Overview of Six Number/Polynomial Sequences Defined by Quadratic Recurrence Relations

Abstract

Six well-known sequences (Fibonacci and Lucas numbers, Pell and Pell–Lucas polynomials, and Chebyshev polynomials) are characterized by quadratic linear recurrence relations. They are unified and reviewed under a common framework. Several useful properties (such as Binet-form expressions, Cassini identities, and Catalan formulae) and remarkable results concerning power sums, ordinary convolutions, and binomial convolutions are presented by employing the symmetric feature, series rearrangements, and the generating function approach. Most of the classical results concerning these six number/polynomial sequences are recorded as consequences.

1. Introduction and Outline

Numerous sequences satisfying linear difference equations have been described in the literature (cf. [1]). By resolving a linear recurrence relation of order two, we introduce two parametric sequences ${\Phi }_n(u,v)$ and ${\Psi }_n(u,v)$. They will provide a common framework for unifying six well-known number/polynomial sequences.

1.1. Quadratic Recurrence Relation

We shall examine the sequence ${\Lambda }_n$ defined by the quadratic recurrence relation

$${\Lambda }_{n+1}=2u{\Lambda }_n+v{\Lambda }_{n-1}\mathrm{with}\mathrm{initial}\mathrm{values}{\Lambda }_0=2a\mathrm{and}{\Lambda }_1=c,$$

where $a,c,u,$ and v are four complex numbers subject to $\Delta =\sqrt{u^2+v}\ne 0$. By manipulating the formal power series

$$\begin{array}{cc}\hfill G\left(y\right)& :=\sum _{n=0}^{\infty }{\Lambda }_n{y}^n=2a+cy+\sum _{n=1}^{\infty }{\Lambda }_{n+1}{y}^{n+1}\hfill \\ & =2a+cy+\sum _{n=1}^{\infty }\left(2u{\Lambda }_n+v{\Lambda }_{n-1}\right)y^{n+1}\hfill \\ & =2a+cy+2uy\left\{G\left(y\right)-2a\right\}+v{y}^{2}G\left(y\right),\hfill \end{array}$$

we derive the rational generating function (cf. [2,3])

$$G\left(y\right)=\sum _{n=0}^{\infty }{\Lambda }_n{y}^n={\displaystyle \frac{2a-4auy+cy}{1-2uy-v{y}^2}}.$$

Applying the partial fraction decomposition

$$G\left(y\right)={\displaystyle \frac{c-2a(u-\Delta )}{2\Delta \{1-y(u+\Delta \left)\right\}}}-{\displaystyle \frac{c-2a(u+\Delta )}{2\Delta \{1-y(u-\Delta \left)\right\}}}$$

and then expanding into the power series, we derive the explicit formula

$${\Lambda }_n={\displaystyle \frac{c-2a(u-\Delta )}{2\Delta }}{\left(u+\Delta \right)}^n-{\displaystyle \frac{c-2a(u+\Delta )}{2\Delta }}{\left(u-\Delta \right)}^n.$$

In particular, we have

$$\begin{array}{cc}\hfill \begin{array}{|c|}\hline a=0\\ \hline\end{array}& {\Lambda }_n⟹{\displaystyle \frac{c}{2\Delta }}{\left(u+\Delta \right)}^n-{\displaystyle \frac{c}{2\Delta }}{\left(u-\Delta \right)}^n,\hfill \\ \hfill \begin{array}{|c|}\hline c=2au\\ \hline\end{array}& {\Lambda }_n⟹a{\left(u+\Delta \right)}^n+a{\left(u-\Delta \right)}^n.\hfill \end{array}$$

1.2. Sequences ${\Phi }_n(u,v)$ and ${\Psi }_n(u,v)$

For brevity, we define

$$\begin{array}{c}\alpha :=\alpha (u,v)=u+\sqrt{u^2+v}\hfill \\ \gamma :=\gamma (u,v)=u-\sqrt{u^2+v}\hfill \end{array}\}\mathrm{with}\alpha \gamma =-v\mathrm{and}\alpha \pm \gamma =\left\{\begin{array}{cc}2u,\hfill & “+”;\hfill \\ 2\Delta ,\hfill & “-”.\hfill \end{array}\right.$$

By dropping the constants a and c from ${\Lambda }_n$, we introduce, for $n\in \mathbb{Z}$, two sequences:

$$\begin{array}{cccc}\hfill {\Phi }_n(u,v)& ={\displaystyle \frac{{(u+\Delta )}^n-{(u-\Delta )}^n}{2\Delta }}\hfill & & ={\displaystyle \frac{{\alpha }^n(u,v)-{\gamma }^n(u,v)}{\alpha (u,v)-\gamma (u,v)}},\hfill \end{array}$$

(1)

$$\begin{array}{cccc}\hfill {\Psi }_n(u,v)& ={(u+\Delta )}^n+{(u-\Delta )}^n\hfill & & ={\alpha }^n(u,v)+{\gamma }^n(u,v).\hfill \end{array}$$

(2)

These two sequences satisfy the common recurrence relation, but with different initial values

$$\begin{array}{cccccc}\hfill & {\Phi }_{n+1}=2u{\Phi }_n+v{\Phi }_{n-1}:\hfill & \hfill & \Phi \left(0\right)=0,\hfill & & \Phi \left(1\right)=1;\hfill \end{array}$$

(3)

$$\begin{array}{cccccc}& {\Psi }_{n+1}=2u{\Psi }_n+v{\Psi }_{n-1}:\hfill & & \Psi \left(0\right)=2,\hfill & \hfill & \Psi \left(1\right)=2u.\hfill \end{array}$$

(4)

Their ordinary and exponential generating functions are as follows:

$$\begin{array}{cccc}\hfill & \sum _{n=0}^{\infty }{y}^n{\Phi }_n(u,v)={\displaystyle \frac{y}{1-2uy-v{y}^2}},\hfill & \hfill & \sum _{n=0}^{\infty }{y}^n{\Psi }_n(u,v)={\displaystyle \frac{2-2uy}{1-2uy-v{y}^2}};\hfill \end{array}$$

(5)

$$\begin{array}{cccc}\hfill & \sum _{n=0}^{\infty }{\displaystyle \frac{y^n}{n!}}{\Phi }_n(u,v)={\displaystyle \frac{e^{y\alpha }-e^{y\gamma }}{2\sqrt{u^2+v}}},\hfill & & \sum _{n=0}^{\infty }{\displaystyle \frac{y^n}{n!}}{\Psi }_n(u,v)=e^{y\alpha }+e^{y\gamma }.\hfill \end{array}$$

(6)

1.3. Six Number/Polynomial Sequences

These parametric sequences ${\Phi }_n(u,v)$ and ${\Psi }_n(u,v)$ are remarkable because they unify the following well-known sequences:

u	v	${\Phi }_n(u,v)$	${\Psi }_n(u,v)$
${\displaystyle \frac{1}{2}}$	1	Fibonacci fumber $F_n$	Lucas number $L_n$
x	1	Pell polynomial $P_n\left(x\right)$	Pell–Lucas polynomial $Q_n\left(x\right)$
x	$-1$	Chebyshev polynomial $U_{n-1}\left(x\right)$	Chebyshev polynomial $2T_n\left(x\right)$

These sequences appear frequently in combinatorics, number theory, and special functions, and can explicitly be expressed as follows:

• Fibonacci and Lucas numbers (cf. Koshy [4] and [5], A000045 and A000032):

  $$F_n={\displaystyle \frac{{\overline{\alpha }}^n-{\overline{\gamma }}^n}{\overline{\alpha }-\overline{\gamma }}}\mathrm{and}{L}_n={\overline{\alpha }}^n+{\overline{\gamma }}^n\mathrm{with}\overline{\alpha },\overline{\gamma }={\displaystyle \frac{1\pm \sqrt{5}}{2}}.$$
• Pell and Pell–Lucas polynomials (cf. [6]):

  $$P_n={\displaystyle \frac{{\widehat{\alpha }}^n-{\widehat{\gamma }}^n}{\alpha -\gamma }}\mathrm{and}{Q}_n={\widehat{\alpha }}^n+{\widehat{\gamma }}^n\mathrm{with}\widehat{\alpha },\widehat{\gamma }=x\pm \sqrt{1+x^2}.$$
• Chebyshev polynomials of the first and second kinds (cf. [7]):

  $$T_n={\displaystyle \frac{{\tilde{\alpha }}^n+{\tilde{\gamma }}^n}{2}}\mathrm{and}{U}_n={\displaystyle \frac{{\tilde{\alpha }}^{n+1}-{\tilde{\gamma }}^{n+1}}{\tilde{\alpha }-\tilde{\gamma }}}\mathrm{with}\tilde{\alpha },\tilde{\gamma }=x\pm \sqrt{x^2-1}.$$

There is a long history as well as vast literature concerning these six sequences (for example, [4,6,7]), which have a wide range of applications in mathematics, physics, computer science, and applied sciences:

• Fibonacci and Lucas numbers occur frequently in mathematics (number theory and primality testing), algorithmic design and analysis in computer sciences, coding theory, recursive methods, and the recognition of regular patterns existing in nature;
• Pell and Pell–Lucas polynomials are important in number theory and Diophantine analysis, recursive constructions, and continued fractions;
• Chebyshev polynomials play fundamental roles in approximation theory and numerical analysis, Fourier series, and special functions.

The aim of this paper is to present a comprehensive overview of them under the common framework of the $\{{\Phi }_n,{\Psi }_n\}$ sequences. As preliminaries, some basic properties will be given in the next section. Then, in Section 3, a number of finite sums and identities will be shown. Closed formulae regarding binomial sums will be derived in Section 4. Finally, we will briefly discuss further prospects.

Each class of summation formulae concerning the $\{{\Phi }_n,{\Psi }_n\}$ sequences will be followed by their corresponding formulae concerning Fibonacci and Lucas numbers, Pell and Pell–Lucas polynomials, and Chebyshev polynomials (of the first and second kinds). Despite attempts to provide a systematic treatment, it would be impossible to cover all aspects of these six sequences due to the enormous quantity of publications addressing them. To limit the length of the article, we have carefully selected and classified the results that we will present, including several new equations, even though it is not our primary concern. The authors hope that this compendium may serve as a reference source for readers in their further investigations.

2. Basic Properties and Preliminary Results

First of all, by making use of the Binet-form expressions (1) and (2), we deduce the two fundamental relations between ${\Phi }_n(u,v)$ and ${\Psi }_n(u,v)$:

$$\begin{array}{cc}\hfill {\Phi }_{n+1}(u,v)+v{\Phi }_{n-1}(u,v)& ={\Psi }_n(u,v),\hfill \\ \hfill {\Psi }_{n+1}(u,v)+v{\Psi }_{n-1}(u,v)& =4{\Delta }^2{\Phi }_n(u,v).\hfill \end{array}$$

They are common generalizations of the following three pairs of well-known equations:

$$\begin{array}{cccc}& F_{n+1}+F_{n-1}=L_n,\hfill & & L_{n+1}+L_{n-1}=5F_n;\hfill \\ & P_{n+1}\left(x\right)+P_{n-1}\left(x\right)=Q_n\left(x\right),\hfill & & Q_{n+1}\left(x\right)+Q_{n-1}\left(x\right)=4(1+x^2)P_n\left(x\right);\hfill \\ & U_{n+1}\left(x\right)-U_{n-1}\left(x\right)=2T_{n+1}\left(x\right),\hfill & & T_{n+1}\left(x\right)-T_{n-1}\left(x\right)=2(x^2-1)U_{n-1}\left(x\right).\hfill \end{array}$$

Then, by manipulating (1) and (2), we can, without difficulty, show further properties of ${\Phi }_n(u,v)$ and ${\Psi }_n(u,v)$, that are highlighted in three theorems.

