{\sqrt{\pi }(k-2m)!m!}$	$\frac{2^{-k}\sqrt{\pi }\left(\frac{1}{2}+k-2m\right)k!}{\mathrm{\Gamma }\left(\frac{3}{2}+k-m\right)m!}$
Ultraspherical polynomial ($C_k^{\left(\lambda \right)}\left(x\right)$)	$\frac{{(-1)}^m{2}^{-1+k-2m}k!\mathrm{\Gamma }(k-m+\lambda )\mathrm{\Gamma }(1+2\lambda )}{(k-2m)!m!\mathrm{\Gamma }(1+\lambda )\mathrm{\Gamma }(k+2\lambda )}$	$\frac{2^{-k+1}(k-2m+\lambda )k!\mathrm{\Gamma }(\lambda +1)\mathrm{\Gamma }(k-2m+2\lambda )}{(k-2m)!m!\mathrm{\Gamma }(2\lambda +1)\mathrm{\Gamma }(1+k-m+\lambda )}$
Fifth kind Chebyshev polynomial ($X_k\left(x\right)$)	$\frac{{\left(-\frac{1}{4}\right)}^m(k-m)!}{m!}\left\{\begin{array}{cc}\frac{k+1}{(k-2m+1)!},\hfill & k\mathrm{even},\hfill \\ \frac{k+2}{(k-2m+2)(k-2m)!},\hfill & k\mathrm{odd}.\hfill \end{array}\right.$	$\frac{4^{-m}}{m!(k-m+1)!}\left\{\begin{array}{cc}(k+1)!,\hfill & k\mathrm{even},\hfill \\ \frac{(1+k-2m)(k+2)!}{(1+k)(2+k-2m)}\hfill & k\mathrm{odd}.\hfill \end{array}\right.$
Sixth kind Chebyshev polynomial ($Y_k\left(x\right)$)	$\frac{{\left(-\frac{1}{4}\right)}^m(k-m+1)!}{m!}\left\{\begin{array}{cc}\frac{1}{{\left((k-2m+1)!\right)}^2},\hfill & k\mathrm{even},\hfill \\ \frac{(2+k)(1+k-2m)}{(1+k)(k-2m+2)!},\hfill & k\mathrm{odd}.\hfill \end{array}\right.$	$\frac{4^{-m}}{m!(k-m+2)!}\left\{\begin{array}{cc}(2+k-2m)(k+1)!,\hfill & k\mathrm{even},\hfill \\ (2+k)(1+k-2m)k!,\hfill & k\mathrm{odd}.\hfill \end{array}\right.$
Hermite polynomial ($H_k\left(x\right)$)	${\displaystyle \frac{{(-1)}^{m}k!2^{k-2m}}{m!(k-2m)!}}$	${\displaystyle \frac{k!}{2^{k}m!(k-2m)!}}$
Third kind Chebyshev polynomial ($V_k\left(x\right)$)	$\frac{2^{k-m}}{(k-m)!}\left\{\begin{array}{cc}\frac{{(-1)}^{\frac{m}{2}}\left(k-\frac{m}{2}\right)!}{\frac{m}{2}!},\hfill & m\mathrm{even},\hfill \\ \frac{{(-1)}^{\frac{1+m}{2}}\left(k-\left(\frac{m+1}{2}\right)\right)!}{\frac{m-1}{2}!},\hfill & m\mathrm{odd}.\hfill \end{array}\right.$	$\frac{1}{2^k}\left\{\begin{array}{cc}\left(\genfrac{}{}{0pt}{}{k}{\frac{m}{2}}\right),\hfill & m\mathrm{even},\hfill \\ \left(\genfrac{}{}{0pt}{}{k}{\frac{m-1}{2}}\right),\hfill & m\mathrm{odd}.\hfill \end{array}\right.$
Fourth kind Chebyshev polynomial ($W_k\left(x\right)$)	$\frac{2^{k-m}}{(k-m)!}\left\{\begin{array}{cc}\frac{{(-1)}^{\frac{m}{2}}\left(k-\frac{m}{2}\right)!}{\frac{m}{2}!},\hfill & m\mathrm{even},\hfill \\ \frac{{(-1)}^{\frac{-1+m}{2}}\left(k-\left(\frac{m+1}{2}\right)\right)!}{\frac{m-1}{2}!},\hfill & m,\mathrm{odd}.\hfill \end{array}\right.$	$\frac{1}{2^k}\left\{\begin{array}{cc}\left(\genfrac{}{}{0pt}{}{k}{\frac{m}{2}}\right),\hfill & m\mathrm{even},\hfill \\ -\left(\genfrac{}{}{0pt}{}{k}{\frac{m-1}{2}}\right),\hfill & m\mathrm{odd}.\hfill \end{array}\right.$

3. High-Order Derivatives and Repeated Integrals of Fibonacci Polynomials

Lemma 1.

For positive integer values of m the following combinatorial identity holds

$$\sum _{j=0}^m\frac{{(-1)}^j(i-j)!}{j!(-j+m)!(i-j-m-q+1)!}=\left(\genfrac{}{}{0pt}{}{m+q-1}{m}\right){(i-m-q+2)}_{q-1}.$$

(14)

Proof. 

First, we extend the summation over j on the lhs of (14) to infinity without affecting the result. By applying Euler’s reflection formula [33] for the gamma function, we make the following substitutions on the left-hand side of (14):

$$\begin{array}{cc}\hfill & \frac{(i-j)!}{(m-j)!}={(-1)}^{m-i}\frac{\mathrm{\Gamma }(j-m)}{\mathrm{\Gamma }(j-i)}=\frac{{(-m)}_j}{{(-i)}_j}\frac{i!}{m!},\hfill \\ \hfill & \frac{1}{(i-j-m-q+1)!}={(-1)}^j\frac{{(m+q-i-1)}_j}{(i-m-q+1)!}.\hfill \end{array}$$

Then the the left side of (14), lhs, can be expressed as

$$\mathrm{lhs}=\frac{i!}{m!(i-m-q+1)!}{}_2{F}_1(-m,m+q-i-1;-i;1).$$

(15)

Now we employ Gauss’s identity for the hypergeometric functions [34] ${}_2{F}_1$ when $z=1$, which gives

$${}_2{F}_1(-m,m+q-i-1;-i;1)=\frac{\mathrm{\Gamma }(1-q)}{\mathrm{\Gamma }(1-q-m)}\frac{\mathrm{\Gamma }(-i)}{\mathrm{\Gamma }(m-i)}=\frac{\mathrm{\Gamma }(q+m)}{\mathrm{\Gamma }\left(q\right)}\frac{\mathrm{\Gamma }(i+1-m)}{\mathrm{\Gamma }(i+1)}.$$

(16)

The last member in (16) was obtained by applying Euler’s reflection formula to the intermediate member. By substituting the last result for the hypergeometric function in (15) and performing a little algebra, we arrive at the rhs of (14). □

Theorem 2.

