Deriving Binomial Convolution Formulas for Horadam Sequences via Context-Free Grammars

Abstract

The Horadam sequence $H_n(a,b;p,q)$ unifies a number of well-known sequences, such as Fibonacci and Lucas sequences. We use the context-free grammars as a new tool to study Horadam sequences. By introducing a set of auxiliary basis polynomials ($v_1,v_2,v_3$) and using the formal derivative associated with the Horadam grammar, we solve the convolution coefficients and provide a unified method to discover convolution formulas associated with binomial coefficients. These results are extended to subsequences with indices $kn$ through a parameterized grammar $G_k$. Using the modified grammar $\widetilde{G_k}$, we derive convolution formulas involving the weighting term ${(-q)}^{n-i}$. Furthermore, applying the proposed framework to $(p,q)$-Fibonacci and $(p,q)$-Lucas sequences, we derive explicit convolution formulas with parameters $(p,q)$. The framework is also applied to derive specific identities for Pell and Pell–Lucas numbers, as well as for Fermat and Fermat–Lucas numbers.

1. Introduction

The Horadam sequence $H_n(a,b;p,q)$ can be defined by

$$H_{n+2}=p{H}_{n+1}+q{H}_n$$

(1)

with the initial values $H_0=a$ and $H_1=b$ [1]. It unifies a number of well-known sequences; see Table 1 for special cases. It can be seen that two types of Horadam sequences, $H_n(0,1;p,q)$ and $H_n(2,p;p,q)$, are special. The sequence $H_n(0,1;p,q)$ is called $(p,q)$-Fibonacci sequence, and $H_n(2,p;p,q)$ is called $(p,q)$-Lucas sequence. In recent years, there have been many extensions of Horadam sequences. Cao and Wang [2] established symmetric inequalities for the reciprocal sums of Fibonacci numbers. Church and Bicknell [3] provided several different binomial convolutions about Fibonacci and Lucas numbers. For $k,n\ge 0$,

$$\sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)F_{ki}{L}_{kn-ki}=2^n{F}_{kn},$$

$$\sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)F_{ki}{F}_{kn-ki}={\displaystyle \frac{1}{5}}(2^n{L}_{kn}-2L_k^n),$$

$$\sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)L_{ki}{L}_{kn-ki}=2^n{L}_{kn}+2L_k^n.$$

Carlitz [4] also showed several binomial convolutions about Fibonacci and Lucas numbers, for example, for $n,k\ge 0$,

$$\sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right){(-1)}^{n-i}{F}_{ki}{F}_{k-1}^{n-i}=F_k^n{F}_n,$$

$$\sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right){(-1)}^{n-i}{L}_{ki}{F}_{k-1}^{n-i}=F_k^n{L}_n.$$

See [5,6,7,8,9,10,11] for more binomial identities about Horadam sequences. Some determinantal formulas and recurrence relations about Horadam sequences were given in [12], and the study of properties of polynomial sequences can be found in [13,14,15,16,17,18].

Table 1. Some special cases of the Horadam sequence.

Name	Parameter Representation
Fibonacci number	$H_n(0,1;1,1)$
Lucas number	$H_n(2,1;1,1)$
Pell number	$H_n(0,1;2,1)$
Pell–Lucas number	$H_n(2,2;2,1)$
Jacobsthal number	$H_n(0,1;1,2)$
Jacobsthal–Lucas number	$H_n(2,1;1,2)$
Fermat number	$H_n(0,1;3,-2)$
Fermat–Lucas number	$H_n(2,3;3,-2)$
balancing number	$H_n(0,1;6,-1)$
modified Lucas-balancing number	$H_n(2,6;6,-1)$
Fibonacci polynomial	$H_n(0,1;x,1)$
Lucas polynomial	$H_n(2,x;x,1)$
Pell polynomial	$H_n(0,1;2x,1)$
Pell–Lucas polynomial	$H_n(2,2x;2x,1)$
Jacobsthal polynomial	$H_n(0,1;1,2x)$
Jacobsthal–Lucas polynomial	$H_n(2,1;1,2x)$
Fermat polynomial	$H_n(0,1;x,-2)$
Fermat–Lucas polynomial	$H_n(2,x;x,-2)$

In [19], we found that the context-free grammar can be used to study Horadam sequences. Many combinatorial polynomials can be studied by grammars, making this method an important tool in combinatorics. The framework of using context-free grammars to generate combinatorial polynomials was proposed by Chen [20]. Using context-free grammars, we developed a general method to deduce determinantal representations of enumerative polynomials [21] and provided three new interpretations of the second-order Eulerian polynomials [22]. The context-free grammar can also be used to discover new combinatorial models and algorithms [16,23], new convolution formulas [23,24], and different expansions and new relationships [16,23,24]. According to [25], an advantage of the grammatical description of a combinatorial sequence is that a recursion of its generating function can be provided by attaching a labeling of the combinatorial object in accordance with the replacement rules of the grammar. According to [26], an advantage of the change in grammars is that the recurrence system of the bi-$\gamma$-coefficients of colored Eulerian polynomials can be deduced and the bi-$\gamma$-positivity can be shown. More applications of context-free grammars in combinatorics can be found in [27,28,29,30,31].

In this paper, we focus on convolution formulas of Horadam sequences associated with binomial coefficients. The paper is organized as follows: In Section 2, we first recall the Horadam grammar, then collect some explicit formulas for Horadam sequences that will be used in the following discussion. In Section 3, we deduce three kinds of binomial convolution formulas. Section 3.1 presents the main result of this paper (Theorem 1). For special sequences, $(p,q)$-Fibonacci sequence and $(p,q)$-Lucas sequence, we give the explicit convolution formulas with parameters $(p,q)$. Section 3.2 introduces additional parameter k and generalizes the results obtained in Section 3.1 via a modified Horadam grammar (Theorem 3). As an application, we give explicit formulas for Pell and Pell–Lucas numbers (Corollary 6), as well as for Fermat and Fermat–Lucas numbers (Corollary 7). Section 3.3 provides a unified way to deduce convolution formulas in the form ${\sum }_{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right){(-q)}^{n-i}{A}_i{B}_k^{n-i}$ (Theorem 5).