2.1. Cassini-like Formulae

Theorem 1 

($m,n\in \mathbb{Z}$).

$$\begin{array}{cc}\hfill \left(a\right)& {\Phi }_{m+n}{\Phi }_{m-n}-{\Phi }_m^2=-{(-v)}^{m-n}{\Phi }_n^2,\hfill \\ \hfill \left(b\right)& {\Psi }_{m+n}{\Psi }_{m-n}-{\Psi }_m^2=4{\Delta }^2{(-v)}^{m-n}{\Phi }_n^2.\hfill \end{array}$$

This theorem unifies the results below for the six number/polynomial sequences in question:

• Fibonacci and Lucas numbers ($m,n\in \mathbb{Z}$):

  $$\begin{array}{cc}& F_{m+n}{F}_{m-n}-F_m^2={(-1)}^{m-n+1}{F}_n^2,\hfill \\ & L_{m+n}{L}_{m-n}-L_m^2=5{(-1)}^{m-n}{F}_n^2.\hfill \end{array}$$
• Pell and Pell–Lucas polynomials ($m,n\in \mathbb{Z}$):

  $$\begin{array}{cc}& P_{m+n}\left(x\right)P_{m-n}\left(x\right)-P_m^2\left(x\right)={(-1)}^{m-n+1}{P}_n^2\left(x\right),\hfill \\ & Q_{m+n}\left(x\right)Q_{m-n}\left(x\right)-Q_m^2\left(x\right)=4{(-1)}^{m-n}(1+x^2)P_n^2\left(x\right).\hfill \end{array}$$
• Chebyshev polynomials ($m,n\in \mathbb{Z}$):

  $$\begin{array}{cc}& U_{m+n}\left(x\right)U_{m-n}\left(x\right)-U_m^2\left(x\right)=-U_{n-1}^2\left(x\right),\hfill \\ & T_{m+n}\left(x\right)T_{m-n}\left(x\right)-T_m^2\left(x\right)=(x^2-1)U_{n-1}^2\left(x\right).\hfill \end{array}$$

2.2. Catalan-like Identities

Theorem 2 

($m,n\in \mathbb{Z}$).

$$\begin{array}{cc}\hfill \left(a\right)& {\Phi }_{m+n}+v^n{\Phi }_{m-n}=\left\{\begin{array}{cc}{\Phi }_m{\Psi }_n,\hfill & \mathrm{even}n;\hfill \\ {\Phi }_n{\Psi }_m,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ \hfill \left(b\right)& {\Phi }_{m+n}-v^n{\Phi }_{m-n}=\left\{\begin{array}{cc}{\Phi }_n{\Psi }_m,\hfill & \mathrm{even}n;\hfill \\ {\Phi }_m{\Psi }_n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ \hfill \left(c\right)& {\Psi }_{m+n}+v^n{\Psi }_{m-n}=\left\{\begin{array}{cc}{\Psi }_m{\Psi }_n,\hfill & \mathrm{even}n;\hfill \\ 4{\Delta }^2{\Phi }_m{\Phi }_n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ \hfill \left(d\right)& {\Psi }_{m+n}-v^n{\Psi }_{m-n}=\left\{\begin{array}{cc}4{\Delta }^2{\Phi }_m{\Phi }_n,\hfill & \mathrm{even}n;\hfill \\ {\Psi }_m{\Psi }_n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

They are common generalizations of well-known results in two variables $\{u,v\}$:

• Fibonacci and Lucas numbers ($m,n\in \mathbb{Z}$):

  $$\begin{array}{cc}& F_{m+n}+F_{m-n}=\left\{\begin{array}{cc}{F}_m{L}_n,\hfill & \mathrm{even}n;\hfill \\ F_n{L}_m,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & F_{m+n}-F_{m-n}=\left\{\begin{array}{cc}{F}_n{L}_m,\hfill & \mathrm{even}n;\hfill \\ F_m{L}_n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & L_{m+n}+L_{m-n}=\left\{\begin{array}{cc}{L}_m{L}_n,\hfill & \mathrm{even}n;\hfill \\ 5F_m{F}_n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & L_{m+n}-L_{m-n}=\left\{\begin{array}{cc}5F_m{F}_n,\hfill & \mathrm{even}n;\hfill \\ L_m{L}_n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$
• Pell and Pell–Lucas polynomials ($m,n\in \mathbb{Z}$):

  $$\begin{array}{cc}& P_{m+n}\left(x\right)+P_{m-n}\left(x\right)=\left\{\begin{array}{cc}{P}_m\left(x\right)Q_n\left(x\right),\hfill & \mathrm{even}n;\hfill \\ P_n\left(x\right)Q_m\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & P_{m+n}\left(x\right)-P_{m-n}\left(x\right)=\left\{\begin{array}{cc}{P}_n\left(x\right)Q_m\left(x\right),\hfill & \mathrm{even}n;\hfill \\ P_m\left(x\right)Q_n\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & Q_{m+n}\left(x\right)+Q_{m-n}\left(x\right)=\left\{\begin{array}{cc}{Q}_m\left(x\right)Q_n\left(x\right),\hfill & \mathrm{even}n;\hfill \\ 4(1+x^2)P_m\left(x\right)P_n\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & Q_{m+n}\left(x\right)-Q_{m-n}\left(x\right)=\left\{\begin{array}{cc}4(1+x^2)P_m\left(x\right)P_n\left(x\right),\hfill & \mathrm{even}n;\hfill \\ Q_m\left(x\right)Q_n\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$
• Chebyshev polynomials ($m,n\in \mathbb{Z}$):

  $$\begin{array}{cc}& U_{m+n}\left(x\right)+{(-1)}^n{U}_{m-n}\left(x\right)=\left\{\begin{array}{cc}2U_m\left(x\right)T_n\left(x\right),\hfill & \mathrm{even}n;\hfill \\ 2U_{n-1}\left(x\right)T_{m+1}\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & U_{m+n}\left(x\right)-{(-1)}^n{U}_{m-n}\left(x\right)=\left\{\begin{array}{cc}2U_{n-1}\left(x\right)T_{m+1}\left(x\right),\hfill & \mathrm{even}n;\hfill \\ 2U_m\left(x\right)T_n\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & T_{m+n}\left(x\right)+{(-1)}^n{T}_{m-n}\left(x\right)=\left\{\begin{array}{cc}2T_m\left(x\right)T_n\left(x\right),\hfill & \mathrm{even}n;\hfill \\ 2(x^2-1)U_{m-1}\left(x\right)U_{n-1}\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & T_{m+n}\left(x\right)-{(-1)}^n{T}_{m-n}\left(x\right)=\left\{\begin{array}{cc}2(x^2-1)U_{m-1}\left(x\right)U_{n-1}\left(x\right),\hfill & \mathrm{even}n;\hfill \\ 2T_m\left(x\right)T_n\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

2.3. Simple Linear Sums

Theorem 3 

($n\in \mathbb{N}$).

$$\begin{array}{cccc}\hfill \left(a\right)& \sum _{k=1}^n{v}^{n-k}{\Phi }_k={\displaystyle \frac{{\Phi }_n+{\Phi }_{n+1}-v^n}{1+2u-v}},\hfill & \hfill & \left(d\right)\sum _{k=1}^n{v}^{n-k}{\Psi }_k={\displaystyle \frac{{\Psi }_n+{\Psi }_{n+1}-2v^n-2u{v}^n}{1+2u-v}},\hfill \\ \hfill \left(b\right)& \sum _{k=1}^n{v}^{n-k}{\Phi }_{2k}={\displaystyle \frac{{\Phi }_{2n+1}-v^n}{2u}},\hfill & & \left(e\right)\sum _{k=1}^n{v}^{n-k}{\Psi }_{2k}={\displaystyle \frac{{\Psi }_{2n+1}-2u{v}^n}{2u}},\hfill \\ \hfill \left(c\right)& \sum _{k=1}^n{v}^{n-k}{\Phi }_{2k-1}={\displaystyle \frac{{\Phi }_{2n}}{2u}};\hfill & & \left(f\right)\sum _{k=1}^n{v}^{n-k}{\Psi }_{2k-1}={\displaystyle \frac{{\Psi }_{2n}-2v^n}{2u}}.\hfill \end{array}$$

Proof. 

As an example, we illustrate below how to confirm Formula (c) by means of the Binet Formula (1). The remaining proofs can be carried out similarly.

$$\begin{array}{cc}\hfill \sum _{k=1}^n{v}^{n-k}{\Phi }_{2k-1}& =\sum _{k=1}^n{v}^{n-k}{\displaystyle \frac{{\alpha }^{2k-1}-{\gamma }^{2k-1}}{\alpha -\gamma }}\hfill \\ & ={\displaystyle \frac{\alpha v^{n-1}(1-{\alpha }^{2n}/v^n)}{(\alpha -\gamma )(1-{\alpha }^2/v)}}-{\displaystyle \frac{\gamma v^{n-1}(1-{\gamma }^{2n}/v^n)}{(\alpha -\gamma )(1-{\gamma }^2/v)}}\hfill \\ & ={\displaystyle \frac{v^n(1-{\gamma }^{2n}/v^n)}{(\alpha -\gamma )(\alpha +\gamma )}}-{\displaystyle \frac{v^n(1-{\alpha }^{2n}/v^n)}{(\alpha -\gamma )(\alpha +\gamma )}}\hfill \\ & ={\displaystyle \frac{{\alpha }^{2n}-{\gamma }^{2n}}{(\alpha -\gamma )(\alpha +\gamma )}}={\displaystyle \frac{{\Phi }_{2n}}{\alpha +\gamma }}={\displaystyle \frac{{\Phi }_{2n}}{2u}}.\hfill \end{array}$$

□

They contain the following finite sums as special cases:

• Fibonacci and Lucas numbers ($n\in \mathbb{N}$):

  $$\begin{array}{cccc}& \sum _{k=1}^n{F}_k=F_{n+2}-1,\hfill & \hfill & \sum _{k=1}^n{L}_k=L_{n+2}-3,\hfill \\ & \sum _{k=1}^n{F}_{2k}=F_{2n+1}-1,\hfill & & \sum _{k=1}^n{L}_{2k}=L_{2n+1}-1,\hfill \\ & \sum _{k=1}^n{F}_{2k-1}=F_{2n};\hfill & & \sum _{k=1}^n{L}_{2k-1}=L_{2n}-2.\hfill \end{array}$$
• Pell and Pell–Lucas polynomials ($n\in \mathbb{N}$):

  $$\begin{array}{cccc}& \sum _{k=1}^n{P}_k\left(x\right)={\displaystyle \frac{P_n\left(x\right)+P_{n+1}\left(x\right)-1}{2x}},\hfill & \hfill & \sum _{k=1}^n{Q}_k\left(x\right)={\displaystyle \frac{Q_n\left(x\right)+Q_{n+1}\left(x\right)-2x-2}{2x}},\hfill \\ & \sum _{k=1}^n{P}_{2k}\left(x\right)={\displaystyle \frac{P_{2n+1}\left(x\right)-1}{2x}},\hfill & & \sum _{k=1}^n{Q}_{2k}\left(x\right)={\displaystyle \frac{Q_{2n+1}\left(x\right)-2x}{2x}},\hfill \\ & \sum _{k=1}^n{P}_{2k-1}\left(x\right)={\displaystyle \frac{P_{2n}\left(x\right)}{2x}};\hfill & & \sum _{k=1}^n{Q}_{2k-1}\left(x\right)={\displaystyle \frac{Q_{2n}\left(x\right)-2}{2x}}.\hfill \end{array}$$
• Chebyshev polynomials ($n\in \mathbb{N}$):

  $$\begin{array}{cccc}& \sum _{k=0}^n{(-1)}^{n-k}{U}_k\left(x\right)=\frac{U_n\left(x\right)+U_{n+1}\left(x\right)+{(-1)}^n}{2+2x},\hfill & \hfill & \sum _{k=1}^n{(-1)}^{n-k}{T}_k\left(x\right)=\frac{T_n\left(x\right)+T_{n+1}\left(x\right)}{2+2x}-\frac{{(-1)}^n}{2},\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}{U}_{2k}\left(x\right)={\displaystyle \frac{U_{2n+1}\left(x\right)}{2x}},\hfill & & \sum _{k=1}^n{(-1)}^{n-k}{T}_{2k}\left(x\right)={\displaystyle \frac{T_{2n+1}\left(x\right)-{(-1)}^{n}x}{2x}},\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}{U}_{2k-1}\left(x\right)={\displaystyle \frac{U_{2n}\left(x\right)-{(-1)}^n}{2x}};\hfill & & \sum _{k=1}^n{(-1)}^{n-k}{T}_{2k-1}\left(x\right)={\displaystyle \frac{T_{2n}\left(x\right)-{(-1)}^n}{2x}}.\hfill \end{array}$$

3. Power Sums and Convolution Identities

In this section, we will evaluate further finite sums of the squares and products of ${\Phi }_n(u,v)$ and ${\Psi }_n(u,v)$. They summarize numerous known identities of the six number/polynomial sequences as particular cases.