For all nonnegative integers i and q, the following result for the q-th derivative of Fibonacci polynomials holds

$$D^q{F}_{n+1}\left(x\right)=\sum _{k=0}^{⌊\frac{n-q}{2}⌋}{(-1)}^k\left(\genfrac{}{}{0pt}{}{k+q-1}{k}\right)(1+n-2k-q){(n-k-q+2)}_{q-1}{F}_{n-2k-q+1}\left(x\right).$$

(17)

Proof. 

We consider the q-th derivative of $F_{n+1}\left(x\right)$ using (2). This has the effect of altering the upper limit of the sum to $\lfloor (n-q)/2\rfloor$, changing the power of x to $n-j-2q$ and altering the binomial factor to $(n-j)!/j!(n-q-2j)!$. Then, we replace $x^{n-j-2q}$ by the inversion formula, namely, (3). Thus, we find that

$$D^q{F}_{n+1}\left(x\right)=\sum _{k=0}^{⌊\frac{n-q}{2}⌋}\frac{(n-k)!}{k!}\sum _{r=0}^{⌊\frac{i-q}{2}⌋-k}\frac{{(-1)}^{r+1}(-1-n+2k+q+2r)}{r!(n-2k-q-r+1)!}{F}_{n-2k-q-2r+1}\left(x\right).$$

Alternatively, we can express the above result as

$$D^q{F}_{n+1}\left(x\right)=\sum _{k=0}^{⌊\frac{n-q}{2}⌋}{(-1)}^{k+1}(-1-n+2k+q)A_{n,q,k}{F}_{n-q-2k+1}\left(x\right),$$

where

$$A_{n,q,k}=\sum _{j=0}^k\frac{{(-1)}^j(n-j)!}{j!(-j+k)!(n-j-k-q+1)!}.$$

From Lemma 1,

$$A_{n,q,k}=\left(\genfrac{}{}{0pt}{}{k+q-1}{k}\right){(n-k-q+2)}_{q-1}.$$

Therefore, we get (17). □

Remark 1.

It should be noted that Theorem 2 is a generalization of the $q=1$ result appearing in [4,15], where it is given as

$$D{F}_{n+1}\left(x\right)=\sum _{k=0}^{\lfloor \frac{n-1}{2}\rfloor }{(-1)}^k(n-2k)F_{n-2k}\left(x\right).$$

Corollary 1.

For all $n\ge q$, the following result applies

$${\displaystyle F_n^{\left(q\right)}=\sum _{k=0}^{⌊\frac{n-q-1}{2}⌋}{(-1)}^k\left(\genfrac{}{}{0pt}{}{k+q-1}{k}\right)(n-2k-q){(n-k-q+1)}_{q-1}{F}_{n-2k-q}.}$$

Proof. 

This result is obtained simply by putting $x=1$ and replacing n by $n-1$ in (17). □

Now, we derive a formula that expresses repeated integrals of the Fibonacci polynomials of degree n explicitly in terms of the Fibonacci polynomials themselves.

Theorem 3.

Let the q times repeated integration of $F_n\left(x\right)$ be denoted by

$${\displaystyle J_n^{\left(q\right)}\left(x\right)={\int }^{\left(q\right)}{F}_{n+1}\left(x\right){\left(dx\right)}^q=\stackrel{qtimes}{\overbrace{\int \int \dots \int }}{F}_{n+1}\left(x\right)\stackrel{qtimes}{\overbrace{dxdx...dx}}.}$$

We have

$$J_n^q\left(x\right)=\sum _{m=0}^q\frac{\left(\genfrac{}{}{0pt}{}{q}{m}\right)(n-2m+q+1)}{{(n-m+1)}_{q+1}}{F}_{n+q-2m+1}\left(x\right)+{\rho }_{q-1}\left(x\right),$$

where ${\rho }_{q-1}\left(x\right)$ is a polynomial whose degree is at most $q-1$.

Proof. 

The polynomial ${\rho }_{q-1}\left(x\right)$ arises from the repeated integration of the constant of integration and its degree can at most be $q-1$. Therefore, to prove Theorem 3, all we need to show is

$$F_{n+1}\left(x\right)=\sum _{m=0}^q\frac{\left(\genfrac{}{}{0pt}{}{q}{m}\right)(n-2m+q+1)}{{(i-m+1)}_{q+1}}{D}^q{F}_{n+q-2m+1}\left(x\right).$$

(18)

Inserting (2) into the rhs of (18), taking q derivatives of the resulting expression and then replacing the power of x by (3), we find that the rhs becomes

$$\mathrm{rhs}=\sum _{m=0}^q\frac{\left(\genfrac{}{}{0pt}{}{q}{m}\right)(n-2m+q+1)}{{(n-m+1)}_{q+1}}\sum _{r=0}^{⌊\frac{n+q}{2}⌋-m}\frac{{(1+n-2m-2r)}_q}{r!}{(1+n-2m+q-2r)}_r{x}^{n-2m-2r}.$$

We can regard the rhs as the product of two separate polynomials, where we replace $m+r$ by ℓ and sum from 0 to $\lfloor n/2\rfloor$. Inside this outer sum, we sum over the products of the coefficients of the polynomials from 0 to ℓ. Consequently, we can express the rhs of (18) as

$$\begin{array}{cc}& \sum _{m=0}^q\frac{(n-2m+q+1)}{{(n-m+1)}_{q+1}}\left(\genfrac{}{}{0pt}{}{q}{m}\right)D^q{F}_{n+q-2m+1}\left(x\right)\hfill \\ & =\sum _{\ell =0}^{⌊\frac{n}{2}⌋}{(1+n-2\ell )}_q\left\{\sum _{p=0}^{\ell }\frac{(1+n-2p+q){(1+n-2\ell +q)}_{\ell -p}}{(\ell -p)!{(1+n-p)}_{q+1}}\left(\genfrac{}{}{0pt}{}{q}{p}\right)\right\}{x}^{n-2\ell }.\hfill \end{array}$$