2. Preliminaries

For the sake of brevity, we shall fix the following notations:

• $H_n$ denotes Horadam sequence $H_n(a,b;p,q)$.
• $A_n$ denotes Horadam sequence $H_n(A_0,A_1;p,q)$,

  $$H_0=A_0,H_1=A_1,\mathrm{and}{H}_{n+2}=p{H}_{n+1}+q{H}_n(n=0,1,2,\cdots ).$$
• $B_n$ denotes Horadam sequence $H_n(B_0,B_1;p,q)$,

  $$H_0=B_0,H_1=B_1,\mathrm{and}{H}_{n+2}=p{H}_{n+1}+q{H}_n(n=0,1,2,\cdots ).$$
• $F_n$ denotes $(p,q)$-Fibonacci sequence $H_n(0,1;p,q)$,

  $$H_0=0,H_1=1,\mathrm{and}{H}_{n+2}=p{H}_{n+1}+q{H}_n(n=0,1,2,\cdots ).$$
• $L_n$ denotes $(p,q)$-Lucas sequence $H_n(2,p;p,q)$,

  $$H_0=2,H_1=p,\mathrm{and}{H}_{n+2}=p{H}_{n+1}+q{H}_n(n=0,1,2,\cdots ).$$
• $\mathrm{\Delta }$ denotes $p^2+4q$.

Let A be an alphabet whose letters are regarded as independent commutative indeterminates. A formal function over A is defined as follows:

(1)

Each letter in A is a formal function.

(2)

If x and y are formal functions, then $x+y$ and $xy$ are all formal functions.

(3)

If x is a formal function and $f\left(t\right)$ is an analytic function in t, then $f\left(x\right)$ is a formal function.

(4)

Every formal function is constructed as above in a finite number of steps.

A context-free grammar G over A consists of substitution rules replacing letters in A by formal functions over A [20]. A rule in a context-free grammar is also said to be a production as in the theory of formal language [32]. We can define a formal derivative $D_G$ with respect to the grammar G as a differential operator on formal functions over A. More precisely, $D_G$ is defined as follows:

(1)

For two formal functions x and y, we have

$$D_G(x+y)=D_G\left(x\right)+D_G\left(y\right),D_G\left(xy\right)=D_G\left(x\right)y+x{D}_G\left(y\right).$$

(2)

For any analytic function $f\left(t\right)$ and formal function x, we have

$$D_{G}f\left(x\right)={\displaystyle \frac{\partial f\left(x\right)}{\partial x}}{D}_G\left(x\right).$$

(3)

For a letter u in A, if there is a rule $u\to v$ in the grammar G, where v is a formal function, then $D_G\left(u\right)=v$; otherwise $D_G\left(u\right)=0$, and u is called a constant or a terminal.

For example, let $G=\{x\to x+y,y\to x\}$ and $D_G$ be the formal derivative with respect to G. Then, we have $D_G\left(x\right)=x+y$, $D_G(x+2y)=D_G\left(x\right)+2D_G\left(y\right)=3x+y$, and $D_G\left(x^{-1}\right)=(-1)x^{-2}{D}_G\left(x\right)=-x^{-1}-x^{-2}y$.

We define $D_G^0\left(x\right)=x$ for any formal function x. It is clear that Leibnitz’s formula still holds for a formal derivative $D_G$,

$$D_G^n\left(xy\right)=\sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)D_G^i\left(x\right)D_G^{n-i}\left(y\right).$$

(2)

The Horadam grammar was introduced in [19]. For frequently used grammars in the paper, we usually mark them with a subscript. We put a superscript label 1 right after G to indicate Horadam grammar throughout the paper, which is defined as follows:

$$G_1=\{x\to px+y,y\to qx\}.$$

(3)

Using Horadam grammar $G_1$, we discovered several identities [19]. At the beginning, we need some useful lemmas for later proof.

Lemma 1 

([19]). Let $H_n$ be Horadam sequence $H_n(a,b;p,q)$. For $n⩾0$, we have

$$D_{G_1}^n(bx+ay)=H_{n+1}x+H_{n}y,$$

and

$$D_{G_1}^n{(bx+ay)|}_{x=0,y=1}=H_n.$$

(4)

In particular,

$$\begin{array}{cc}\hfill D_{G_1}^n{\left(x\right)|}_{x=0,y=1}& =F_n,\hfill \\ \hfill D_{G_1}^n{(px+2y)|}_{x=0,y=1}& =L_n.\hfill \end{array}$$

Lemma 2 

([19]). Let $F_n$ be $(p,q)$-Fibonacci sequence $H_n(0,1;p,q)$. For $n,m⩾1$,

$$F_{n+m-1}=F_n{F}_m+q{F}_{n-1}{F}_{m-1}.$$

Lemma 3. 

For $n,k⩾1$, we have

$$F_{kn+1}+q{F}_{kn-1}=L_{kn}.$$

Proof. 