3.1. Quadratic Sums

Theorem 4 

($n\in \mathbb{N}$).

$$\begin{array}{cc}\hfill \left(a\right)& \sum _{k=0}^n{v}^{n-k}{\Phi }_k^2={\displaystyle \frac{{\Phi }_n{\Phi }_{n+1}}{2u}},\hfill \\ \hfill \left(b\right)& \sum _{k=0}^n{v}^{n-k}{\Psi }_k^2={\displaystyle \frac{{\Psi }_n{\Psi }_{n+1}}{2u}}+2v^n;\hfill \\ \hfill \left(c\right)& \sum _{k=0}^n{(-v)}^{n-k}{\Phi }_k^2={\displaystyle \frac{{\Phi }_{n+1}{\Psi }_n}{4{\Delta }^2}}-{\displaystyle \frac{n+1}{2{\Delta }^2}}{(-v)}^n,\hfill \\ \hfill \left(d\right)& \sum _{k=0}^n{(-v)}^{n-k}{\Psi }_k^2={\Phi }_{n+1}{\Psi }_n+2(n+1){(-v)}^n.\hfill \end{array}$$

Proof. 

As an example, we give an induction proof of Formula (a). Recalling the recurrence relation (3), we have

$$\begin{array}{cc}\hfill {\Phi }_n{\Phi }_{n+1}& =2u{\Phi }_n^2+v{\Phi }_n{\Phi }_{n-1}\hfill \\ & =2u({\Phi }_n^2+v{\Phi }_{n-1}^2)+v^2{\Phi }_{n-1}{\Phi }_{n-2}\hfill \\ & =2u({\Phi }_n^2+v{\Phi }_{n-1}^2+v^2{\Phi }_{n-2}^2)+v^3{\Phi }_{n-2}{\Phi }_{n-3}.\hfill \end{array}$$

Iterating this process n times gives Formula (a). □

The corresponding formulae related to the six number/polynomial sequences are displayed as follows:

• Fibonacci and Lucas numbers ($n\in \mathbb{N}$):

  $$\begin{array}{cc}{\displaystyle \sum _{k=0}^n}{F}_k^2=F_n{F}_{n+1},\hfill & {\displaystyle \sum _{k=0}^n}{(-1)}^{n-k}{F}_k^2={\displaystyle \frac{F_{n+1}{L}_n}{5}}-{\displaystyle \frac{2(n+1)}{5}}{(-1)}^n,\hfill \\ {\displaystyle \sum _{k=0}^n}{L}_k^2=L_n{L}_{n+1}+2;\hfill & {\displaystyle \sum _{k=0}^n}{(-1)}^{n-k}{L}_k^2=F_{n+1}{L}_n+2(n+1){(-1)}^n.\hfill \end{array}$$
• Pell and Pell–Lucas polynomials ($n\in \mathbb{N}$):

  $$\begin{array}{cc}{\displaystyle \sum _{k=0}^n}{P}_k^2\left(x\right)={\displaystyle \frac{P_n\left(x\right)P_{n+1}\left(x\right)}{2x}},\hfill & {\displaystyle \sum _{k=0}^n}{(-1)}^{n-k}{P}_k^2\left(x\right)={\displaystyle \frac{P_{n+1}\left(x\right)Q_n\left(x\right)}{4(1+x^2)}}-{\displaystyle \frac{{(-1)}^n(n+1)}{2(1+x^2)}},\hfill \\ {\displaystyle \sum _{k=0}^n}{Q}_k^2\left(x\right)={\displaystyle \frac{Q_n\left(x\right)Q_{n+1}\left(x\right)}{2x}}+2;\hfill & {\displaystyle \sum _{k=0}^n}{(-1)}^{n-k}{Q}_k^2\left(x\right)=P_{n+1}\left(x\right)Q_n\left(x\right)+2(n+1){(-1)}^n.\hfill \end{array}$$
• Chebyshev polynomials ($n\in \mathbb{N}$):

  $$\begin{array}{cccc}& \sum _{k=0}^n{(-1)}^{n-k}{U}_k^2\left(x\right)={\displaystyle \frac{U_n\left(x\right)U_{n+1}\left(x\right)}{2x}},\hfill & \hfill & \sum _{k=0}^n{(-1)}^{n-k}{T}_k^2\left(x\right)={\displaystyle \frac{T_n\left(x\right)T_{n+1}\left(x\right)}{2x}}+{\displaystyle \frac{{(-1)}^n}{2}},\hfill \\ & \sum _{k=0}^n{T}_k^2\left(x\right)={\displaystyle \frac{U_n\left(x\right)T_n\left(x\right)}{2}}+{\displaystyle \frac{n+1}{2}};\hfill & & \sum _{k=0}^n{U}_k^2\left(x\right)={\displaystyle \frac{U_{n+1}\left(x\right)T_{n+1}\left(x\right)}{2(x^2-1)}}-{\displaystyle \frac{n+2}{2(x^2-1)}}.\hfill \end{array}$$

3.2. Double Product Sums

By means of the generating functions in (5), it is almost routine to establish the following convolution identities.

Theorem 5 

($n\in \mathbb{N}$).

$$\begin{array}{cc}\hfill \left(a\right)& \sum _{k=0}^n{\Phi }_k{\Phi }_{n-k}={\displaystyle \frac{(n+1){\Psi }_n-2{\Phi }_{n+1}}{4{\Delta }^2}},\hfill \\ \hfill \left(b\right)& \sum _{k=0}^n{\Phi }_k{\Psi }_{n-k}=(n+1){\Phi }_n,\hfill \\ \hfill \left(c\right)& \sum _{k=0}^n{\Psi }_k{\Psi }_{n-k}=(n+1){\Psi }_n+2{\Phi }_{n+1}.\hfill \end{array}$$

Proof. 

Let $\left[y^k\right]\Omega \left(y\right)$ stand for the coefficient of $y^k$ in the formal power series $\Omega \left(y\right)$, and $D_y$ for the derivative operator with respect to y. According to the generating function relations displayed in (5), the convolution sum in (b) can expressed as

$$\begin{array}{cc}\hfill \sum _{k=0}^n{\Phi }_k{\Psi }_{n-k}& =\left[y^n\right]\left\{{\displaystyle \frac{y(2-2uy)}{{(1-2uy-v{y}^2)}^2}}\right\}\hfill \\ & =\left[y^n\right]\left\{{\displaystyle \frac{y}{1-2uy-v{y}^2}}+y{D}_y\left({\displaystyle \frac{y}{1-2uy-v{y}^2}}\right)\right\}\hfill \\ & ={\Phi }_n+n{\Phi }_n,\hfill \end{array}$$

which confirms Formula (b). The proofs of the other identities in the theorem can be provided analogously, and the interested reader can work them out as exercises.□

Their particular cases are exhibited as follows:

• Fibonacci and Lucas numbers ($n\in \mathbb{N}$):

  $$\begin{array}{cc}& \sum _{k=0}^n{F}_k{F}_{n-k}={\displaystyle \frac{(n+1)L_n-2F_{n+1}}{5}},\hfill \\ & \sum _{k=0}^n{F}_k{L}_{n-k}=(n+1)F_n,\hfill \\ & \sum _{k=0}^n{L}_k{L}_{n-k}=(n+1)L_n+2F_{n+1}.\hfill \end{array}$$
• Pell and Pell–Lucas polynomials ($n\in \mathbb{N}$):

  $$\begin{array}{cc}& \sum _{k=0}^n{P}_k\left(x\right)P_{n-k}\left(x\right)={\displaystyle \frac{(n+1)Q_n\left(x\right)-2P_{n+1}\left(x\right)}{4(1+x^2)}},\hfill \\ & \sum _{k=0}^n{P}_k\left(x\right)Q_{n-k}\left(x\right)=(n+1)P_n\left(x\right),\hfill \\ & \sum _{k=0}^n{Q}_k\left(x\right)Q_{n-k}\left(x\right)=(n+1)Q_n\left(x\right)+2P_{n+1}\left(x\right).\hfill \end{array}$$
• Chebyshev polynomials ($n\in \mathbb{N}$):

  $$\begin{array}{cc}& \sum _{k=0}^n{U}_k\left(x\right)U_{n-k}\left(x\right)={\displaystyle \frac{(n+3)T_{n+2}\left(x\right)-U_{n+2}\left(x\right)}{2(x^2-1)}},\hfill \\ & \sum _{k=0}^n{U}_k\left(x\right)T_{n-k}\left(x\right)={\displaystyle \frac{n+2}{2}}{U}_n\left(x\right),\hfill \\ & \sum _{k=0}^n{T}_k\left(x\right)T_{n-k}\left(x\right)={\displaystyle \frac{n+1}{2}}{T}_n\left(x\right)+{\displaystyle \frac{U_n\left(x\right)}{2}}.\hfill \end{array}$$

3.3. Duplication Product Sums

Theorem 6 

($m\in \mathbb{Z},n\in \mathbb{N}$ and $\delta =0,1$).

$$\begin{array}{cc}\hfill \left(a\right)& \sum _{k=0}^n{\Phi }_{m+2k}{\Phi }_{\delta +2n-2k}={\displaystyle \frac{n+1}{4{\Delta }^2}}{\Psi }_{m+2n+\delta }-{\displaystyle \frac{{(-v)}^{\delta }}{8u{\Delta }^2}}{\Phi }_{2n+2}{\Psi }_{m-\delta },\hfill \\ \hfill \left(b\right)& \sum _{k=0}^n{\Psi }_{m+2k}{\Psi }_{\delta +2n-2k}=(n+1){\Psi }_{m+2n+\delta }+{\displaystyle \frac{{(-v)}^{\delta }}{2u}}{\Phi }_{2n+2}{\Psi }_{m-\delta },\hfill \\ \hfill \left(c\right)& \sum _{k=0}^n{\Phi }_{m+2k}{\Psi }_{\delta +2n-2k}=(n+1){\Phi }_{m+2n+\delta }+{\displaystyle \frac{{(-v)}^{\delta }}{2u}}{\Phi }_{2n+2}{\Phi }_{m-\delta }.\hfill \end{array}$$

Proof. 