(19)

To derive a formula for the inner sum in (19), we set

$$M_{n,q,\ell }=\sum _{p=0}^{\ell }\frac{(1+n-2p+q){(1+n-2\ell +q)}_{\ell -p}}{(\ell -p)!{(1+n-p)}_{q+1}}\left(\genfrac{}{}{0pt}{}{q}{p}\right).$$

We insert the above sum into Maple and apply Zeilberger’s algorithm [26,27] via the sumrecursion routine leads to the following recurrence relation:

$$(\ell +1)(n-\ell )M_{n,q,\ell +1}=(n-2\ell +q-1)(n-2\ell +q)M_{n,q,\ell },$$

with the initial value given by

$$M_{n,q,0}=\frac{n!}{(n+q)!}.$$

This is solved easily, yielding

$$M_{n,q,\ell }=\frac{(n-\ell )!}{\ell !(n-2\ell +q)!}.$$

Thus, the sum over p in (19) simplifies to

$$\sum _{p=0}^{\ell }\frac{(1+n-2p+q){(1+n-2\ell +q)}_{\ell -p}}{(\ell -p)!{(1+n-p)}_{q+1}}\left(\genfrac{}{}{0pt}{}{q}{p}\right)=\frac{(n-\ell )!}{\ell !(n-2\ell +q)!}.$$

Inserting the above sum into (19) gives

$$\sum _{m=0}^q\frac{(n-2m+q+1)}{{(i-m+1)}_{q+1}}\left(\genfrac{}{}{0pt}{}{q}{m}\right)D^q{F}_{n+q-2m+1}\left(x\right)=\sum _{\ell =0}^{⌊\frac{n}{2}⌋}\frac{{(n-2\ell +1)}_{\ell }}{\ell !}{x}^{n-2\ell }=F_{n+1}\left(x\right),$$

where the last member was obtained by replacing n by $i+1$ in (2). Therefore, we arrive at the lhs of (18). □

4. Connection Formulas between Different Polynomials

This section is focused on developing some connection formulas between Fibonacci polynomials and some orthogonal polynomials.

4.1. Connection Formulas Involving Fibonacci and Symmetric Orthogonal Polynomials

Here, we need both the power series representation for ultraspherical polynomials and the inversion formula for Fibonacci polynomials given by (3). In addition, we require the following lemma:

Lemma 2.

For positive integer values of k, the following combinatorial identity holds

$$\begin{array}{cc}\hfill & \sum _{m=0}^{\lfloor k/2\rfloor }\frac{{(-1)}^m{2}^{k-2m-1}\mathrm{\Gamma }(k-m+\lambda )}{m!(k-2m)!}\sum _{t=0}^{\lfloor k/2\rfloor -m}\frac{{(-1)}^t}{t!}\hfill \\ \hfill & \times (k+1-2m-2t){(k-2m-t+2)}_{t-1}\times F_{k-2m-2t+1}\left(x\right)\hfill \\ \hfill & =\sum _{m=0}^{\lfloor k/2\rfloor }{(-1)}^m(k+1-2m)\left(\sum _{r=0}^m\frac{2^{k-2r-1}\mathrm{\Gamma }(k-r+\lambda )}{(m-r)!r!(k-m-r+1)!}\right)F_{k-2m+1}\left(x\right).\hfill \end{array}$$

Proof. 

The proof is Lemma 2 can be obtained if the right-hand side is expanded and rearranged after a series of algebraic computations. □

Theorem 4.

The connection formula for ultraspherical polynomials expressed in terms of the Fibonacci polynomials is given by

$$\begin{array}{cc}\hfill C_k^{\left(\lambda \right)}\left(x\right)& =\frac{2^{-1+k+2\lambda }k!\mathrm{\Gamma }\left(\frac{1}{2}+\lambda \right)\mathrm{\Gamma }(k+\lambda )}{\sqrt{\pi }\mathrm{\Gamma }(k+2\lambda )}\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{{(-1)}^m(1+k-2m)}{m!(k-m+1)!}{F}_{k-2m+1}\left(x\right)\hfill \\ & \times {}_2{F}_1\left.\left(\begin{array}{c}-m,-1-k+m\\ 1-k-\lambda \end{array}\right|-\frac{1}{4}\right),k\ge 0.\hfill \end{array}$$

(20)

Proof. 

The coefficients for the representation of the ultraspherical or Gegenbauer polynomials, $C_k^{\left(\lambda \right)}\left(x\right)$, in powers of x are displayed in Table 1. To obtain the connection formula involving Fibonacci polynomials, the power form representation of the ultraspherical polynomials [19] is utilized along with the inversion formula for Fibonacci polynomials (3) to get

$$\begin{array}{cc}\hfill C_k^{\left(\lambda \right)}\left(x\right)& =\frac{k!\mathrm{\Gamma }(1+2\lambda )}{\mathrm{\Gamma }(1+\lambda )\mathrm{\Gamma }(k+2\lambda )}\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{{(-1)}^m{2}^{-1+k-2m}\mathrm{\Gamma }(k-m+\lambda )}{m!(k-2m)!}\sum _{t=0}^{⌊\frac{k}{2}⌋-m}\frac{{(-1)}^t}{t!}\hfill \\ & \times (1+k-2m-2t){(2+k-2m-t)}_{t-1}{F}_{k-2m-2t+1}\left(x\right).\hfill \end{array}$$

From Lemma 2, the above result can be expressed alternatively as

$$\begin{array}{cc}\hfill C_k^{\left(\lambda \right)}\left(x\right)& =\frac{k!\mathrm{\Gamma }(1+2\lambda )}{\mathrm{\Gamma }(1+\lambda )\mathrm{\Gamma }(k+2\lambda )}\sum _{m=0}^{⌊\frac{k}{2}⌋}{(-1)}^m(1+k-2m)F_{k-2m+1}\left(x\right)\hfill \\ \hfill & \times \sum _{r=0}^m\frac{2^{-1+k-2r}\mathrm{\Gamma }(k-r+\lambda )}{(m-r)!r!(k-m-r+1)!}.\hfill \end{array}$$

(21)

By extending the summation over r to infinity and applying the reflection formula for the gamma function [35], we get

$$\sum _{r=0}^m\frac{2^{-1+k-2r}\mathrm{\Gamma }(k-r+\lambda )}{(m-r)!r!(1+k-m-r)!}=\frac{2^{-1+k}\mathrm{\Gamma }(k+\lambda )}{(k-m+1)!m!}{}_2{F}_1\left.\left(\begin{array}{c}-m,m-1-k\\ 1-k-\lambda \end{array}\right|-\frac{1}{4}\right).$$

By inserting the above result into (21), we arrive at (20). □

Corollary 2.