From Lemma 1, we get that $F_{kn+1}=D_{G_1}^{kn+1}\left(x\right){{|}_{x=0,y=1},F_{kn-1}=D_{G_1}^{kn-1}\left(x\right)|}_{x=0,y=1}$, and $L_{kn}=D_{G_1}^{kn}{(px+2y)|}_{x=0,y=1}$. Since

$$\begin{array}{cc}\hfill D_{G_1}^{kn+1}\left(x\right)+q{D}_{G_1}^{kn-1}\left(x\right)& =D_{G_1}^{kn-1}(D_{G_1}^2\left(x\right)+qx)\hfill \\ \hfill & =D_{G_1}^{kn-1}((p^2+2q)x+py)\hfill \\ \hfill & =D_{G_1}^{kn-1}(L_{2}x+L_{1}y)\hfill \\ \hfill & =D_{G_1}^{kn}(L_{1}x+L_{0}y)\hfill \\ \hfill & =D_{G_1}^{kn}(px+2y),\hfill \end{array}$$

taking $x=0,y=1$ in both sides of the above equation, we can obtain the desired result. □

When $k=1$, by Lemma 3, we see that $F_{n+1}+q{F}_{n-1}=L_n$.

3. Convolution Formulas of Horadam Sequences

In this section, we provide a framework to discover binomial convolution formulas of Horadam sequences. Section 3.1 deduces convolution formulas involving the $(p,q)$-Fibonacci sequence $F_n$ and the $(p,q)$-Lucas sequence $L_n$. Section 3.2 generalizes the result of Section 3.1 involving $F_{kn}$ and $L_{kn}$.

In Section 3.1 and Section 3.2, we always assume $\mathrm{\Delta }\ne 0$.

3.1. Convolution Formulas Related to $F_n$ and $L_n$

Let $A_n$ be the Horadam sequences $H_n(A_0,A_1;p,q)$ and $B_n$ be $H_n(B_0,B_1;p,q)$, respectively. By Leibnitz formula (2) and Equation (4), we have

$$\begin{array}{cc}\hfill & D_{G_1}^n\left((A_{1}x+A_{0}y)(B_{1}x+B_{0}y)\right){|}_{x=0,y=1}\hfill \\ \hfill =& \sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)D_{G_1}^i(A_{1}x+A_{0}y){{|}_{x=0,y=1}{D}_{G_1}^{n-i}(B_{1}x+B_{0}y)|}_{x=0,y=1}\hfill \\ \hfill =& \sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)A_i{B}_{n-i}.\hfill \end{array}$$

(5)

It follows that if we find the sequence corresponding to

$$D_{G_1}^n\left((A_{1}x+A_{0}y)(B_{1}x+B_{0}y)\right){|}_{x=0,y=1};$$

then, we can discover the convolution formulas. Since it is difficult to find the sequences corresponding to $D_{G_1}^n\left(x^2\right){{|}_{x=0,y=1},D_{G_1}^n\left(xy\right)|}_{x=0,y=1}$ and $D_{G_1}^n\left(y^2\right){|}_{x=0,y=1}$, we choose another three polynomials $v_1,v_2$ and $v_3$ to express $(A_{1}x+A_{0}y)(B_{1}x+B_{0}y)$. We recall the special sequences $F_n$ and $L_n$,

$$F_0=0,F_1=1,F_2=p,F_3=p^2+q,\cdots$$

$$L_0=2,L_1=p,L_2=p^2+2q,L_3=p^3+3pq,\cdots$$

Notice that $(1x+0y)(px+2y)=p{x}^2+2xy$ is a special polynomial since

$$p{x}^2+2xy=F_2{x}^2+2F_{1}xy+F_0{y}^2$$

and

$$D_{G_1}(p{x}^2+2xy)=2((p^2+q)x^2+2pxy+y^2)=2(F_3{x}^2+2F_{2}xy+F_1{y}^2).$$

For a nonzero polynomial v, if there exists a number $\lambda$ such that $D_{G_1}\left(v\right)=\lambda v$, then v is also a special polynomial. From the above discussion about special polynomials, we choose the following polynomials: $v_1,v_2$ and $v_3$. Let

$$\left\{\begin{array}{cc}\hfill v_1& =p{x}^2+2xy,\hfill \\ \hfill v_2& =(p^2+q)x^2+2pxy+y^2,\hfill \\ \hfill v_3& =-q{x}^2+pxy+y^2,\hfill \end{array}\right.$$

(6)

which satisfy $D_{G_1}\left(v_1\right)=2v_2$ and $D_{G_1}\left(v_3\right)=p{v}_3$. We can verify the determinant of the coefficient matrix

$$\left|\begin{array}{ccc}p& p^2+q& -q\\ 2& 2p& p\\ 0& 1& 1\end{array}\right|=-p^2-4q=-\mathrm{\Delta }\ne 0,$$

which follows that $x^2$, $xy$ and $y^2$ can be represented as linear combinations of $v_1,v_2$ and $v_3$. Therefore, the product of linear forms, $(A_{1}x+A_{0}y)(B_{1}x+B_{0}y)$, can be linearly expressed in terms of $v_1,v_2$ and $v_3$. Moreover, the sequences corresponding to $D_{G_1}^n\left(v_1\right){{|}_{x=0,y=1},D_{G_1}^n\left(v_2\right)|}_{x=0,y=1}$ and $D_{G_1}^n\left(v_3\right){|}_{x=0,y=1}$ can be given as follows.

Lemma 4. 

For $n⩾0$, we have

(i) 

$D_{G_1}^n\left(v_1\right){|}_{x=0,y=1}=2^n{F}_n$;

(ii) 

$D_{G_1}^n\left(v_2\right){|}_{x=0,y=1}=2^n{F}_{n+1}$;

(iii) 

$D_{G_1}^n\left(v_3\right){|}_{x=0,y=1}=p^n$.

Proof. 