They can be proved by using the Binet formulae (1) and (2). For example, Formula (c) can be shown as follows:

$$\begin{array}{cc}& \sum _{k=0}^n{\Phi }_{m+2k}{\Psi }_{\delta +2n-2k}=\sum _{k=0}^n{\displaystyle \frac{{\alpha }^{m+2k}-{\gamma }^{m+2k}}{\alpha -\gamma }}\times \left\{{\alpha }^{\delta +2n-2k}+{\gamma }^{\delta +2n-2k}\right\}\hfill \\ & ={\displaystyle \frac{1}{\alpha -\gamma }}\sum _{k=0}^n\left\{{\alpha }^{m+2n+\delta }-{\gamma }^{m+2n+\delta }+{\alpha }^m{\gamma }^{2n+\delta }{\left({\displaystyle \frac{\alpha }{\gamma }}\right)}^{2k}-{\gamma }^m{\alpha }^{2n+\delta }{\left({\displaystyle \frac{\gamma }{\alpha }}\right)}^{2k}\right\}\hfill \\ & =(n+1){\Phi }_{m+2n+\delta }+{\displaystyle \frac{{\alpha }^m{\gamma }^{\delta }-{\gamma }^m{\alpha }^{\delta }}{\alpha -\gamma }}\times {\displaystyle \frac{{\alpha }^{2n+2}-{\gamma }^{2n+2}}{2u(\alpha -\gamma )}}\hfill \\ & =(n+1){\Phi }_{m+2n+\delta }+{\displaystyle \frac{{(-v)}^{\delta }}{2u}}{\Phi }_{2n+2}{\Phi }_{m-\delta }.\hfill \end{array}$$

□

• Fibonacci and Lucas numbers ($m\in \mathbb{Z},n\in \mathbb{N}$ and $\delta =0,1$):

  $$\begin{array}{cc}& \sum _{k=0}^n{F}_{m+2k}{F}_{\delta +2n-2k}={\displaystyle \frac{n+1}{5}}{L}_{m+2n+\delta }-{\displaystyle \frac{{(-1)}^{\delta }}{5}}{F}_{2n+2}{L}_{m-\delta },\hfill \\ & \sum _{k=0}^n{L}_{m+2k}{L}_{\delta +2n-2k}=(n+1)L_{m+2n+\delta }+{(-1)}^{\delta }{F}_{2n+2}{L}_{m-\delta },\hfill \\ & \sum _{k=0}^n{F}_{m+2k}{L}_{\delta +2n-2k}=(n+1)F_{m+2n+\delta }+{(-1)}^{\delta }{F}_{2n+2}{F}_{m-\delta }.\hfill \end{array}$$
• Pell and Pell–Lucas polynomials ($m\in \mathbb{Z},n\in \mathbb{N}$ and $\delta =0,1$):

  $$\begin{array}{cc}& \sum _{k=0}^n{P}_{m+2k}\left(x\right)P_{\delta +2n-2k}\left(x\right)=(n+1){\displaystyle \frac{Q_{m+2n+\delta }\left(x\right)}{4(1+x^2)}}-{(-1)}^{\delta }{\displaystyle \frac{P_{2n+2}\left(x\right)Q_{m-\delta }\left(x\right)}{8x(1+x^2)}},\hfill \\ & \sum _{k=0}^n{Q}_{m+2k}\left(x\right)Q_{\delta +2n-2k}\left(x\right)=(n+1)Q_{m+2n+\delta }\left(x\right)+{(-1)}^{\delta }{\displaystyle \frac{P_{2n+2}\left(x\right)Q_{m-\delta }\left(x\right)}{2x}},\hfill \\ & \sum _{k=0}^n{P}_{m+2k}\left(x\right)Q_{\delta +2n-2k}\left(x\right)=(n+1)P_{m+2n+\delta }\left(x\right)+{(-1)}^{\delta }{\displaystyle \frac{P_{2n+2}\left(x\right)P_{m-\delta }\left(x\right)}{2x}}.\hfill \end{array}$$
• Chebyshev polynomials ($m\in \mathbb{Z},n\in \mathbb{N}$ and $\delta =0,1$):

  $$\begin{array}{cc}& \sum _{k=0}^n{U}_{m+2k}\left(x\right)U_{\delta +2n-2k}\left(x\right)=(n+1){\displaystyle \frac{T_{m+2n+2+\delta }\left(x\right)}{2(x^2-1)}}-{\displaystyle \frac{U_{2n+1}\left(x\right)T_{m-\delta }\left(x\right)}{4x(x^2-1)}},\hfill \\ & \sum _{k=0}^n{T}_{m+2k}\left(x\right)T_{\delta +2n-2k}\left(x\right)=(n+1){\displaystyle \frac{T_{m+2n+\delta }\left(x\right)}{2}}+{\displaystyle \frac{U_{2n+1}\left(x\right)T_{m-\delta }\left(x\right)}{4x}},\hfill \\ & \sum _{k=0}^n{U}_{m+2k}\left(x\right)T_{\delta +2n-2k}\left(x\right)=(n+1){\displaystyle \frac{U_{m+2n+\delta }\left(x\right)}{2}}+{\displaystyle \frac{U_{2n+1}\left(x\right)U_{m-\delta }\left(x\right)}{4x}}.\hfill \end{array}$$

3.4. Triple Product Sums

Theorem 7 

($n\in {\mathbb{N}}_0$).

$$\begin{array}{cc}\hfill \left(a\right)\sum _{i+j+k=n}{\Phi }_i{\Phi }_j{\Phi }_k& ={\displaystyle \frac{{\Phi }_n}{4{\Delta }^2}}\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{2}}\right)-{\displaystyle \frac{3n{\Psi }_{n+1}+6v{\Phi }_n}{16{\Delta }^4}},\hfill \\ \hfill \left(b\right)\sum _{i+j+k=n}{\Phi }_i{\Phi }_j{\Psi }_k& ={\displaystyle \frac{{\Psi }_n}{4{\Delta }^2}}\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{2}}\right)-{\displaystyle \frac{n+2}{4{\Delta }^2}}{\Phi }_{n+1},\hfill \\ \hfill \left(c\right)\sum _{i+j+k=n}{\Phi }_i{\Psi }_j{\Psi }_k& ={\Phi }_n\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{2}}\right)+{\displaystyle \frac{n{\Psi }_{n+1}+2v{\Phi }_n}{4{\Delta }^2}},\hfill \\ \hfill \left(d\right)\sum _{i+j+k=n}{\Psi }_i{\Psi }_j{\Psi }_k& ={\Psi }_n\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{2}}\right)+3(n+2){\Phi }_{n+1}.\hfill \end{array}$$

Proof. 

As a sample proof, we show Formula (d) by means of recursive constructions. Applying Formula (c) and then (b) in Theorem 5, we can proceed with

$$\begin{array}{cc}\hfill \sum _{i+j+k=n}{\Psi }_i{\Psi }_j{\Psi }_k& =\sum _{k=0}^n{\Psi }_{n-k}\sum _{i+j=k}{\Psi }_i{\Psi }_j\hfill \\ & =\sum _{k=0}^n{\Psi }_{n-k}\left\{2{\Phi }_{k+1}+(k+1){\Psi }_k\right\}\hfill \\ & =2(n+2){\Phi }_{n+1}+\sum _{k=0}^n(k+1){\Psi }_k{\Psi }_{n-k}\hfill \\ & =2(n+2){\Phi }_{n+1}+\sum _{k=0}^n(n-k+1){\Psi }_k{\Psi }_{n-k}\begin{array}{|c|}\hline k\to n-k\\ \hline\end{array}\hfill \\ & =2(n+2){\Phi }_{n+1}+{\displaystyle \frac{n+2}{2}}\sum _{k=0}^n{\Psi }_k{\Psi }_{n-k}\hfill \\ & ={\Psi }_n\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{2}}\right)+3(n+2){\Phi }_{n+1},\hfill \end{array}$$

where the last line follows again from Theorem 5(c). □

• Fibonacci and Lucas numbers ($n\in {\mathbb{N}}_0$):

  $$\begin{array}{cc}\hfill \sum _{i+j+k=n}{F}_i{F}_j{F}_k& =\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{2}}\right){\displaystyle \frac{F_n}{5}}-{\displaystyle \frac{3n{L}_{n+1}+6F_n}{25}},\hfill \\ \hfill \sum _{i+j+k=n}{F}_i{F}_j{L}_k& =\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{2}}\right){\displaystyle \frac{L_n}{5}}-{\displaystyle \frac{n+2}{5}}{F}_{n+1},\hfill \\ \hfill \sum _{i+j+k=n}{F}_i{L}_j{L}_k& =\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{2}}\right)F_n+{\displaystyle \frac{n{L}_{n+1}+2F_n}{5}},\hfill \\ \hfill \sum _{i+j+k=n}{L}_i{L}_j{L}_k& =\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{2}}\right)L_n+3(n+2)F_{n+1}.\hfill \end{array}$$
• Pell and Pell–Lucas polynomials ($n\in {\mathbb{N}}_0$):

  $$\begin{array}{cc}\hfill \sum _{i+j+k=n}{P}_i\left(x\right)P_j\left(x\right)P_k\left(x\right)& =\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{2}}\right){\displaystyle \frac{P_n\left(x\right)}{4(1+x^2)}}-{\displaystyle \frac{3n{Q}_{n+1}\left(x\right)+6P_n\left(x\right)}{16{(1+x^2)}^2}},\hfill \\ \hfill \sum _{i+j+k=n}{P}_i\left(x\right)P_j\left(x\right)Q_k\left(x\right)& =\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{2}}\right){\displaystyle \frac{Q_n\left(x\right)}{4(1+x^2)}}-{\displaystyle \frac{n+2}{4(1+x^2)}}{P}_{n+1}\left(x\right),\hfill \\ \hfill \sum _{i+j+k=n}{P}_i\left(x\right)Q_j\left(x\right)Q_k\left(x\right)& =\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{2}}\right)P_n\left(x\right)+{\displaystyle \frac{n{Q}_{n+1}\left(x\right)+2P_n\left(x\right)}{4(1+x^2)}},\hfill \\ \hfill \sum _{i+j+k=n}{Q}_i\left(x\right)Q_j\left(x\right)Q_k\left(x\right)& =\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{2}}\right)Q_n\left(x\right)+3(n+2)P_{n+1}\left(x\right).\hfill \end{array}$$
• Chebyshev polynomials ($n\in {\mathbb{N}}_0$):

  $$\begin{array}{cc}\hfill \sum _{i+j+k=n}{U}_i\left(x\right)U_j\left(x\right)U_k\left(x\right)& =\left({\displaystyle \genfrac{}{}{0pt}{}{n+5}{2}}\right){\displaystyle \frac{U_{n+2}\left(x\right)}{4(x^2-1)}}-{\displaystyle \frac{3(n+3)T_{n+4}\left(x\right)-3U_{n+2}\left(x\right)}{8{(x^2-1)}^2}},\hfill \\ \hfill \sum _{i+j+k=n}{U}_i\left(x\right)U_j\left(x\right)T_k\left(x\right)& =\left({\displaystyle \genfrac{}{}{0pt}{}{n+4}{2}}\right){\displaystyle \frac{T_{n+2}\left(x\right)}{4(x^2-1)}}-{\displaystyle \frac{n+4}{8(x^2-1)}}{U}_{n+2}\left(x\right),\hfill \\ \hfill \sum _{i+j+k=n}{U}_i\left(x\right)T_j\left(x\right)T_k\left(x\right)& =\left({\displaystyle \genfrac{}{}{0pt}{}{n+3}{2}}\right){\displaystyle \frac{U_n\left(x\right)}{4}}+{\displaystyle \frac{(n+1)T_{n+2}\left(x\right)-U_n\left(x\right)}{8(x^2-1)}},\hfill \\ \hfill \sum _{i+j+k=n}{T}_i\left(x\right)T_j\left(x\right)T_k\left(x\right)& =\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{2}}\right){\displaystyle \frac{T_n\left(x\right)}{4}}+{\displaystyle \frac{3(n+2)}{8}}{U}_n\left(x\right).\hfill \end{array}$$

4. Closed Formulae of Binomial Sums

In this section, we shall evaluate binomial sums weighted by the functions ${\Phi }_n$ and/or ${\Psi }_n$. This can mainly be accomplished by utilizing the binomial theorem and/or by manipulating the exponential generating functions displayed in (6). Since the proofs are carried out by the standard generating function approach (cf. Comtet [2] and Wilf [3]), we confine ourselves to offering one sample proof for each theorem without reproducing the full details.