From Theorem 4, the solutions to the connection problems involving: (1) Legendre and Fibonacci polynomials, (2) Chebyshev of the first kind and Fibonacci polynomials and (3) Chebyshev of the second kind and Fibonacci polynomials are, respectively,

$$P_k\left(x\right)=\frac{2^k\mathrm{\Gamma }\left(\frac{1}{2}+k\right)}{\sqrt{\pi }}\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{{(-1)}^m(1+k-2m)}{m!(k-m+1)!}{}_2{F}_1\left.\left(\begin{array}{c}-m,-1-k+m\\ \frac{1}{2}-k\end{array}\right|-\frac{1}{4}\right)F_{k-2m+1}\left(x\right),$$

$$T_k\left(x\right)=2^{-1+k}k!\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{{(-1)}^m(1+k-2m)}{m!(k-m+1)!}{}_2{F}_1\left.\left(\begin{array}{c}-m,-1-k+m\\ 1-k\end{array}\right|-\frac{1}{4}\right)F_{k-2m+1}\left(x\right),$$

and

$$U_k\left(x\right)=2^{k}k!\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{{(-1)}^m(1+k-2m)}{m!(k-m+1)!}{}_2{F}_1\left.\left(\begin{array}{c}-m,-1-k+m\\ -k\end{array}\right|-\frac{1}{4}\right)F_{k-2m+1}\left(x\right).$$

We can reverse the approach adopted in Theorem 4 and Corollary 2 by replacing the powers of x in the Fibonacci polynomials by the corresponding inverse formulas for symmetric orthogonal polynomials in Table 1. We shall refer to these as inverse connection formulas.

Theorem 5.

The solution to the Fibonacci–Legendre inverse connection problem is

$$F_{k+1}\left(x\right)=2^{-k}\sqrt{\pi }k!\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{(k-2m+1/2)}{\mathrm{\Gamma }\left(\frac{3}{2}+k-m\right)m!}{}_2{F}_1\left.\left(\begin{array}{c}-m,-\frac{1}{2}-k+m\\ -k\end{array}\right|-4\right)P_{k-2m}\left(x\right).$$

Proof. 

The proof is similar to the proof of Theorem 4. □

Theorem 6.

The Chebyshev of the fifth-kind Fibonacci and Chebyshev of the sixth-kind Fibonacci connection formulas are given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill X_{2k+ϵ}\left(x\right)& =(2k+ϵ)!\sum _{m=0}^k\frac{{(-1)}^m(2k-2m+ϵ+1)}{m!(2k-m+ϵ+1)!}\hfill \\ \hfill & \times {}_3{F}_2\left.\left(\begin{array}{c}-m,-ϵ-\frac{1}{2}-k,-ϵ-1-2k+m\\ -ϵ+\frac{1}{2}-k,-ϵ-2k\end{array}\right|-\frac{1}{4}\right)F_{2k-2m+ϵ+1}\left(x\right),\hfill \end{array}\end{array}$$

(22)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Y_{2k+ϵ}\left(x\right)& =(2k+ϵ)!\sum _{m=0}^k\frac{{(-1)}^m(2k-2m+ϵ+1)}{m!(2k-m+ϵ+1)!}\hfill \\ \hfill & \times {}_3{F}_2\left.\left(\begin{array}{c}-m,-ϵ-\frac{1}{2}-k,-ϵ-1-2k+m\\ -ϵ+\frac{1}{2}-k,-ϵ-2k-1\end{array}\right|-\frac{1}{4}\right)F_{2k-2m+ϵ+1}\left(x\right),\hfill \end{array}\end{array}$$

(23)

and $ϵ=0,1$.

Proof. 

As in previous proofs, we shall only prove one of these results. We will make use of the even-order degree of fifth-kind Chebyshev polynomials to get

$$X_{2k}\left(x\right)=(2k+1)\sum _{r=0}^k\frac{{(-1)}^r{2}^{-2r}(2k-r)!}{(1+2k-2r)!r!}{x}^{2k-2r}.$$

Inserting the inversion formula of the Fibonacci polynomials given in (3) leads to

$$\begin{array}{cc}\hfill X_{2k}\left(x\right)=& (2k+1)\sum _{r=0}^k\frac{{(-1)}^r{2}^{-2r}(2k-r)!}{(1+2k-2r)!r!}\hfill \\ \hfill & \sum _{m=0}^{k-r}\frac{{(-1)}^m(1+2k-2m-2r){(2+2k-m-2r)}_{-1+m}}{m!}{F}_{2k-2m-2r+1}\left(x\right).\hfill \end{array}$$

(24)

Equation (24) can be written alternatively in the following form

$$\begin{array}{cc}\hfill X_{2k}\left(x\right)=& (2k+1)\sum _{m=0}^k\left({-1)}^m(2k-2m+1\right)\times \hfill \\ \hfill & \sum _{p=0}^m\frac{4^{-p}(2k-p)!}{(2k-2p+1)(2k-m-p+1)!p!(m-p)!}{F}_{2k-2m+1}\left(x\right),\hfill \end{array}$$

(25)

that can be written again in the following form

$$\begin{array}{cc}\hfill X_{2k}\left(x\right)=& \left(2k\right)!\sum _{m=0}^k\frac{{(-1)}^m(1+2k-2m)}{(1+2k-m)!m!}\times \hfill \\ & {}_3{F}_2\left.\left(\begin{array}{c}-\frac{1}{2}-k,-m,-1-2k+m\\ \frac{1}{2}-k,-2k\end{array}\right|-\frac{1}{4}\right)F_{2k-2m+1}\left(x\right).\hfill \end{array}$$

The remaining results in the theorem can be proved in a similar manner. □

Corollary 3.