First, we claim that

$$D_{G_1}^n\left(v_1\right)=2^n{F}_{n+2}{x}^2+2^{n+1}{F}_{n+1}xy+2^n{F}_n{y}^2.$$

(7)

Note that

$$D_{G_1}\left(v_1\right)=D_{G_1}(p{x}^2+2xy)=2(p^2+q)x^2+2^{2}pxy+2y^2.$$

Thus the assertion holds for $n=1$. By induction, we assume that the assertion holds for $n=m-1$. Since $D_{G_1}\left(x^2\right)=2p{x}^2+2xy$, $D_{G_1}\left(xy\right)=q{x}^2+pxy+y^2$, and $D_{G_1}\left(y^2\right)=2qxy$, we have

$$\begin{array}{cc}\hfill D_{G_1}^m\left(v_1\right)& =D_{G_1}\left(D_{G_1}^{m-1}\left(v_1\right)\right)\hfill \\ \hfill & =D_{G_1}(2^{m-1}{F}_{m+1}{x}^2+2^m{F}_{m}xy+2^{m-1}{F}_{m-1}{y}^2)\hfill \\ \hfill & =2^{m-1}{F}_{m+1}{D}_{G_1}\left(x^2\right)+2^m{F}_m{D}_{G_1}\left(xy\right)+2^{m-1}{F}_{m-1}{D}_{G_1}\left(y^2\right)\hfill \\ \hfill & =2^m(p{F}_{m+1}+q{F}_m)x^2+2^m(F_{m+1}+p{F}_m+q{F}_{m-1})xy+2^m{F}_m{y}^2\hfill \\ \hfill & =2^m{F}_{m+2}{x}^2+2^{m+1}{F}_{m+1}xy+2^m{F}_m{y}^2.\hfill \end{array}$$

Therefore the assertion is true for $n=m$, as desired.

Taking $x=0,y=1$ on both sides of (7), we can obtain $\left(i\right)$. Since $D_{G_1}\left(v_1\right)=2v_2$, we obtain $D_{G_1}^n\left(v_2\right)={\displaystyle \frac{1}{2}}{D}_{G_1}^{n+1}\left(v_1\right)$ and hence $\left(ii\right)$ holds.

At last, since $D_{G_1}\left(v_3\right)=p{v}_3$, we can get $D_{G_1}^n\left(v_3\right)=p^n{v}_3$. Combining this identity and $v_3{|}_{x=0,y=1}=1$, we obtain $\left(iii\right)$. □

Now we give the main result of this paper.

Theorem 1. 

Let $A_n$ denote the Horadam sequence $H_n(A_0,A_1;p,q)$ and let $B_n$ denote the Horadam sequence $H_n(B_0,B_1;p,q)$, respectively. Set $\Delta =p^2+4q$. We have

$$\begin{array}{cc}\hfill & \sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)A_i{B}_{n-i}\hfill \\ \hfill =& {\displaystyle \frac{-p{A}_1{B}_1+(A_0{B}_1+A_1{B}_0)(p^2+2q)-A_0{B}_0(p^3+3pq)}{\Delta }}{2}^n{F}_n\hfill \\ \hfill +& {\displaystyle \frac{2A_1{B}_1-(A_0{B}_1+A_1{B}_0)p+A_0{B}_0(p^2+2q)}{\Delta }}{2}^n{F}_{n+1}\hfill \\ \hfill +& {\displaystyle \frac{-2A_1{B}_1+(A_0{B}_1+A_1{B}_0)p+A_0{B}_{0}2q}{\Delta }}{p}^n.\hfill \end{array}$$

Proof. 

By (6), we see that

$$\begin{array}{cc}\hfill x^2& =-{\displaystyle \frac{p}{\Delta }}{v}_1+{\displaystyle \frac{2}{\Delta }}{v}_2-{\displaystyle \frac{2}{\Delta }}{v}_3,\hfill \\ \hfill xy& ={\displaystyle \frac{p^2+2q}{\Delta }}{v}_1-{\displaystyle \frac{p}{\Delta }}{v}_2+{\displaystyle \frac{p}{\Delta }}{v}_3,\hfill \\ \hfill y^2& =-{\displaystyle \frac{p^3+3pq}{\Delta }}{v}_1+{\displaystyle \frac{p^2+2q}{\Delta }}{v}_2+{\displaystyle \frac{2q}{\Delta }}{v}_3.\hfill \end{array}$$

Substituting the above three equations into $(A_{1}x+A_{0}y)(B_{1}x+B_{0}y)$ yields

$$\begin{array}{cc}\hfill & (A_{1}x+A_{0}y)(B_{1}x+B_{0}y)\hfill \\ \hfill =& {\displaystyle \frac{-p{A}_1{B}_1+(A_0{B}_1+A_1{B}_0)(p^2+2q)-A_0{B}_0(p^3+3pq)}{\Delta }}{v}_1\hfill \\ \hfill +& {\displaystyle \frac{2A_1{B}_1-(A_0{B}_1+A_1{B}_0)p+A_0{B}_0(p^2+2q)}{\Delta }}{v}_2\hfill \\ \hfill +& {\displaystyle \frac{-2A_1{B}_1+(A_0{B}_1+A_1{B}_0)p+A_0{B}_{0}2q}{\Delta }}{v}_3.\hfill \end{array}$$

Then, by Lemma 4, we get

$$\begin{array}{cc}\hfill & D_{G_1}^n\left((A_{1}x+A_{0}y)(B_{1}x+B_{0}y)\right){|}_{x=0,y=1}\hfill \\ \hfill =& {\displaystyle \frac{-p{A}_1{B}_1+(A_0{B}_1+A_1{B}_0)(p^2+2q)-A_0{B}_0(p^3+3pq)}{\Delta }}{2}^n{F}_n\hfill \\ \hfill +& {\displaystyle \frac{2A_1{B}_1-(A_0{B}_1+A_1{B}_0)p+A_0{B}_0(p^2+2q)}{\Delta }}{2}^n{F}_{n+1}\hfill \\ \hfill +& {\displaystyle \frac{-2A_1{B}_1+(A_0{B}_1+A_1{B}_0)p+A_0{B}_{0}2q}{\Delta }}{p}^n.\hfill \end{array}$$

Combining with (5), we can obtain the desired result. □

Substituting the initial values of Horadam sequences in Theorem 1 yields the following corollary.