4.1. Binomial Linear Sums

Theorem 8 

($m\in \mathbb{Z}$ and $n\in {\mathbb{N}}_0$).

$$\begin{array}{cc}\hfill \left(a\right)& \sum _{k=0}^n{\left({\displaystyle \frac{v}{2u}}\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_{m+k}={\displaystyle \frac{{\Phi }_{m+2n}}{{\left(2u\right)}^n}},\hfill \\ \hfill \left(b\right)& \sum _{k=0}^n{\left({\displaystyle \frac{v}{2u}}\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_{m+k}={\displaystyle \frac{{\Psi }_{m+2n}}{{\left(2u\right)}^n}};\hfill \\ \hfill \left(c\right)& \sum _{k=0}^n{\left(-2u\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_{m+k}={\displaystyle \frac{v^m{\Phi }_{n-m}}{{(-1)}^{m+n-1}}},\hfill \\ \hfill \left(d\right)& \sum _{k=0}^n{\left(-2u\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_{m+k}={\displaystyle \frac{v^m{\Psi }_{n-m}}{{(-1)}^{m+n}}}.\hfill \end{array}$$

Proof. 

By means of the binomial theorem, we can evaluate the sum in (a):

$$\begin{array}{cc}\hfill \sum _{k=0}^n{\left({\displaystyle \frac{v}{2u}}\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_{m+k}& =\sum _{k=0}^n{\left({\displaystyle \frac{v}{2u}}\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\displaystyle \frac{{\alpha }^{m+k}-{\gamma }^{m+k}}{\alpha -\gamma }}\hfill \\ & ={\displaystyle \frac{{\alpha }^m{\left(\frac{v}{2u}+\alpha \right)}^n-{\gamma }^m{\left(\frac{v}{2u}+\gamma \right)}^n}{\alpha -\gamma }}\hfill \\ & ={\displaystyle \frac{{\alpha }^{m+2n}-{\gamma }^{m+2n}}{{\left(2u\right)}^n(\alpha -\gamma )}}={\displaystyle \frac{{\Phi }_{m+2n}}{{\left(2u\right)}^n}},\hfill \end{array}$$

where the last line is justified by

$${\alpha }^2=2u\alpha +v\mathrm{and}{\gamma }^2=2u\gamma +v.$$

This completes the proof for Formula (a). The remaining three formulae can be proved similarly.□

• Fibonacci and Lucas numbers ($m\in \mathbb{Z}$ and $n\in {\mathbb{N}}_0$):

  $$\begin{array}{cc}& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_{m+k}=F_{m+2n},\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)L_{m+k}=L_{m+2n};\hfill \\ & \sum _{k=0}^n{\left(-1\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_{m+k}={(-1)}^{m+n-1}{F}_{n-m},\hfill \\ & \sum _{k=0}^n{\left(-1\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)L_{m+k}={(-1)}^{m+n}{L}_{n-m}.\hfill \end{array}$$
• Pell and Pell–Lucas polynomials ($m\in \mathbb{Z}$ and $n\in {\mathbb{N}}_0$):

  $$\begin{array}{cc}& \sum _{k=0}^n{\left({\displaystyle \frac{1}{2x}}\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_{m+k}\left(x\right)={\displaystyle \frac{P_{m+2n}\left(x\right)}{{\left(2x\right)}^n}},\hfill \\ & \sum _{k=0}^n{\left({\displaystyle \frac{1}{2x}}\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Q_{m+k}\left(x\right)={\displaystyle \frac{Q_{m+2n}\left(x\right)}{{\left(2x\right)}^n}};\hfill \\ & \sum _{k=0}^n{\left(-2x\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_{m+k}\left(x\right)={(-1)}^{m+n-1}{P}_{n-m}\left(x\right),\hfill \\ & \sum _{k=0}^n{\left(-2x\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Q_{m+k}\left(x\right)={(-1)}^{m+n}{Q}_{n-m}\left(x\right).\hfill \end{array}$$
• Chebyshev polynomials ($m\in \mathbb{Z}$ and $n\in {\mathbb{N}}_0$):

  $$\begin{array}{cc}& \sum _{k=0}^n{\left(-{\displaystyle \frac{1}{2x}}\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_{m+k}\left(x\right)={\displaystyle \frac{U_{m+2n}\left(x\right)}{{\left(2x\right)}^n}},\hfill \\ & \sum _{k=0}^n{\left(-{\displaystyle \frac{1}{2x}}\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)T_{m+k}\left(x\right)={\displaystyle \frac{T_{m+2n}\left(x\right)}{{\left(2x\right)}^n}};\hfill \\ & \sum _{k=0}^n{\left(-2x\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_{m+k}\left(x\right)={(-1)}^{n-1}{U}_{n-m-2}\left(x\right),\hfill \\ & \sum _{k=0}^n{\left(-2x\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)T_{m+k}\left(x\right)={(-1)}^n{T}_{n-m}\left(x\right).\hfill \end{array}$$

4.2. Binomial Duplicate Sums

Theorem 9 

($m\in \mathbb{Z}$ and $n\in {\mathbb{N}}_0$).

$$\begin{array}{cc}\hfill \left(a\right)& \sum _{k=0}^n{(-v)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_{m+2k}={\left(2u\right)}^n{\Phi }_{m+n},\hfill \\ \hfill \left(b\right)& \sum _{k=0}^n{(-v)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_{m+2k}={\left(2u\right)}^n{\Psi }_{m+n};\hfill \\ \hfill \left(c\right)& \sum _{k=0}^n{v}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_{m+2k}=\left\{\begin{array}{cc}{\Phi }_{m+n}{\left(2\Delta \right)}^n,\hfill & \mathrm{even}n;\hfill \\ {\Psi }_{m+n}{\left(2\Delta \right)}^{n-1},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ \hfill \left(d\right)& \sum _{k=0}^n{v}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_{m+2k}=\left\{\begin{array}{cc}{\Psi }_{m+n}{\left(2\Delta \right)}^n,\hfill & \mathrm{even}n;\hfill \\ {\Phi }_{m+n}{\left(2\Delta \right)}^{n+1},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

Proof. 

As an example, we present a proof below for Formula (b). The others can be proved analogously. In fact, we have

$$\begin{array}{cc}\hfill \sum _{k=0}^n{(-v)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_{m+2k}& =\sum _{k=0}^n{(-v)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left({\alpha }^{m+2k}+{\gamma }^{m+2k}\right)\hfill \\ & ={\alpha }^m{\left({\alpha }^2-v\right)}^n+{\gamma }^m{\left({\gamma }^2-v\right)}^n\hfill \\ & ={\left(2u\right)}^n{\alpha }^{m+n}+{\left(2u\right)}^n{\gamma }^{m+n}={\left(2u\right)}^n{\Psi }_{m+n},\hfill \end{array}$$

where we have made use of the equalities

$${\alpha }^2-v=2u\alpha \mathrm{and}{\gamma }^2-v=2u\gamma .$$

□

• Fibonacci and Lucas numbers ($m\in \mathbb{Z}$ and $n\in {\mathbb{N}}_0$):

  $$\begin{array}{cc}& \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_{m+2k}=F_{m+n},\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)L_{m+2k}=L_{m+n};\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_{m+2k}=\left\{\begin{array}{cc}{\sqrt{5}}^n{F}_{m+n},\hfill & \mathrm{even}n;\hfill \\ {\sqrt{5}}^{n-1}{L}_{m+n},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)L_{m+2k}=\left\{\begin{array}{cc}{\sqrt{5}}^n{L}_{m+n},\hfill & \mathrm{even}n;\hfill \\ {\sqrt{5}}^{n+1}{F}_{m+n},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$
  The first two formulae were discovered by Carlitz (1967, cf. [4], page 163).
• Pell and Pell–Lucas polynomials ($m\in \mathbb{Z}$ and $n\in {\mathbb{N}}_0$):

  $$\begin{array}{cc}& \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_{m+2k}\left(x\right)={\left(2x\right)}^n{P}_{m+n}\left(x\right),\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Q_{m+2k}\left(x\right)={\left(2x\right)}^n{Q}_{m+n}\left(x\right);\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_{m+2k}\left(x\right)=\left\{\begin{array}{cc}{P}_{m+n}\left(x\right){\left(2\sqrt{1+x^2}\right)}^n,\hfill & \mathrm{even}n;\hfill \\ Q_{m+n}\left(x\right){\left(2\sqrt{1+x^2}\right)}^{n-1},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Q_{m+2k}\left(x\right)=\left\{\begin{array}{cc}{Q}_{m+n}\left(x\right){\left(2\sqrt{1+x^2}\right)}^n,\hfill & \mathrm{even}n;\hfill \\ P_{m+n}\left(x\right){\left(2\sqrt{1+x^2}\right)}^{n+1},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$
• Chebyshev polynomials ($m\in \mathbb{Z}$ and $n\in {\mathbb{N}}_0$):

  $$\begin{array}{cc}& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_{m+2k}\left(x\right)={\left(2x\right)}^n{U}_{m+n}\left(x\right),\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)T_{m+2k}\left(x\right)={\left(2x\right)}^n{T}_{m+n}\left(x\right);\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_{m+2k}\left(x\right)=\left\{\begin{array}{cc}{U}_{m+n}\left(x\right){\left(2\sqrt{x^2-1}\right)}^n,\hfill & \mathrm{even}n;\hfill \\ 2T_{m+n+1}\left(x\right){\left(2\sqrt{x^2-1}\right)}^{n-1},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)T_{m+2k}\left(x\right)=\left\{\begin{array}{cc}{T}_{m+n}\left(x\right){\left(2\sqrt{x^2-1}\right)}^n,\hfill & \mathrm{even}n;\hfill \\ {\displaystyle \frac{1}{2}}{U}_{m+n-1}\left(x\right){\left(2\sqrt{x^2-1}\right)}^{n+1},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

4.3. Binomial Quadruplicate Sums

Theorem 10 

($m\in \mathbb{Z}$ and $n\in \mathbb{N}$).

$$\begin{array}{cc}\hfill \left(a\right)& \sum _{k=0}^n{v}^{2m(n-k)}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_{4mk}={\Phi }_{2mn}{\Psi }_{2m}^n,\hfill \\ \hfill \left(b\right)& \sum _{k=0}^n{v}^{2m(n-k)}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_{4mk}={\Psi }_{2mn}{\Psi }_{2m}^n;\hfill \\ \hfill \left(c\right)& \sum _{k=0}^n{(-v^{2m})}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_{4mk}=\left\{\begin{array}{cc}{(2\Delta )}^n{\Phi }_{2mn}{\Phi }_{2m}^n,\hfill & \mathrm{even}n;\hfill \\ {(2\Delta )}^{n-1}{\Psi }_{2mn}{\Phi }_{2m}^n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ \hfill \left(d\right)& \sum _{k=0}^n{(-v^{2m})}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_{4mk}=\left\{\begin{array}{cc}{(2\Delta )}^n{\Psi }_{2mn}{\Phi }_{2m}^n,\hfill & \mathrm{even}n;\hfill \\ {(2\Delta )}^{n+1}{\Phi }_{2mn}{\Phi }_{2m}^n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

Proof. 