Two trigonometric identities arising from Theorem 6 are:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & 2^{2k}\left(2k\right)!\sum _{m=0}^k\frac{{(-1)}^m(1+2k-2m)}{m!(2k-m+1)!}{}_3{F}_2\left.\left(\begin{array}{c}-m,-\frac{1}{2}-k,-1-2k+m\\ \frac{1}{2}-k,-2k\end{array}\right|-\frac{1}{4}\right)\hfill \\ \hfill & \times F_{2k-2m+1}(cos\theta )=sec\theta cos\left((2k+1)\theta \right),\hfill \end{array}\end{array}$$

(26)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & 2^{2k}\left(2k\right)!\sum _{m=0}^k\frac{{(-1)}^m(1+2k-2m)}{m!(2k-m+1)!}{}_3{F}_2\left.\left(\begin{array}{c}-m,-\frac{1}{2}-k,-1-2k+m\\ -1-2k,\frac{1}{2}-k\end{array}\right|-\frac{1}{4}\right)\hfill \\ & \times F_{2k-2m+1}(cos\theta )=csc2\theta sin\left((2k+2)\theta \right).\hfill \end{array}\end{array}$$

(27)

Proof. 

By setting $ϵ$ equal to zero in the connection formulas of Theorem 6, i.e., (22) and (23), and introducing the trigonometric representations for $X_{2k}\left(x\right)$ and $Y_{2k}\left(x\right)$ given by (11) and (12), respectively, we eventually arrive at (26) and (27). □

Theorem 7.

The Fibonacci–Chebyshev of the sixth-kind connection formula is given by

$$F_{2k+1}\left(x\right)=\sum _{m=0}^k\frac{\left(2k-2m+2\right)}{2^{2m}m!}\frac{\mathrm{\Gamma }(2k+2)}{\mathrm{\Gamma }(2k-m+3)}{}_3{F}_2\left.\left(\begin{array}{c}-m,\frac{1}{2}-k,m-2-2k\\ -\frac{1}{2}-k,-2k\end{array}\right|-4\right)Y_{2k-2m}\left(x\right).$$

Proof. 

First, we replace n by $2k+1$ and j by m in (2). Then, $x^{2k-2m}$ is replaced by (9) with ${\overline{A}}_{m,k}$ given in the seventh row of the third column in Table 1 and ${\varphi }_{k-2m}$ given by the sixth kind of Chebyshev polynomial. Thus, we obtain

$$F_{2k+1}\left(x\right)=\sum _{m=0}^k\left(\genfrac{}{}{0pt}{}{2k-m}{m}\right)\sum _{j=0}^{k-m}\frac{(2k-2m-2j+2)}{2^{2j}j!}\frac{(2k-2m+1)!}{(2k-2m-j+2)!}{Y}_{2k-2m-2j}\left(x\right).$$

We re-arrange the above result by separating each value of m. Therefore, we substitute m by $k-m-j$ in the sums, in which case we extract all the terms corresponding to $j=0,1,2,\cdots$. Alternatively, we can collect all the terms given by the upper limit in the sums and the next highest terms and so on. In either case, the resulting sums can be expressed as

$$F_{2k+1}\left(x\right)=\sum _{m=0}^k(2m+2)Y_{2m}\left(x\right)\sum _{j=0}^{k-m}\frac{(2k-j)!}{2^{2k-2m-2j}j!}\frac{(2k+1-2j)}{(k-m-j)!(k+m+2-j)!}.$$

Now, we replace m by $k-m$ and use the following identities

$$(2k+1-2j)=(2k+1){(-k+1/2)}_j/{(-k-1/2)}_j,\frac{1}{(m-j)!}=\frac{{(-1)}^j}{m!}{(-m)}_j,$$

and

$$\frac{\mathrm{\Gamma }(2k-j+1)}{\mathrm{\Gamma }(2k-m-j+3)}={(-1)}^m\frac{{(m-2k-2)}_j}{{(-2k)}_j}\frac{\mathrm{\Gamma }(m-2k-2)}{\mathrm{\Gamma }(-2k)}.$$

By using more algebra and extending the sum over j to infinity, we finally obtain the third result in the theorem. □

Other connection formulas are presented in

Table 2. Solutions to inverse connection problems.

	Inverse Connection Formulas
Fibonacci-ultrapsherical	$\begin{array}{cc}{F}_{k+1}\left(x\right)\hfill & {\displaystyle =\frac{2^{1-k-2\lambda }\sqrt{\pi }k!}{\mathrm{\Gamma }\left(\lambda +1/2\right)}\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{(k-2m+\lambda )}{m!(k-2m)!}}\hfill \\ & {\displaystyle \times \frac{\mathrm{\Gamma }(k-2m+2\lambda )}{\mathrm{\Gamma }(1+k-m+\lambda )}{}_2{F}_1\left.\left(\begin{array}{c}-m,-k+m-\lambda \\ -k\end{array}\right|-4\right)\times C_{k-2m}^{\left(\lambda \right)}\left(x\right),}\hfill \end{array}$
Fibonacci–Chebyshev of the first kind	${\displaystyle F_{k+1}\left(x\right)=2^{1-k}k!\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{1}{m!(k-m)!}{}_2{F}_1\left.\left(\begin{array}{c}-m,-k+m\\ -k\end{array}\right|-4\right)T_{k-2m}\left(x\right)}$
Fibonacci–Chebyshev of the second kind	${\displaystyle F_{k+1}\left(x\right)=2^{-k}k!\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{(k-2m+1)}{m!(k-m+1)!}{}_2{F}_1\left.\left(\begin{array}{c}-m,-1-k+m\\ -k\end{array}\right|-4\right)U_{k-2m}\left(x\right)}$
Fibonacci–Hermite	${\displaystyle F_{k+1}\left(x\right)=\frac{1}{2^k}\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{k!}{m!(k-2m)!}{}_1{F}_1(-m;-k;4)H_{k-2m}\left(x\right)}$
Hermite–Fibonacci	${\displaystyle H_k\left(x\right)=k!\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{{(-1)}^m{2}^{k-2m}}{m!(k-2m)!}{}_1{F}_1(-m;2+k-2m;-4)F_{k-2m+1}\left(x\right)}$
Fibonacci–Chebyshev of the fifth kind	${\displaystyle \begin{array}{cc}& {\displaystyle F_{2k+1}\left(x\right)=k!\sum _{m=0}^k\frac{{\left(k-m+\frac{3}{2}\right)}_m}{(k-m)!m!{(2+2k-2m)}_m}{}_3{F}_2\left.\left(\begin{array}{c}-m,\frac{1}{2}-k,-1-2k+m\\ -\frac{1}{2}-k,-2k\end{array}\right|-4\right)X_{2k-2m}\left(x\right)}\hfill \\ & \begin{array}{cc}{F}_{2k+2}\left(x\right)\hfill & {\displaystyle =(2k+3)(2k+1)!\sum _{m=0}^k\frac{k+1-m}{2^{2m-1}(2k-2m+3)m!(2k-m+2)!}}\hfill \\ & \times {}_3{F}_2\left.\left(\begin{array}{c}-m,-\frac{1}{2}-k,-2-2k+m\\ -1-2k,-\frac{3}{2}-k\end{array}\right|-4\right)X_{2k-2m+1}\left(x\right);\hfill \end{array}\hfill \end{array}}$
Fibonacci–Chebyshev of the sixth kind	$\begin{array}{cc}{F}_{2k+2}\left(x\right)\hfill & {\displaystyle =(2k+3)(2k+1)!\sum _{m=0}^k\frac{k-m+1}{2^{2m-1}m!(2k-m+3)!}}\hfill \\ & {\displaystyle \times {}_3{F}_2\left.\left(\begin{array}{c}-m,-\frac{1}{2}-k,-3-2k+m\\ -1-2k,-\frac{3}{2}-k\end{array}\right|-4\right)Y_{2k-2m+1}\left(x\right).}\hfill \end{array}$