Corollary 1. 

For $(p,q)$-Fibonacci sequence $F_n$ and $(p,q)$-Lucas sequence $L_n$, we have

(i) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)F_i{L}_{n-i}=2^n{F}_n}$;

(ii) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)F_i{F}_{n-i}=\frac{-2^{n}p}{\Delta }{F}_n+\frac{2^{n+1}}{\Delta }{F}_{n+1}-\frac{2p^n}{\Delta }}$;

(iii) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)L_i{L}_{n-i}=-2^{n}p{F}_n+2^{n+1}{F}_{n+1}+2p^n}$.

Using Lemma 3, we can obtain the following result from the above corollary.

Corollary 2. 

We have

(i) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)F_i{F}_{n-i}=\frac{2^n}{\Delta }{L}_n-\frac{2p^n}{\Delta }}$;

(ii) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)L_i{L}_{n-i}=2^n{L}_n+2p^n}$.

3.2. Convolution Formulas Related to $F_{kn}$ and $L_{kn}$

In the previous subsection, we have obtained some convolution formulas related to $F_n$ and $L_n$ using Horadam grammar (3), which can be rewritten as

$$G_1=\{x\to F_{2}x+F_{1}y,y\to q{F}_{1}x+q{F}_{0}y\}.$$

where $F_0=0$, $F_1=1$ and $F_2=p$ are $(p,q)$-Fibonacci numbers. Now, we add a parameter k and consider the grammar

$$G_k=\{x\to F_{k+1}x+F_{k}y,y\to q{F}_{k}x+q{F}_{k-1}y\}.$$

where $F_{k-1}$, $F_k$ and $F_{k+1}$ are all $(p,q)$-Fibonacci numbers.

Theorem 2. 

For $n⩾1$, we have

(i) 

$D_{G_k}^n\left(x\right)=F_{kn+1}x+F_{kn}y$;

(ii) 

$D_{G_k}^n\left(y\right)=q{F}_{kn}x+q{F}_{kn-1}y$;

(iii) 

$D_{G_k}^n(bx+ay)=(b{F}_{kn+1}+aq{F}_{kn})x+(b{F}_{kn}+aq{F}_{kn-1})y$.

Proof. 

First, we prove (i) by induction. Note that

$$D_{G_k}\left(x\right)=F_{k+1}x+F_{k}y.$$

The result holds for $n=1$. By induction, assume that the result holds for $n=m-1$. Then, we calculate

$$\begin{array}{cc}\hfill D_{G_k}^m\left(x\right)& =D_{G_k}\left(D_{G_k}^{m-1}\left(x\right)\right)\hfill \\ & =D_{G_k}(F_{k(m-1)+1}x+F_{k(m-1)}y)\hfill \\ & =F_{k(m-1)+1}{D}_{G_k}\left(x\right)+F_{k(m-1)}{D}_{G_k}\left(y\right)\hfill \\ & =F_{k(m-1)+1}(F_{k+1}x+F_{k}y)+F_{k(m-1)}(q{F}_{k}x+q{F}_{k-1}y)\hfill \\ & =(F_{k(m-1)+1}{F}_{k+1}+q{F}_{k(m-1)}{F}_k)x+(F_{k(m-1)+1}{F}_k+q{F}_{k(m-1)}{F}_{k-1})y\hfill \\ & =F_{km+1}x+F_{km}y,\hfill \end{array}$$

where the last equation follows from Lemma 2. It shows that the result still holds for $n=m$. Similarly, we can obtain ($ii$) by induction. The result ($iii$) follows from (i) and ($ii$). This completes the proof. □

By (1) and Lemma 3, we can get $p{F}_{kn+1}+2q{F}_{kn}=F_{kn+2}+q{F}_{kn}=L_{kn+1}$. Using the result of Theorem 2 ($iii$) with $a=2$ and $b=p$, we have the following corollary.

Corollary 3. 

For $n⩾1$, we have

$$D_{G_k}^n(px+2y)=L_{kn+1}x+L_{kn}y.$$

Next, we consider to use the Leibnitz formula

$$\begin{array}{cc}\hfill & D_{G_k}^n\left((A_{1}x+A_{0}y)(B_{1}x+B_{0}y)\right){|}_{x=0,y=1}\hfill \\ \hfill =& \sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)D_{G_k}^i(A_{1}x+A_{0}y){{|}_{x=0,y=1}{D}_{G_k}^{n-i}(B_{1}x+B_{0}y)|}_{x=0,y=1}.\hfill \end{array}$$

(8)

to generate a new convolution formula. Taking $x=0,y=1$ in Theorem 2$\left(iii\right)$, we can see that the right-hand side of Equation (8) is equal to

$$\sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)(A_1{F}_{ki}+q{A}_0{F}_{ki-1})(B_1{F}_{kn-ki}+q{B}_0{F}_{kn-ki-1}),$$

which can be written as

$$\sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\sum _{j,l=0}^1{A}_j{B}_l{q}^{2-j-l}{F}_{ki-1+j}{F}_{kn-ki-1+l}.$$

Therefore we need to find the sequence corresponding to

$$D_{G_k}^n\left((A_{1}x+A_{0}y)(B_{1}x+B_{0}y)\right){|}_{x=0,y=1}.$$

We employ the same basis polynomials $v_1,v_2$ and $v_3$ defined in Equation (6) to solve the above problem. From $D_{G_k}\left(v_1\right)=2F_{k+2}{x}^2+2^2{F}_{k+1}xy+2F_k{y}^2$, $D_{G_k}\left(v_2\right)=2F_{k+3}{x}^2+2^2{F}_{k+2}xy+2F_{k+1}{y}^2$, and $D_{G_k}\left(v_3\right)=L_k{v}_3$, we can derive the following lemma. The proof is similar to that of Lemma 4, so we omit it.