The summation formula in (c) can be validated as follows:

$$\begin{array}{cc}\hfill \sum _{k=0}^n{(-v^{2m})}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_{4mk}& =\sum _{k=0}^n{(-v^{2m})}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\displaystyle \frac{{\alpha }^{4mk}-{\gamma }^{4mk}}{\alpha -\gamma }}\hfill \\ & ={\displaystyle \frac{{({\alpha }^{4m}-v^{2m})}^n-{({\gamma }^{4m}-v^{2m})}^n}{\alpha -\gamma }}\hfill \\ & ={\displaystyle \frac{{\alpha }^{2mn}{({\alpha }^{2m}-{\gamma }^{2m})}^n-{\gamma }^{2mn}{({\gamma }^{2m}-{\alpha }^{2m})}^n}{\alpha -\gamma }}\hfill \\ & ={\displaystyle \frac{{({\alpha }^{2m}-{\gamma }^{2m})}^n({\alpha }^{2mn}-{(-1)}^n{\gamma }^{2mn})}{\alpha -\gamma }}\hfill \\ & =\left\{\begin{array}{cc}{(2\Delta )}^n{\Phi }_{2mn}{\Phi }_{2m}^n,\hfill & \mathrm{even}n;\hfill \\ {(2\Delta )}^{n-1}{\Psi }_{2mn}{\Phi }_{2m}^n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

The other three identities can be proved similarly. □

• Fibonacci and Lucas numbers ($m\in \mathbb{Z}$ and $n\in \mathbb{N}$):

  $$\begin{array}{cc}& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_{4mk}=F_{2mn}{L}_{2m}^n,\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)L_{4mk}=L_{2mn}{L}_{2m}^n;\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_{4mk}=\left\{\begin{array}{cc}{\sqrt{5}}^n{F}_{2mn}{F}_{2m}^n,\hfill & \mathrm{even}n;\hfill \\ {\sqrt{5}}^{n-1}{L}_{2mn}{F}_{2m}^n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)L_{4mk}=\left\{\begin{array}{cc}{\sqrt{5}}^n{L}_{2mn}{F}_{2m}^n,\hfill & \mathrm{even}n;\hfill \\ {\sqrt{5}}^{n+1}{F}_{2mn}{F}_{2m}^n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$
  The first formula was found by Hoggatt (1968; see Koshy [4], page 163).
• Pell and Pell–Lucas polynomials ($m\in \mathbb{Z}$ and $n\in \mathbb{N}$):

  $$\begin{array}{cc}& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_{4mk}\left(x\right)=P_{2mn}\left(x\right)Q_{2m}^n\left(x\right),\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Q_{4mk}\left(x\right)=Q_{2mn}\left(x\right)Q_{2m}^n\left(x\right);\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_{4mk}\left(x\right)=\left\{\begin{array}{cc}{\left(2\sqrt{1+x^2}\right)}^n{P}_{2mn}\left(x\right)P_{2m}^n\left(x\right),\hfill & \mathrm{even}n;\hfill \\ {\left(2\sqrt{1+x^2}\right)}^{n-1}{Q}_{2mn}\left(x\right)P_{2m}^n\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Q_{4mk}\left(x\right)=\left\{\begin{array}{cc}{\left(2\sqrt{1+x^2}\right)}^n{Q}_{2mn}\left(x\right)P_{2m}^n\left(x\right),\hfill & \mathrm{even}n;\hfill \\ {\left(2\sqrt{1+x^2}\right)}^{n+1}{P}_{2mn}\left(x\right)P_{2m}^n\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$
• Chebyshev polynomials ($m\in \mathbb{Z}$ and $n\in \mathbb{N}$):

  $$\begin{array}{cc}& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_{4mk}\left(x\right)=2^n{U}_{2mn}\left(x\right)T_{2m}^n\left(x\right),\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)T_{4mk}\left(x\right)=2^n{T}_{2mn}\left(x\right)T_{2m}^n\left(x\right);\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_{4mk}\left(x\right)=2^n\times \left\{\begin{array}{cc}{\sqrt{x^2-1}}^n{U}_{2mn}\left(x\right)U_{2m-1}^n\left(x\right),\hfill & \mathrm{even}n;\hfill \\ {\sqrt{x^2-1}}^{n-1}{T}_{2mn+1}\left(x\right)U_{2m-1}^n\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)T_{4mk}\left(x\right)=2^n\times \left\{\begin{array}{cc}{\sqrt{x^2-1}}^n{T}_{2mn}\left(x\right)U_{2m-1}^n\left(x\right),\hfill & \mathrm{even}n;\hfill \\ {\sqrt{x^2-1}}^{n+1}{U}_{2mn-1}\left(x\right)U_{2m-1}^n\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

4.4. Binomial Convolution Sums

Theorem 11 

($m\in \mathbb{Z}$ and $n\in {\mathbb{N}}_0$).

$$\begin{array}{cc}\hfill \left(a\right)& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_{mk}{\Phi }_{mn-mk}={\displaystyle \frac{2^{n-1}{\Psi }_{mn}-{\Psi }_m^n}{2{\Delta }^2}},\hfill \\ \hfill \left(b\right)& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_{mk}{\Psi }_{mn-mk}=2^n{\Phi }_{mn},\hfill \\ \hfill \left(c\right)& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_{mk}{\Psi }_{mn-mk}=2^n{\Psi }_{mn}+2{\Psi }_m^n.\hfill \end{array}$$

Proof. 

Recalling (6), we have the exponential generating function

$$\sum _{k=0}^{\infty }{\displaystyle \frac{y^k}{k!}}{\Phi }_{mk}={\displaystyle \frac{e^{y{\alpha }^m}-e^{y{\gamma }^m}}{2\Delta }}.$$

According to the Cauchy product, the binomial sum in (a) can be expressed as

$$\begin{array}{cc}\hfill \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_{mk}{\Phi }_{mn-mk}& =n!\left[y^n\right]{\left\{{\displaystyle \frac{e^{y{\alpha }^m}-e^{y{\gamma }^m}}{2\Delta }}\right\}}^2\hfill \\ & =n!\left[y^n\right]\left\{{\displaystyle \frac{e^{2y{\alpha }^m}+e^{2y{\gamma }^m}-2e^{y{\alpha }^m+y{\gamma }^m}}{4{\Delta }^2}}\right\}\hfill \\ & ={\displaystyle \frac{2^n({\alpha }^{mn}+{\gamma }^{mn})-2{({\alpha }^m+{\gamma }^m)}^n}{4{\Delta }^2}}\}\hfill \\ & ={\displaystyle \frac{2^{n-1}{\Psi }_{mn}-{\Psi }_m^n}{2{\Delta }^2}}.\hfill \end{array}$$

This confirms Identity (a). Analogously, (b) and (c) can be proved.□

• Fibonacci and Lucas numbers ($m\in \mathbb{Z}$ and $n\in {\mathbb{N}}_0$):

  $$\begin{array}{cc}& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_{mk}{F}_{mn-mk}={\displaystyle \frac{2^n{L}_{mn}-2L_m^n}{5}},\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_{mk}{L}_{mn-mk}=2^n{F}_{mn},\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)L_{mk}{L}_{mn-mk}=2^n{L}_{mn}+2L_m^n.\hfill \end{array}$$
• Pell and Pell–Lucas polynomials ($m\in \mathbb{Z}$ and $n\in {\mathbb{N}}_0$: [8,9]):

  $$\begin{array}{cc}& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_{mk}\left(x\right)P_{mn-mk}\left(x\right)={\displaystyle \frac{2^{n-1}{Q}_{mn}\left(x\right)-Q_m^n\left(x\right)}{2(1+x^2)}},\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_{mk}\left(x\right)Q_{mn-mk}\left(x\right)=2^n{P}_{mn}\left(x\right),\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Q_{mk}\left(x\right)Q_{mn-mk}\left(x\right)=2^n{Q}_{mn}\left(x\right)+2Q_m^n\left(x\right).\hfill \end{array}$$
• Chebyshev polynomials ($m\in \mathbb{Z}$ and $n\in {\mathbb{N}}_0$):

  $$\begin{array}{cc}& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_{mk}\left(x\right)U_{mn-mk}\left(x\right)={\displaystyle \frac{2^{n-1}}{x^2-1}}\left\{T_{mn+2}\left(x\right)-T_m^n\left(x\right)\right\},\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_{mk}\left(x\right)T_{mn-mk}\left(x\right)=2^{n-1}\left\{U_{mn}\left(x\right)+U_m^n\left(x\right)\right\},\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)T_{mk}\left(x\right)T_{mn-mk}\left(x\right)=2^{n-1}\left\{T_{mn}\left(x\right)+T_m^n\left(x\right)\right\}.\hfill \end{array}$$

4.5. Alternating Convolution Sums

Theorem 12 

($m\in \mathbb{Z}$ and $n\in \mathbb{N}$).

$$\begin{array}{cc}\hfill \left(a\right)& \sum _{k=0}^n{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_{mk}{\Phi }_{mn-mk}={\Phi }_m^n\times \left\{\begin{array}{cc}-2{(2\Delta )}^{n-2},\hfill & \mathrm{even}n;\hfill \\ 0,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ \hfill \left(b\right)& \sum _{k=0}^n{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_{mk}{\Psi }_{mn-mk}={\Phi }_m^n\times \left\{\begin{array}{cc}2{(2\Delta )}^n,\hfill & \mathrm{even}n;\hfill \\ 0,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ \hfill \left(c\right)& \sum _{k=0}^n{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_{mk}{\Psi }_{mn-mk}={\Phi }_m^n\times \left\{\begin{array}{cc}0,\hfill & \mathrm{even}n;\hfill \\ -2{(2\Delta )}^{n-1},\hfill & \mathrm{even}n.\hfill \end{array}\right.\hfill \end{array}$$

Proof. 

This theorem can also be proved by the generating function method. We provide only (b) as an example because the others can be proved in exactly the same manner. In view of (6), the following generating functions hold:

$$\sum _{k=0}^{\infty }{(\pm 1)}^k{\displaystyle \frac{y^k}{k!}}{\Psi }_{mk}=e^{\pm y{\alpha }^m}+e^{\pm y{\gamma }^m}.$$

Then, the binomial sum in (b) can be evaluated in closed form as follows:

$$\begin{array}{cc}\hfill \sum _{k=0}^n{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_{mk}{\Psi }_{mn-mk}& =n!\left[y^n\right]\left\{e^{y{\alpha }^m}+e^{y{\gamma }^m}\right\}\times \left\{e^{-y{\alpha }^m}+e^{-y{\gamma }^m}\right\}\hfill \\ & =n!\left[y^n\right]\left\{2+e^{y{\alpha }^m-y{\gamma }^m}+e^{y{\gamma }^m-y{\alpha }^m}\right\}\hfill \\ & ={\left({\alpha }^m-{\gamma }^m\right)}^n(1+{(-1)}^n)\hfill \\ & ={\Phi }_m^n\times \left\{\begin{array}{cc}2{(2\Delta )}^n,\hfill & \mathrm{even}n;\hfill \\ 0,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

□

• Fibonacci and Lucas numbers ($m\in \mathbb{Z}$ and $n\in \mathbb{N}$):

  $$\begin{array}{cc}& \sum _{k=0}^n{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_{mk}{F}_{mn-mk}={\left(\sqrt{5}{F}_m\right)}^n\times \left\{\begin{array}{cc}-{\displaystyle \frac{2}{5}},\hfill & \mathrm{even}n;\hfill \\ 0,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)L_{mk}{L}_{mn-mk}={\left(\sqrt{5}{F}_m\right)}^n\times \left\{\begin{array}{cc}2,\hfill & \mathrm{even}n;\hfill \\ 0,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_{mk}{L}_{mn-mk}={\left(\sqrt{5}{F}_m\right)}^n\times \left\{\begin{array}{cc}0,\hfill & \mathrm{even}n;\hfill \\ {\displaystyle \frac{-2}{\sqrt{5}}},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$
• Pell and Pell–Lucas polynomials ($m\in \mathbb{Z}$ and $n\in \mathbb{N}$: [10,11,12]):

  $$\begin{array}{cc}& \sum _{k=0}^n{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_{mk}\left(x\right)P_{mn-mk}\left(x\right)=P_m^n\left(x\right)\times \left\{\begin{array}{cc}-2{\left(2\sqrt{1+x^2}\right)}^{n-2},\hfill & \mathrm{even}n;\hfill \\ 0,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Q_{mk}\left(x\right)Q_{mn-mk}\left(x\right)=P_m^n\left(x\right)\times \left\{\begin{array}{cc}2{\left(2\sqrt{1+x^2}\right)}^n,\hfill & \mathrm{even}n;\hfill \\ 0,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_{mk}\left(x\right)Q_{mn-mk}\left(x\right)=P_m^n\left(x\right)\times \left\{\begin{array}{cc}0,\hfill & \mathrm{even}n;\hfill \\ -2{\left(2\sqrt{1+x^2}\right)}^{n-1},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$
• Chebyshev polynomials ($m\in \mathbb{Z}$ and $n\in \mathbb{N}$):

  $$\begin{array}{cc}& \sum _{k=0}^n{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_{mk}\left(x\right)U_{mn-mk}\left(x\right)=U_{m-1}^n\left(x\right)\times \left\{\begin{array}{cc}-2{\left(2\sqrt{x^2-1}\right)}^{n-2},\hfill & \mathrm{even}n;\hfill \\ 0,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)T_{mk}\left(x\right)T_{mn-mk}\left(x\right)=U_{m-1}^n\left(x\right)\times \left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}{\left(2\sqrt{x^2-1}\right)}^n,\hfill & \mathrm{even}n;\hfill \\ 0,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_{mk}\left(x\right)T_{mn-mk}\left(x\right)=U_{m-1}^n\left(x\right)\times \left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}{\left(2\sqrt{x^2-1}\right)}^n,\hfill & \mathrm{even}n;\hfill \\ -x{\left(2\sqrt{x^2-1}\right)}^{n-1},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