. Particularly, the first five formulas can be determined by replacing

$$x^{k-2j}=\sqrt{\pi }\sum _{m=0}^{\lfloor k/2\rfloor -j}{2}^{-k+2m-2j}(k-2j-2m+1/2)\frac{(k-2m)!}{j!\mathrm{\Gamma }(k-2j-m+3/2)m!}{P}_{k-2j-2m}\left(x\right).$$

(28)

in the corresponding columns of Table 1 and following the same procedure as in Theorem 4. Observe that (28) is obtained from (9) and the fourth row in the third column of Table 1. The results corresponding to Fibonacci–Chebyshev inverse formulas are obtained by the same procedure as in Theorem 7.

4.2. Connection Formulas Involving Non-Symmetric Polynomials

Theorem 8.

The connection formulas between the Chebyshev polynomials of the third and fourth kinds in terms of the Fibonacci polynomials are, respectively, given by

$$\begin{array}{cc}\hfill V_k\left(x\right)=& 2^{k}k!\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{{(-1)}^m(k-2m+1)}{m!(k-m+1)!}{}_2{F}_1\left.\left(\begin{array}{c}-m,m-k-1\\ -k\end{array}\right|-\frac{1}{4}\right)F_{k-2m+1}\left(x\right)\hfill \\ \hfill & +2^{k-1}(k-1)!\sum _{m=0}^{⌊\frac{k-1}{2}⌋}\frac{{(-1)}^m(2m-k)}{(k-m)!m!}{}_2{F}_1\left.\left(\begin{array}{c}-m,m-k\\ 1-k\end{array}\right|-\frac{1}{4}\right)F_{k-2m}\left(x\right),\hfill \end{array}$$

(29)

and

$$\begin{array}{cc}\hfill & W_k\left(x\right)=2^{k}k!\sum _{m=0}^{\lfloor \frac{k}{2}\rfloor }\frac{{(-1)}^m(k-2m+1)}{m!(k-m+1)!}{}_2{F}_1\left.\left(\begin{array}{c}-m,m-k-1\\ -k\end{array}\right|-\frac{1}{4}\right)F_{k-2m+1}\left(x\right)\hfill \\ \hfill & -2^{k-1}(k-1)!\sum _{m=0}^{⌊\frac{k-1}{2}⌋}\frac{{(-1)}^m(2m-k)}{(k-m)!m!}{}_2{F}_1\left.\left(\begin{array}{c}-m,m-k\\ 1-k\end{array}\right|-\frac{1}{4}\right)F_{k-2m}\left(x\right).\hfill \end{array}$$

(30)

Proof. 

The proof follows by the same approach used in Theorems 5 and 6, except that (10) are used instead of (9). □

Corollary 4.

The following trigonometric identities can be derived:

$$\begin{array}{cc}& 2^{k}k!\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{{(-1)}^m(k-2m+1)}{m!(k-m+1)!}{}_2{F}_1\left.\left(\begin{array}{c}-m,m-k-1\\ -k\end{array}\right|-\frac{1}{4}\right)F_{k-2m+1}(cos\theta )\hfill \\ & +2^{k-1}(k-1)!\sum _{m=0}^{⌊\frac{k-1}{2}⌋}\frac{{(-1)}^m(-k+2m)}{(k-m)!m!}{}_2{F}_1\left.\left(\begin{array}{c}-m,m-k\\ 1-k\end{array}\right|-\frac{1}{4}\right)F_{k-2m}(cos\theta )\hfill \\ & {\displaystyle =\frac{cos\left(\left(j+\frac{1}{2}\right)\theta \right)}{cos\left(\frac{\theta }{2}\right)},}\hfill \end{array}$$

$$\begin{array}{cc}& 2^{k}k!\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{{(-1)}^m(k-2m+1)}{m!(k-m+1)!}{}_2{F}_1\left.\left(\begin{array}{c}-m,m-k-1\\ -k\end{array}\right|-\frac{1}{4}\right)F_{k-2m+1}(cos\theta )\hfill \\ & -2^{k-1}(k-1)!\sum _{m=0}^{⌊\frac{k-1}{2}⌋}\frac{{(-1)}^m(2m-k)}{(k-m)!m!}{}_2{F}_1\left.\left(\begin{array}{c}-m,m-k\\ 1-k\end{array}\right|-\frac{1}{4}\right)F_{k-2m}(cos\theta )\hfill \\ & {\displaystyle =\frac{sin\left(\left(j+\frac{1}{2}\right)\theta \right)}{sin\left(\frac{\theta }{2}\right)},}\hfill \end{array}$$

and

$$2^{k}k!\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{{(-1)}^m(k-2m+1)}{m!(k-m+1)!}{F}_{k-2m+1}(cos\theta ){}_2{F}_1\left.\left(\begin{array}{c}-m,m-k-1\\ -k\end{array}\right|-\frac{1}{4}\right)=\frac{sin\left((j+1)\theta \right)}{sin\theta }.$$

Proof. 

The above results are obtained simply by replacing x by $cos\theta$ in (29) and (30) and then equating them to their respective trigonometric representations presented in Section 2. The third result follows by summing the two results in the corollary according to (13). Alternatively, it can be obtained by replacing x by $cos\theta$ in the third result of Corollary 3 and using the trigonometric representation of the Chebyshev polynomials of the second kind also in Section 2. □

Theorem 9.