Lemma 5. 

For $n⩾1$, we have

(i) 

$D_{G_k}^n\left(v_1\right){|}_{x=0,y=1}=2^n{F}_{kn}$;

(ii) 

$D_{G_k}^n\left(v_2\right){|}_{x=0,y=1}=2^n{F}_{kn+1}$;

(iii) 

$D_{G_k}^n\left(v_3\right){|}_{x=0,y=1}=L_k^n$.

Using the same manner with Theorem 1, we can obtain the following theorem by Theorem 2, Corollary 3 and Lemma 5:

Theorem 3. 

Let $A_n$ denote the Horadam sequence $H_n(A_0,A_1;p,q)$ and let $B_n$ denote the Horadam sequence $H_n(B_0,B_1;p,q)$, respectively. Set $\Delta =p^2+4q$. We have

$$\begin{array}{cc}\hfill & \sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\sum _{j,l=0}^1{A}_j{B}_l{q}^{2-j-l}{F}_{ki-1+j}{F}_{kn-ki-1+l}\hfill \\ \hfill =& {\displaystyle \frac{-p{A}_1{B}_1+(A_0{B}_1+A_1{B}_0)(p^2+2q)-A_0{B}_0(p^3+3pq)}{\Delta }}{2}^n{F}_{kn}\hfill \\ \hfill +& {\displaystyle \frac{2A_1{B}_1-(A_0{B}_1+A_1{B}_0)p+A_0{B}_0(p^2+2q)}{\Delta }}{2}^n{F}_{kn+1}\hfill \\ \hfill +& {\displaystyle \frac{-2A_1{B}_1+(A_0{B}_1+A_1{B}_0)p+A_0{B}_{0}2q}{\Delta }}{L}_k^n.\hfill \end{array}$$

Corollary 4. 

For the $(p,q)$-Fibonacci sequence $F_n$ and $(p,q)$-Lucas sequence $L_n$, we have

(i) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)F_{ki}{L}_{kn-ki}=2^n{F}_{kn}}$;

(ii) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)F_{ki}{F}_{kn-ki}=\frac{-2^{n}p}{\Delta }{F}_{kn}+\frac{2^{n+1}}{\Delta }{F}_{kn+1}-\frac{2}{\Delta }{L}_k^n}$;

(iii) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)L_{ki}{L}_{kn-ki}=-2^{n}p{F}_{kn}+2^{n+1}{F}_{kn+1}+2L_k^n}$.

By Lemma 3, we have $-p{F}_{kn}+2F_{kn+1}=q{F}_{kn-1}+F_{kn+1}=L_{kn}$. So we obtain the following corollary:

Corollary 5. 

For the $(p,q)$-Fibonacci sequence $F_n$ and $(p,q)$-Lucas sequence $L_n$, we have

(i) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)F_{ki}{F}_{kn-ki}=\frac{2^n}{\Delta }{L}_{kn}-\frac{2}{\Delta }{L}_k^n}$;

(ii) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)L_{ki}{L}_{kn-ki}=2^n{L}_{kn}+2L_k^n}$.

Many sequences, including Fibonacci and Lucas numbers, Jacobsthal and Jacobsthal–Lucas numbers, and all other sequences shown in Table 1, are special cases of $F_n$ and $L_n$. For simplicity, we only give convolution formulas for Pell and Pell–Lucas numbers, as well as for Fermat and Fermat–Lucas numbers as applications of Theorem 3, as readers may derive other formulas by analogy.

Corollary 6. 

Let $P_n$ be the Pell number $H_n(0,1;2,1)$ and $Q_n$ be the Pell–Lucas number $H_n(2,2;2,1)$, respectively. We have

(i) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)P_{ki}{Q}_{kn-ki}=2^n{P}_{kn}}$;

(ii) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)P_{ki}{P}_{kn-ki}=-2^{n-2}{P}_{kn}+2^{n-2}{P}_{kn+1}-\frac{1}{4}{Q}_k^n}$;

(iii) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)Q_{ki}{Q}_{kn-ki}=-2^{n+1}{P}_{kn}+2^{n+1}{P}_{kn+1}+2Q_k^n}$.

Corollary 7. 

Let $S_n$ be the Fermat number $H_n(0,1;3,-2)$ and $T_n$ be the Fermat–Lucas number $H_n(2,3;3,-2)$, respectively. We have

(i) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)S_{ki}{T}_{kn-ki}=2^n{S}_{kn}}$;

(ii) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)S_{ki}{S}_{kn-ki}=-3·2^n{S}_{kn}+2^{n+1}{S}_{kn+1}-2T_k^n}$;

(iii) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)T_{ki}{T}_{kn-ki}=-3·2^n{S}_{kn}+2^{n+1}{S}_{kn+1}+2T_k^n}$.

Remark 1. 