4.6. Binomial Square Sums

Theorem 13 

($n\in \mathbb{N}$).

$$\begin{array}{cc}\hfill \left(a\right)& \sum _{k=0}^n{v}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_k^2=\left\{\begin{array}{cc}{(2\Delta )}^{n-2}{\Psi }_n,\hfill & \mathrm{even}n;\hfill \\ {(2\Delta )}^{n-1}{\Phi }_n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ \hfill \left(b\right)& \sum _{k=0}^n{v}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_k^2=\left\{\begin{array}{cc}{(2\Delta )}^n{\Psi }_n,\hfill & \mathrm{even}n;\hfill \\ {(2\Delta )}^{n+1}{\Phi }_n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ \hfill \left(c\right)& \sum _{k=0}^n{(-v)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_k^2={\displaystyle \frac{{\left(2u\right)}^n{\Psi }_n-2{(-2v)}^n}{4{\Delta }^2}},\hfill \\ \hfill \left(d\right)& \sum _{k=0}^n{(-v)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_k^2={\left(2u\right)}^n{\Psi }_n+2{(-2v)}^n.\hfill \end{array}$$

Proof. 

As a sample proof, we demonstrate (a) by making use of the binomial theorem. Specifically, we have

$$\begin{array}{cc}\hfill \sum _{k=0}^n{v}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_k^2& =\sum _{k=0}^n{v}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\left\{{\displaystyle \frac{{\alpha }^k-{\gamma }^k}{\alpha -\gamma }}\right\}}^2\hfill \\ & =\sum _{k=0}^n{v}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\displaystyle \frac{{\alpha }^{2k}+{\gamma }^{2k}-2{(-v)}^k}{{(2\Delta )}^2}}\hfill \\ & ={\displaystyle \frac{{({\alpha }^2+v)}^n+{({\gamma }^2+v)}^n}{{(2\Delta )}^2}}\hfill \\ & ={\displaystyle \frac{{\alpha }^n{(\alpha -\gamma )}^n+{\gamma }^n{(\gamma -\alpha )}^n}{{(2\Delta )}^2}}\hfill \\ & ={\displaystyle \frac{{(\alpha -\gamma )}^n\left({\alpha }^n+{(-1)}^n{\gamma }^n\right)}{{(2\Delta )}^2}}\hfill \\ & =\left\{\begin{array}{cc}{(2\Delta )}^{n-2}{\Psi }_n,\hfill & \mathrm{even}n;\hfill \\ {(2\Delta )}^{n-1}{\Phi }_n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

□

• Fibonacci and Lucas numbers ($n\in \mathbb{N}$):

  $$\begin{array}{cc}& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_k^2=\left\{\begin{array}{cc}{\sqrt{5}}^{n-2}{L}_n,\hfill & \mathrm{even}n;\hfill \\ {\sqrt{5}}^{n-1}{F}_n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)L_k^2=\left\{\begin{array}{cc}{\sqrt{5}}^n{L}_n,\hfill & \mathrm{even}n;\hfill \\ {\sqrt{5}}^{n+1}{F}_n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_k^2={\displaystyle \frac{L_n-2{(-2)}^n}{5}},\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)L_k^2=L_n+2{(-2)}^n.\hfill \end{array}$$
• Pell and Pell–Lucas polynomials ($n\in \mathbb{N}$):

  $$\begin{array}{cc}& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_k^2\left(x\right)=\left\{\begin{array}{cc}{\left(2\sqrt{1+x^2}\right)}^{n-2}{Q}_n\left(x\right),\hfill & \mathrm{even}n;\hfill \\ {\left(2\sqrt{1+x^2}\right)}^{n-1}{P}_n\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Q_k^2\left(x\right)=\left\{\begin{array}{cc}{\left(2\sqrt{1+x^2}\right)}^n{Q}_n\left(x\right),\hfill & \mathrm{even}n;\hfill \\ {\left(2\sqrt{1+x^2}\right)}^{n+1}{P}_n\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_k^2\left(x\right)={\displaystyle \frac{{\left(2x\right)}^n{Q}_n\left(x\right)-2{(-2)}^n}{4(1+x^2)}},\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Q_k^2\left(x\right)={\left(2x\right)}^n{Q}_n\left(x\right)+2{(-2)}^n.\hfill \end{array}$$
• Chebyshev polynomials ($n\in \mathbb{N}$):

  $$\begin{array}{cc}& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_k^2\left(x\right)={\displaystyle \frac{{\left(2x\right)}^n{T}_{n+2}\left(x\right)-2^n}{2(x^2-1)}},\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)T_k^2\left(x\right)={\left(2x\right)}^n{\displaystyle \frac{T_n\left(x\right)}{2}}+2^{n-1};\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_k^2\left(x\right)=\left\{\begin{array}{cc}2{\left(2\sqrt{x^2-1}\right)}^{n-2}{T}_{n+2}\left(x\right),\hfill & \mathrm{even}n;\hfill \\ {\left(2\sqrt{x^2-1}\right)}^{n-1}{U}_{n+1}\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)T_k^2\left(x\right)=\left\{\begin{array}{cc}{\left(2\sqrt{x^2-1}\right)}^n{\displaystyle \frac{T_n\left(x\right)}{2}},\hfill & \mathrm{even}n;\hfill \\ {\left(2\sqrt{x^2-1}\right)}^{n+1}{\displaystyle \frac{U_{n-1}\left(x\right)}{4}},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

4.7. Binomial Cubic Sums

Theorem 14 

($n\in {\mathbb{N}}_0$).

$$\begin{array}{cc}\hfill \left(a\right)& \sum _{k=0}^n{(-v)}^{3n-3k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_{2k}^3={\displaystyle \frac{{\Psi }_3^n}{4{\Delta }^2}}{\Phi }_{3n}-{\displaystyle \frac{3{\left(2u{v}^2\right)}^n}{4{\Delta }^2}}{\Phi }_n,\hfill \\ \hfill \left(b\right)& \sum _{k=0}^n{(-v)}^{3n-3k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_{2k}^3={\Psi }_3^n{\Psi }_{3n}+3{\left(2u{v}^2\right)}^n{\Psi }_n;\hfill \\ \hfill \left(c\right)& \sum _{k=0}^n{v}^{3n-3k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_{2k}^3=\left\{\begin{array}{cc}{(2\Delta )}^{n-2}\left\{{\Phi }_3^n{\Phi }_{3n}-3v^{2n}{\Phi }_n\right\},\hfill & \mathrm{even}n;\hfill \\ {(2\Delta )}^{n-3}\left\{{\Phi }_3^n{\Psi }_{3n}-3v^{2n}{\Psi }_n\right\},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ \hfill \left(d\right)& \sum _{k=0}^n{v}^{3n-3k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_{2k}^3=\left\{\begin{array}{cc}{(2\Delta )}^n\left\{{\Phi }_3^n{\Psi }_{3n}+3v^{2n}{\Psi }_n\right\},\hfill & \mathrm{even}n;\hfill \\ {(2\Delta )}^{n+1}\left\{{\Phi }_3^n{\Phi }_{3n}+3v^{2n}{\Phi }_n\right\},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

Proof. 

We give a proof of (b) as an example. The others can be proved analogously. In fact, the binomial sum in (b) can be manipulated and then evaluated as follows:

$$\begin{array}{cc}\hfill \sum _{k=0}^n{(-v)}^{3n-3k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_{2k}^3& =\sum _{k=0}^n{(-v)}^{3n-3k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\left({\alpha }^{2k}+{\gamma }^{2k}\right)}^3\hfill \\ & =\sum _{k=0}^n{(-v)}^{3n-3k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left\{{\alpha }^{6k}+{\gamma }^{6k}+3{\left(v\alpha \right)}^{2k}+3{\left(v\gamma \right)}^{2k}\right\}\hfill \\ & ={\left({\alpha }^6-v^3\right)}^n+{\left({\gamma }^6-v^3\right)}^n+3v^{2n}{({\alpha }^2-v)}^n+3v^{2n}{({\gamma }^2-v)}^n\hfill \\ & =\left({\alpha }^{3n}+{\gamma }^{3n}\right){\left({\alpha }^3+{\gamma }^3\right)}^n+3{\left(2u\right)}^n{v}^{2n}\left({\alpha }^n+{\gamma }^n\right)\hfill \\ & ={\Psi }_3^n{\Psi }_{3n}+3{\left(2u{v}^2\right)}^n{\Psi }_n.\hfill \end{array}$$

□

• Fibonacci and Lucas numbers ($n\in {\mathbb{N}}_0$):

  $$\begin{array}{cc}& \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_{2k}^3={\displaystyle \frac{1}{5}}{L}_3^n{F}_{3n}-{\displaystyle \frac{3}{5}}{F}_n,\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)L_{2k}^3=L_3^n{L}_{3n}+3L_n;\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_{2k}^3=\left\{\begin{array}{cc}{\sqrt{5}}^{n-2}\left\{F_3^n{F}_{3n}-3F_n\right\},\hfill & \mathrm{even}n;\hfill \\ {\sqrt{5}}^{n-3}\left\{L_3^n{F}_{3n}-3L_n\right\},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)L_{2k}^3=\left\{\begin{array}{cc}{\sqrt{5}}^n\left\{F_3^n{L}_{3n}+3L_n\right\},\hfill & \mathrm{even}n;\hfill \\ {\sqrt{5}}^{n+1}\left\{F_3^n{F}_{3n}+3F_n\right\},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$
• Pell and Pell–Lucas polynomials ($n\in {\mathbb{N}}_0$):

  $$\begin{array}{cc}& \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_{2k}^3\left(x\right)={\displaystyle \frac{Q_3^n\left(x\right)}{4(1+x^2)}}{P}_{3n}\left(x\right)-{\displaystyle \frac{3{\left(2x\right)}^n}{4(1+x^2)}}{P}_n\left(x\right),\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Q_{2k}^3\left(x\right)=Q_3^n\left(x\right)Q_{3n}\left(x\right)+3{\left(2x\right)}^n{Q}_n\left(x\right);\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_{2k}^3\left(x\right)=\left\{\begin{array}{cc}{\left(2\sqrt{1+x^2}\right)}^{n-2}\left\{P_3^n\left(x\right)P_{3n}\left(x\right)-3P_n\left(x\right)\right\},\hfill & \mathrm{even}n;\hfill \\ {\left(2\sqrt{1+x^2}\right)}^{n-3}\left\{Q_3^n\left(x\right)P_{3n}\left(x\right)-3Q_n\left(x\right)\right\},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Q_{2k}^3\left(x\right)=\left\{\begin{array}{cc}{\left(2\sqrt{1+x^2}\right)}^n\left\{P_3^n\left(x\right)Q_{3n}\left(x\right)+3Q_n\left(x\right)\right\},\hfill & \mathrm{even}n;\hfill \\ {\left(2\sqrt{1+x^2}\right)}^{n+1}\left\{P_3^n\left(x\right)P_{3n}\left(x\right)+3P_n\left(x\right)\right\},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$
• Chebyshev polynomials ($n\in {\mathbb{N}}_0$):