The connection formulas between the Fibonacci polynomials and the Chebyshev polynomials of the third and fourth kinds are, respectively, given by

$$\begin{array}{cc}\hfill F_{k+1}\left(x\right)& {\displaystyle =\frac{k!}{2^k}\{\sum _{m=0}^{⌊\frac{k}{2}⌋}\frac{1}{m!(k-m)!}{V}_{k-2m}\left(x\right){}_2{F}_1\left.\left(\begin{array}{c}-m,m-k\\ -k\end{array}\right|-4\right)+\sum _{m=0}^{⌊\frac{k-1}{2}⌋}\frac{1}{m!(k-m)!}}\hfill \\ & \times {}_2{F}_1\left.\left(\begin{array}{c}-m,m-k\\ -k\end{array}\right|-4\right)V_{k-2m+1}\left(x\right)\}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill F_{k+1}\left(x\right)& {\displaystyle =\frac{k!}{2^k}\{\sum _{m=0}^{\lfloor \frac{k}{2}\rfloor }\frac{1}{m!(k-m)!}{W}_{k-2m}\left(x\right){}_2{F}_1\left(\left.\begin{array}{c}-m,m-k\\ -k\end{array}\right|-4\right)+\sum _{m=0}^{\lfloor \frac{k-1}{2}\rfloor }\frac{1}{m!(k-m)!}}\hfill \\ & \left.\times {}_2{F}_1\left(\begin{array}{c}-m,m-k\\ -k\end{array}\right|-4\right)W_{k-2m+1}\left(x\right)\}.\hfill \end{array}$$

Proof. 

Here, we shall describe the proof, leaving the reader to fill in the details. Basically, the results in the theorem can be proved by following Theorem 5, except that instead of using (9) to replace the powers of x in (2), we use (10), where the coefficients ${\overline{B}}_{m,k}$ appear in the last two rows of the third column in Table 1. As in the previous theorem, the even and odd values of the first sum have to be studied separately, which results in two different Gaussian hypergeometric functions appearing in the results for the Fibonacci polynomials when they are expressed in terms of either the Chebyshev polynomials of the third kind or Chebyshev polynomials of the fourth kind. □

5. Linear Products of Fibonacci and Orthogonal Polynomials

In this section, we derive new formulas in terms of the Fibonacci polynomials for the products of linear powers of the Fibonacci polynomials with orthogonal polynomials.

Theorem 10.

For all nonnegative integers j and k, the linear product of the Fibonacci polynomials, $F_{j+1}\left(x\right)$, with the ultraspherical polynomials, $C_k^{\left(\lambda \right)}\left(x\right)$ can be expressed as

$$\begin{array}{cc}\hfill F_{j+1}\left(x\right)C_k^{\left(\lambda \right)}\left(x\right)=& \frac{2^{-1+k+2\lambda }\mathrm{\Gamma }\left(\frac{1}{2}+\lambda \right)\mathrm{\Gamma }(k+\lambda )}{\sqrt{\pi }\mathrm{\Gamma }(k+2\lambda )}\times \hfill \\ & \sum _{p=0}^k{(-1)}^p\left(\genfrac{}{}{0pt}{}{k}{p}\right){}_2{F}_1\left.\left(\begin{array}{c}-p,p-k\\ 1-k-\lambda \end{array}\right|-\frac{1}{4}\right)F_{j+k-2p+1}\left(x\right).\hfill \end{array}$$

Proof. 

By expressing the ultraspherical polynomials, $C_k^{\left(\lambda \right)}\left(x\right)$ in powers of x, i.e., by introducing the coefficient in the fifth row of the second column in Table 1 into (7), we find that

$$F_{j+1}\left(x\right)C_k^{\left(\lambda \right)}\left(x\right)=\frac{k!\mathrm{\Gamma }(2\lambda +1)}{\mathrm{\Gamma }(\lambda +1)\mathrm{\Gamma }(k+2\lambda )}\sum _{r=0}^{⌊\frac{k}{2}⌋}\frac{{(-1)}^r{2}^{k-2r-1}\mathrm{\Gamma }(k-r+\lambda )}{(k-2r)!r!}{x}^{k-2r}{F}_{j+1}\left(x\right).$$

At this stage, we introduce the moments formula for the Fibonacci polynomials, namely (4), into the above result. This yields

$$\begin{array}{cc}\hfill F_{j+1}\left(x\right)C_k^{\left(\lambda \right)}\left(x\right)=& \frac{k!\mathrm{\Gamma }(2\lambda +1)}{\mathrm{\Gamma }(1+\lambda )\mathrm{\Gamma }(k+2\lambda )}\sum _{r=0}^{⌊\frac{k}{2}⌋}\frac{{(-1)}^r{2}^{k-2r-1}\mathrm{\Gamma }(k-r+\lambda )}{(k-2r)!r!}\hfill \\ & \times \sum _{s=0}^{k-2r}{(-1)}^s\left(\genfrac{}{}{0pt}{}{k-2r}{s}\right)F_{j+k-2r-2s+1}\left(x\right).\hfill \end{array}$$

Next, we expand the above result for each value of s. As we have done in other proofs appearing in this paper, we re-arrange the terms. This results in the following double sum:

$$F_{j+1}\left(x\right)C_k^{\left(\lambda \right)}\left(x\right)=\frac{k!\mathrm{\Gamma }(1+2\lambda )}{\mathrm{\Gamma }(1+\lambda )\mathrm{\Gamma }(k+2\lambda )}\sum _{p=0}^j{(-1)}^p\left\{\sum _{r=0}^p\frac{2^{k-2r-1}\left(\genfrac{}{}{0pt}{}{k-2r}{p-r}\right)\mathrm{\Gamma }(k-r+\lambda )}{(k-2r)!r!}\right\}{F}_{j+k-2p+1}\left(x\right).$$

To simplify the sum over r, we introduce the following identities [34]:

$$\frac{\mathrm{\Gamma }(k-r+\lambda )}{\mathrm{\Gamma }(k-r-p+1)}=\frac{{(p-k)}_r}{{(1-k-\lambda )}_r}\frac{\mathrm{\Gamma }(k+\lambda )}{\mathrm{\Gamma }(k-p+1)},$$

and

$$\frac{1}{\mathrm{\Gamma }(p-r+1)}=\frac{{(-1)}^r}{p!}{(-p)}_r,$$

therefore, we can write

$$F_{j+1}\left(x\right)C_k^{\left(\lambda \right)}\left(x\right)=\frac{2^{k-1}\mathrm{\Gamma }\left(2\lambda +1\right)\mathrm{\Gamma }(k+\lambda )}{\mathrm{\Gamma }(\lambda +1)\mathrm{\Gamma }(k+2\lambda )}\sum _{p=0}^k{(-1)}^p\left(\genfrac{}{}{0pt}{}{k}{p}\right){}_2{F}_1\left.\left(\begin{array}{c}-p,p-k\\ 1-k-\lambda \end{array}\right|-\frac{1}{4}\right)F_{j+k-2p+1}\left(x\right).$$

Applying the duplication formula [35] to $\mathrm{\Gamma }(2\lambda +1)$ in the above equation, we obtain the result given in Theorem 10. □

Corollary 5.