In Section 3.1 and Section 3.2, we assume $\Delta =p^2+4q\ne 0$. This is a necessary condition for the invertibility of the coefficient matrix used to define $v_1,v_2$ and $v_3$ in Equation (6). When $\Delta =0$, the method fails. But for $v_1,v_2$ and $v_3$ in Equation (6) and Theorem 2, Corollary 3 and Lemma 5 always hold, regardless of whether Δ is zero. Namely, the following equations are valid for all values of p and q:

$$\begin{array}{cc}\hfill D_{G_k}^n{\left(x\right)|}_{x=0,y=1}& =F_{kn},\hfill \\ \hfill D_{G_k}^n{(px+2y)|}_{x=0,y=1}& =L_{kn},\hfill \\ \hfill D_{G_k}^n\left(v_1\right){|}_{x=0,y=1}& =2^n{F}_{kn},\hfill \\ \hfill D_{G_k}^n\left(v_3\right){|}_{x=0,y=1}& =L_k^n.\hfill \end{array}$$

Note that $v_1=p{x}^2+2xy=x(px+2y)$ and if $\Delta =0$, then $v_3=-q{x}^2+pxy+y^2={\displaystyle \frac{1}{4}}{(px+2y)}^2$. Thus, using the Formula (8) with $A_{1}x+A_{0}y=x$, $B_{1}x+B_{0}y=px+2y$, we have

$$\sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)F_{ki}{L}_{kn-ki}=2^n{F}_{kn}.$$

(9)

Using the Formula (8) with $A_{1}x+A_{0}y=px+2y$, $B_{1}x+B_{0}y=px+2y$, we can drive

$$\sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)L_{ki}{L}_{kn-ki}=2^2{L}_k^n.$$

(10)

Since $F_n={\displaystyle \frac{n{p}^n}{2^{n-1}}}$ and $L_n={\displaystyle \frac{p^n}{2^{n-1}}}$ when $\Delta =0$, the two convolution Formulas (9) and (10) are consistent with the formulas of Corollary 4 $\left(i\right)$ and $\left(iii\right)$, respectively.

To sum up, the formulas of Corollary 4 $\left(i\right)$ and $\left(iii\right)$ always hold, regardless of whether Δ is zero.

For example, let $M_n$ denote the modified Oresme number $H_n(0,1;1,-{\displaystyle \frac{1}{4}})$ defined by

$$M_0=0,M_1=1,\mathrm{and}{M}_{n+2}=M_{n+1}-{\displaystyle \frac{1}{4}}{M}_n(n=0,1,2,\cdots ),$$

and let $R_n$ denote the Oresme–Lucas number $H_n(2,1;1,-{\displaystyle \frac{1}{4}})$ defined by

$$R_0=2,R_1=1,\mathrm{and}{R}_{n+2}=R_{n+1}-{\displaystyle \frac{1}{4}}{R}_n(n=0,1,2,\cdots ).$$

The sequence $M_n$ begins with

$$0,1,1,{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{5}{16}},{\displaystyle \frac{3}{16}},\cdots ,{\displaystyle \frac{n}{2^{n-1}}},\cdots$$

The sequence $R_n$ begins with

$$2,1,{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{8}},{\displaystyle \frac{1}{16}},{\displaystyle \frac{1}{32}},\cdots ,{\displaystyle \frac{1}{2^{n-1}}},\cdots$$

We can verify the Formulas (9) and (10). For example, when $n=3$, $k=2$, we have

$$\left({\displaystyle \genfrac{}{}{0pt}{}{3}{0}}\right)M_0{R}_6+\left({\displaystyle \genfrac{}{}{0pt}{}{3}{1}}\right)M_2{R}_4+\left({\displaystyle \genfrac{}{}{0pt}{}{3}{2}}\right)M_4{R}_2+\left({\displaystyle \genfrac{}{}{0pt}{}{3}{3}}\right)M_6{R}_0=2^3{M}_6.$$

$$\left({\displaystyle \genfrac{}{}{0pt}{}{3}{0}}\right)R_0{R}_6+\left({\displaystyle \genfrac{}{}{0pt}{}{3}{1}}\right)R_2{R}_4+\left({\displaystyle \genfrac{}{}{0pt}{}{3}{2}}\right)R_4{R}_2+\left({\displaystyle \genfrac{}{}{0pt}{}{3}{3}}\right)R_6{R}_0=2^2{R}_2^3.$$

3.3. Convolution Formulas Related to ${(-q)}^{n-i}$

Motivated by Fibonacci convolution identities involving the term ${(-1)}^{n-i}$ in [4], we extend the study of these identities to Horadam sequences, focusing on convolution formulas weighted by ${(-q)}^{n-i}$. We introduce a new context-free grammar to generate such formulas. Let

$$\widetilde{G_k}=\{x\to F_{k+1}x+F_{k}y,y\to q{F}_{k}x+q{F}_{k-1}y,z\to -q{F}_{k-1}z\}.$$

The substitution rules on $\{x,y\}$ in $\widetilde{G_k}$ is the same with $G_k$, and hence by Theorem 2, we have

$$D_{\widetilde{G_k}}^n\left(x\right)=F_{kn+1}x+F_{kn}y,D_{\widetilde{G_k}}^n\left(y\right)=q{F}_{kn}x+q{F}_{kn-1}y.$$

(11)

The rule $z\to -q{F}_{k-1}z$ in the grammar $\widetilde{G_k}$ leads that $D_{\widetilde{G_k}}\left(z\right)=-q{F}_{k-1}z$ and $D_{\widetilde{G_k}}^2\left(z\right)=D_{\widetilde{G_k}}\left(D_{\widetilde{G_k}}\left(z\right)\right)=D_{\widetilde{G_k}}(-q{F}_{k-1}z)=-q{F}_{k-1}{D}_{\widetilde{G_k}}\left(z\right)={(-q)}^2{F}_{k-1}^{2}z$, and one can easily check that

$$D_{\widetilde{G_k}}^n\left(z\right)={(-q)}^n{F}_{k-1}^{n}z.$$

(12)

Theorem 4. 

For $n⩾1$, we have

(i) 

$D_{\widetilde{G_k}}^n\left(xz\right)=F_k^n(F_{n+1}xz+F_{n}yz)$;

(ii) 

$D_{\widetilde{G_k}}^n\left(yz\right)=q{F}_k^n(F_{n}xz+F_{n-1}yz)$;

(iii) 

$D_{\widetilde{G_k}}^n\left((bx+ay)z\right)=F_k^n(b{F}_{n+1}+aq{F}_n)xz+F_k^n(b{F}_n+aq{F}_{n-1})yz$.