  $$\begin{array}{cc}& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_{2k}^3\left(x\right)={\displaystyle \frac{2^n{T}_3^n\left(x\right)}{4(x^2-1)}}{U}_{3n+2}\left(x\right)-{\displaystyle \frac{3{\left(2x\right)}^n}{4(x^2-1)}}{U}_n\left(x\right),\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)T_{2k}^3\left(x\right)=2^{n-2}{T}_3^n\left(x\right)T_{3n}\left(x\right)+{\displaystyle \frac{3}{4}}{\left(2x\right)}^n{T}_n\left(x\right);\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_{2k}^3\left(x\right)=\left\{\begin{array}{cc}{\left(2\sqrt{x^2-1}\right)}^{n-2}\left\{U_2^n\left(x\right)U_{3n+2}\left(x\right)-3U_n\left(x\right)\right\},\hfill & \mathrm{even}n;\hfill \\ 2{\left(2\sqrt{x^2-1}\right)}^{n-3}\left\{U_2^n\left(x\right)T_{3n+3}\left(x\right)-3T_{n+1}\left(x\right)\right\},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)T_{2k}^3\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{{\left(2\sqrt{x^2-1}\right)}^n}{4}}\left\{U_2^n\left(x\right)T_{3n}\left(x\right)+3T_n\left(x\right)\right\},\hfill & \mathrm{even}n;\hfill \\ {\displaystyle \frac{{\left(2\sqrt{x^2-1}\right)}^{n+1}}{8}}\left\{U_2^n\left(x\right)U_{3n-1}\left(x\right)+3U_{n-1}\left(x\right)\right\},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

4.8. Binomial Quartic Sums

Theorem 15 

($n\in \mathbb{N}$).

$$\begin{array}{cc}\hfill \left(a\right)& \sum _{k=0}^n{v}^{2n-2k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_k^4={\displaystyle \frac{{\Psi }_{2n}{\Psi }_2^n-4{(-2uv)}^n{\Psi }_n+6{\left(2v^2\right)}^n}{16{\Delta }^4}},\hfill \\ \hfill \left(b\right)& \sum _{k=0}^n{v}^{2n-2k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_k^4={\Psi }_{2n}{\Psi }_2^n+4{(-2uv)}^n{\Psi }_n+6{\left(2v^2\right)}^n;\hfill \\ \hfill \left(c\right)& \sum _{k=0}^n{(-v^2)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_k^4=\left\{\begin{array}{cc}{\displaystyle \frac{{\Psi }_{2n}}{16{\Delta }^4}}{(4u\Delta )}^n-{\displaystyle \frac{{\Psi }_n}{4{\Delta }^4}}{(2v\Delta )}^n,\hfill & \mathrm{even}n;\hfill \\ {\displaystyle \frac{{\Phi }_{2n}}{8{\Delta }^3}}{(4u\Delta )}^n+{\displaystyle \frac{{\Phi }_n}{2{\Delta }^3}}{(2v\Delta )}^n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ \hfill \left(d\right)& \sum _{k=0}^n{(-v^2)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Psi }_k^4=\left\{\begin{array}{cc}{\Psi }_{2n}{(4u\Delta )}^n+4{\Psi }_n{(2v\Delta )}^n,\hfill & \mathrm{even}n;\hfill \\ 2\Delta {\Phi }_{2n}{(4u\Delta )}^n-8\Delta {\Phi }_n{(2v\Delta )}^n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

Proof. 

We prove Formula (c) by reformulating the binomial sum and then evaluating it in closed form as illustrated below:

$$\begin{array}{cc}\hfill \sum _{k=0}^n{(-v^2)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\Phi }_k^4& =\sum _{k=0}^n{(-v^2)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\left\{{\displaystyle \frac{{\alpha }^k-{\gamma }^k}{\alpha -\gamma }}\right\}}^4\hfill \\ & =\sum _{k=0}^n{(-v^2)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\displaystyle \frac{{\alpha }^{4k}+{\gamma }^{4k}-4{(-v{\alpha }^2)}^k-4{(-v{\gamma }^2)}^k+6v^{2k}}{{(2\Delta )}^4}}\hfill \\ & ={\displaystyle \frac{{({\alpha }^4-v^2)}^n+{({\gamma }^4-v^2)}^n-4{(-v^2-v{\alpha }^2)}^n-4{(-v^2-v{\gamma }^2)}^n}{{(2\Delta )}^4}}\hfill \\ & ={\displaystyle \frac{{(4u\Delta )}^n\left({\alpha }^{2n}+{(-1)}^n{\gamma }^{2n}\right)-4{(-2v\Delta )}^n({\alpha }^n+{(-1)}^n{\gamma }^n)}{{(2\Delta )}^4}}\hfill \\ & =\left\{\begin{array}{cc}{\displaystyle \frac{{\Psi }_{2n}}{16{\Delta }^4}}{(4u\Delta )}^n-{\displaystyle \frac{{\Psi }_n}{4{\Delta }^4}}{(2v\Delta )}^n,\hfill & \mathrm{even}n;\hfill \\ {\displaystyle \frac{{\Phi }_{2n}}{8{\Delta }^3}}{(4u\Delta )}^n+{\displaystyle \frac{{\Phi }_n}{2{\Delta }^3}}{(2v\Delta )}^n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

The other three identities in the theorem can be shown similarly. □

• Fibonacci and Lucas numbers ($m\in \mathbb{Z}$ and $n\in \mathbb{N}$):

  $$\begin{array}{cc}& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_k^4={\displaystyle \frac{3^n{L}_{2n}-4{(-1)}^n{L}_n+6\times 2^n}{25}};\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)L_k^4=3^n{L}_{2n}+4{(-1)}^n{L}_n+6\times 2^n;\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)F_k^4={\left(\sqrt{5}\right)}^n\times \left\{\begin{array}{cc}{\displaystyle \frac{L_{2n}}{25}}-{\displaystyle \frac{4L_n}{25}},\hfill & \mathrm{even}n;\hfill \\ {\displaystyle \frac{F_{2n}}{5\sqrt{5}}}+{\displaystyle \frac{4F_n}{5\sqrt{5}}},\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)L_k^4={\left(\sqrt{5}\right)}^n\times \left\{\begin{array}{cc}{L}_{2n}+4L_n,\hfill & \mathrm{even}n;\hfill \\ \sqrt{5}{F}_{2n}-4\sqrt{5}{F}_n,\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$
• Pell and Pell–Lucas polynomials ($m\in \mathbb{Z}$ and $n\in \mathbb{N}$):

  $$\begin{array}{cc}& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_k^4\left(x\right)={\displaystyle \frac{Q_{2n}\left(x\right){(2+4x^2)}^n-4{(-2x)}^n{Q}_n\left(x\right)+6\times 2^n}{16{(1+x^2)}^2}},\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Q_k^4\left(x\right)=Q_{2n}\left(x\right){(2+4x^2)}^n+4{(-2x)}^n{Q}_n\left(x\right)+6\times 2^n;\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_k^4\left(x\right)={\left(2\sqrt{1+x^2}\right)}^{n-3}\times \left\{\begin{array}{cc}{\displaystyle \frac{Q_{2n}\left(x\right)}{2\sqrt{1+x^2}}}{\left(2x\right)}^n-{\displaystyle \frac{2Q_n\left(x\right)}{\sqrt{1+x^2}}},\hfill & \mathrm{even}n;\hfill \\ P_{2n}\left(x\right){\left(2x\right)}^n+4P_n\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)Q_k^4\left(x\right)={\left(2\sqrt{1+x^2}\right)}^{n+1}\times \left\{\begin{array}{cc}{\displaystyle \frac{Q_{2n}\left(x\right)}{2\sqrt{1+x^2}}}{\left(2x\right)}^n+{\displaystyle \frac{2Q_n\left(x\right)}{\sqrt{1+x^2}}},\hfill & \mathrm{even}n;\hfill \\ P_{2n}\left(x\right){\left(2x\right)}^n-4P_n\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$
• Chebyshev polynomials ($m\in \mathbb{Z}$ and $n\in \mathbb{N}$):

  $$\begin{array}{cc}& \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_k^4\left(x\right)={\displaystyle \frac{T_{2n+4}\left(x\right){(2x^2-1)}^n-4x^n{T}_{n+2}\left(x\right)+3}{2^{3-n}{(x^2-1)}^2}};\hfill \\ & \sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)T_k^4\left(x\right)=2^{n-3}\left\{T_{2n}\left(x\right){(2x^2-1)}^n+4x^n{T}_n\left(x\right)+3\right\},\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)U_k^4\left(x\right)={\left(2\sqrt{x^2-1}\right)}^{n-3}\times \left\{\begin{array}{cc}{\displaystyle \frac{{\left(2x\right)}^n{T}_{2n+4}\left(x\right)}{\sqrt{x^2-1}}}-{\displaystyle \frac{4T_{n+2}\left(x\right)}{\sqrt{x^2-1}}},\hfill & \mathrm{even}n;\hfill \\ U_{2n+3}\left(x\right){\left(2x\right)}^n-4U_{n+1}\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \\ & \sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)T_k^4\left(x\right)={\displaystyle \frac{{\left(2\sqrt{x^2-1}\right)}^{n+1}}{16}}\times \left\{\begin{array}{cc}{\displaystyle \frac{{\left(2x\right)}^n{T}_{2n}\left(x\right)}{\sqrt{x^2-1}}}+{\displaystyle \frac{4{(-1)}^n{T}_n\left(x\right)}{\sqrt{x^2-1}}},\hfill & \mathrm{even}n;\hfill \\ U_{2n-1}\left(x\right){\left(2x\right)}^n-4{(-1)}^n{U}_{n-1}\left(x\right),\hfill & \mathrm{odd}n.\hfill \end{array}\right.\hfill \end{array}$$

5. Conclusions and Further Prospects

We have presented a comprehensive collection of the summation formulae for two (universal) sequences, ${\Phi }_n(u,v)$ and ${\Psi }_n(u,v)$. Detailed proofs are offered for only a few of them, as most of them can be validated without difficulty by employing recurrence relations, Binet-form expressions, and generating functions. To ensure accuracy, all the displayed equations have been verified by the computer algebra system Wolfram Mathematica (version 11). Our formulae unify numerous identities for the following six number/polynomial sequences:

• Fibonacci and Lucas numbers mainly collected in Koshy’s monograph [4] and scattered through the literature (cf. [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]);
• Pell and Pell–Lucas polynomials mainly collected in Koshy’s monograph [6] and scattered through the literature (cf. [8,10,11,28,29,30,31,32,33]);
• Chebyshev polynomials of the first and second kinds mainly collected in Mason and Handscomb’s monograph [7] and scattered through the literature (cf. [34,35,36,37,38,39,40,41,42,43]).

It should be pointed out that there are important topics not covered in this paper. For instance, the following five themes:

• Reciprocal sums (concerning Fibonacci/Lucas numbers [44,45,46,47]);
• Multiple convolution sums (concerning Fibonacci/Lucas numbers [48,49,50,51] and Chebyshev polynomials [52,53,54]);
• Power sums of higher orders (concerning Chebyshev polynomials [37,39,55,56]);
• Binomial (finite and infinite) sums involving integer (floor and ceiling) functions [30,46,57,58,59,60,61];
• Fibonomial/Lucanomial coefficients and related sums connected to the q-series theory (cf. [62,63,64,65,66]).

The authors believe that these topics are worthy of extensive investigation. Interested readers are encouraged to carry out further explorations.