For all nonnegative integers, j and k, the following results hold

$$\begin{array}{cc}\hfill F_{j+1}\left(x\right)P_k\left(x\right)=& \frac{2^k\mathrm{\Gamma }\left(\frac{1}{2}+k\right)}{\sqrt{\pi }k!}\sum _{p=0}^k{(-1)}^p\left(\genfrac{}{}{0pt}{}{k}{p}\right){}_2{F}_1\left.\left(\begin{array}{c}-p,p-k\\ \frac{1}{2}-k\end{array}\right|-\frac{1}{4}\right)F_{j+k-2p+1}\left(x\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill F_{j+1}\left(x\right)T_k\left(x\right)=& 2^{-1+k}\sum _{p=0}^k{(-1)}^p\left(\genfrac{}{}{0pt}{}{k}{p}\right){}_2{F}_1\left.\left(\begin{array}{c}-p,p-k\\ 1-k\end{array}\right|-\frac{1}{4}\right)F_{j+k-2p+1}\left(x\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill F_{j+1}\left(x\right)U_k\left(x\right)=& 2^k\sum _{p=0}^k{(-1)}^p\left(\genfrac{}{}{0pt}{}{k}{p}\right){}_2{F}_1\left.\left(\begin{array}{c}-p,p-k\\ -k\end{array}\right|-\frac{1}{4}\right)F_{j+k-2p+1}\left(x\right),\hfill \end{array}$$

$$\begin{array}{cc}& F_{i+1}\left(x\right)X_{2j}\left(x\right)=\sum _{p=0}^{2j}{(-1)}^p\left(\genfrac{}{}{0pt}{}{2j}{p}\right){}_3{F}_2\left.\left(\begin{array}{c}-p,-\frac{1}{2}-j,p-2j\\ \frac{1}{2}-j,-2j\end{array}\right|-\frac{1}{4}\right)F_{i+2j-2p+1}\left(x\right),\hfill \end{array}$$

$$\begin{array}{cc}& F_{i+1}\left(x\right)X_{2j+1}\left(x\right)=\sum _{p=0}^{2j+1}{(-1)}^p\left(\genfrac{}{}{0pt}{}{1+2j}{p}\right){}_3{F}_2\left.\left(\begin{array}{c}-p,-\frac{3}{2}-j,p-2j-1\\ -1-2j,-\frac{1}{2}-j\end{array}\right|-\frac{1}{4}\right)F_{i+2j-2p+2}\left(x\right),\hfill \end{array}$$

$$\begin{array}{cc}& F_{i+1}\left(x\right)Y_{2j}\left(x\right)=\sum _{p=0}^{2j}{(-1)}^p\left(\genfrac{}{}{0pt}{}{2j}{p}\right){}_3{F}_2\left.\left(\begin{array}{c}-p,-\frac{1}{2}-j,p-2j\\ -1-2j,\frac{1}{2}-j\end{array}\right|-\frac{1}{4}\right)F_{i+2j-2p+1}\left(x\right),\hfill \end{array}$$

$$\begin{array}{cc}& F_{i+1}\left(x\right)Y_{2j+1}\left(x\right)=\sum _{p=0}^{2j+1}{(-1)}^p\left(\genfrac{}{}{0pt}{}{1+2j}{p}\right){}_3{F}_2\left.\left(\begin{array}{c}-p,-\frac{3}{2}-j,p-2j-1\\ -2-2j,-\frac{1}{2}-j\end{array}\right|-\frac{1}{4}\right)F_{i+2j-2p+2}\left(x\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill F_{j+1}\left(x\right)V_k\left(x\right)& =2^k\sum _{p=0}^k{(-1)}^p\left(\genfrac{}{}{0pt}{}{k}{p}\right){}_2{F}_1\left.\left(\begin{array}{c}-p,-k+p\\ -k\end{array}\right|-\frac{1}{4}\right)F_{j+k-2p+1}\left(x\right)\hfill \\ & -2^{-1+k}\sum _{p=0}^{k-1}{(-1)}^p\left(\genfrac{}{}{0pt}{}{-1+k}{p}\right){}_2{F}_1\left.\left(\begin{array}{c}-p,p-k+1\\ 1-k\end{array}\right|-\frac{1}{4}\right)F_{j+k-2p}\left(x\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill F_{j+1}\left(x\right)W_k\left(x\right)& =2^k\sum _{p=0}^k{(-1)}^p\left(\genfrac{}{}{0pt}{}{k}{p}\right){}_2{F}_1\left.\left(\begin{array}{c}-p,p-k\\ -k\end{array}\right|-\frac{1}{4}\right)F_{j+k-2p+1}\left(x\right)\hfill \\ & +2^{-1+k}\sum _{p=0}^{k-1}{(-1)}^p\left(\genfrac{}{}{0pt}{}{-1+k}{p}\right){}_2{F}_1\left.\left(\begin{array}{c}-p,p-k+1\\ 1-k\end{array}\right|-\frac{1}{4}\right)F_{j+k-2p}\left(x\right).\hfill \end{array}$$

Proof. 

The proofs for the above results follow the proof in Theorem 10 except that the corresponding results in the second column of Table 1 represent the starting point for each proof. Note that combining the last two results in the corollary according to (13) also yield the third result in the corollary. □

6. Conclusions

This paper began with the derivation of new formulas for the higher-order derivatives and anti-derivatives of the Fibonacci polynomials. With the aid of these results, several interesting connection formulas involving the Fibonacci polynomials and both types of orthogonal polynomials (symmetric and non-symmetric) were derived. These were then followed by the corresponding inverse connection formulas. Finally, by using similar methods, we were also able to derive several formulas for the linear products of the Fibonacci polynomials with orthogonal polynomials.