Proof. 

First, we show (i) by induction. Note that

$$D_{\widetilde{G_k}}\left(xz\right)=(F_{k+1}-q{F}_{k-1})xz+F_{k}yz=F_k(pxz+yz)=F_k(F_{2}xz+F_{1}yz).$$

The result holds for $n=1$. By induction, assume that the result holds for $n=m-1$. Then, since

$$D_{\widetilde{G_k}}\left(yz\right)=(q{F}_{k}x+q{F}_{k-1}y)z-q{F}_{k-1}yz=q{F}_{k}xz,$$

(13)

we calculate

$$\begin{array}{cc}\hfill D_{\widetilde{G_k}}^m\left(xz\right)& =D_{\widetilde{G_k}}\left(D_{\widetilde{G_k}}^{m-1}\left(xz\right)\right)\hfill \\ & =D_{\widetilde{G_k}}\left(F_k^{m-1}(F_{m}xz+F_{m-1}yz)\right)\hfill \\ & =F_k^{m-1}(F_m{D}_{\widetilde{G_k}}\left(xz\right)+F_{m-1}{D}_{\widetilde{G_k}}\left(yz\right))\hfill \\ & =F_k^{m-1}(F_m{F}_k(F_{2}xz+F_{1}yz)+F_{m-1}q{F}_{k}xz)\hfill \\ & =F_k^m((F_m{F}_2+q{F}_{m-1})xz+F_m{F}_{1}yz)\hfill \\ & =F_k^m(F_{m+1}xz+F_{m}yz),\hfill \end{array}$$

where the last equation follows from Lemma 2. It shows that (i) still holds for $n=m$. By (13), we can get $D_{\widetilde{G_k}}^n\left(yz\right)=q{F}_k{D}_{\widetilde{G_k}}^{n-1}\left(xz\right)$, which yields ($ii$). Combining (i) and ($ii$), we obtain ($iii$). This completes the proof. □

Now, we use the Leibnitz formula to generate new convolution formulas.

Theorem 5. 

Let $F_n$ denote the $(p,q)$-Fibonacci sequence and let $L_n$ denote the $(p,q)$-Lucas sequence, respectively. We have

$$\sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right){(-q)}^{n-i}(b{F}_{ki}+aq{F}_{ki-1})F_{k-1}^{n-i}=F_k^n(b{F}_n+aq{F}_{n-1}).$$

Proof. 

We consider the following formula:

$$D_{\widetilde{G_k}}^n{\left((bx+ay)z\right)|}_{x=0,y=z=1}=\sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)D_{\widetilde{G_k}}^i(bx+ay){{|}_{x=0,y=z=1}{D}_{\widetilde{G_k}}^{n-i}\left(z\right)|}_{x=0,y=z=1}.$$

(14)

For the right-hand side of (14), by (11) and (12), we have

$$D_{\widetilde{G_k}}^i{(bx+ay)|}_{x=0,y=z=1}=b{F}_{ki}+aq{F}_{ki-1}$$

and

$$D_{\widetilde{G_k}}^{n-i}{\left(z\right)|}_{x=0,y=z=1}={(-q)}^{n-i}{F}_{k-1}^{n-i}.$$

For the left-hand side of (14), by Theorem 4 $\left(iii\right)$, we have

$$D_{\widetilde{G_k}}^n{\left((bx+ay)z\right)|}_{x=0,y=z=1}=F_k^n(b{F}_n+aq{F}_{n-1}).$$

Thus we obtain the desired formula. □

Corollary 8. 

Let $F_n$ denote the $(p,q)$-Fibonacci sequence and let $L_n$ denote the $(p,q)$-Lucas sequence, respectively. We have the following:

(i) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right){(-q)}^{n-i}{F}_{ki}{F}_{k-1}^{n-i}=F_k^n{F}_n}$;

(ii) 

${\displaystyle \sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right){(-q)}^{n-i}{L}_{ki}{F}_{k-1}^{n-i}=F_k^n{L}_n}$.

Proof. 

When $a=0$ and $b=1$, the formula $\left(i\right)$ is easily obtained. When $a=2$ and $b=p$, we have $b{F}_n+aq{F}_{n-1}=p{F}_n+2q{F}_{n-1}=F_{n+1}+q{F}_{n-1}=L_n$ by Equation (1) and Lemma 3, and hence we can obtain the desired formula $\left(ii\right)$. □

4. Conclusions

This paper uses context-free grammars as a new tool to investigate Horadam sequences. We have provided a unified approach to discover convolution formulas involving $H_n(A_0,A_1;p,q)$ and $H_n(B_0,B_1;p,q)$ with binomial coefficients. All convolution formulas in this paper are derived from Leibnitz’s formula, hence in the form

$$\sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)A_i{B}_{n-i}.$$

For special sequences, $H_n(0,1;p,q)$ (denoted as $F_n$ for brevity) and $H_n(2,p;p,q)$ (denoted as $L_n$ for brevity), we have deduced the explicit formulas with parameters $(p,q)$. Our results apply to parameterized sequences of various types, whereas some authors only proved them for one or a few specific sequences, including those in [3,4,5,18,30]. To the best of our knowledge (compare with the alternative known result), Theorems 1, 3 and 5 are new.

Our result highlights the capacity of grammars in exploring new formulas. Distinct grammars can be used to derive different formulas. This is demonstrated in the paper by the change in grammars from $G_k$ to $\widetilde{G_k}$, which is used to obtain the formulas in Theorem 5. Constructing new variations in the grammar could be a pathway to discovering identities for other weighted polynomial sequences. Indeed, many combinatorial polynomials can be studied by grammars, making this method an important tool in combinatorics. It may be interesting to consider multivariate Fibonacci polynomials using grammars, and some formulas could be deduced, for example, the generalized Cassini formula.