\documentclass[12pt,reqno]{article}

\usepackage[usenames]{color}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amscd}
\usepackage{graphicx}

\usepackage[colorlinks=true,
linkcolor=webgreen,
filecolor=webbrown,
citecolor=webgreen]{hyperref}

\definecolor{webgreen}{rgb}{0,.5,0}
\definecolor{webbrown}{rgb}{.6,0,0}

\usepackage{color}
\usepackage{fullpage}
\usepackage{float}

\usepackage{psfig}
\usepackage{graphics}
\usepackage{latexsym}
\usepackage{epsf}
\usepackage{breakurl}

\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{.1in}
\setlength{\evensidemargin}{.1in}
\setlength{\topmargin}{-.1in}
\setlength{\textheight}{8.4in}

\newcommand{\seqnum}[1]{\href{https://oeis.org/#1}{\rm \underline{#1}}}

\newcommand{\pf}{{\rm PF}}
\newcommand{\tp}{{\rm TP}}
\newcommand{\emd}{{\rm EMD}}
\def\R{\mathbb{R}}
\def\N{\mathbb{N}}


\begin{document}

\begin{center}
\epsfxsize=4in
\leavevmode\epsffile{logo129.eps}
\end{center}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}


\begin{center}
\vskip 1cm{\LARGE\bf 
Identities and Hadamard Product of the \\
\vskip .04in
Generalized Fibonacci, Lucas, Catalan, \\
\vskip .1in
and Harmonic Numbers 
}
\vskip 1cm
\large
Vladimir V. Kruchinin and Maria Y. Perminova\\
Tomsk State University of Control Systems and Radioelectronics\\
Tomsk \\
Russian Federation\\ 
\href{mailto:kru@ie.tusur.ru}{\tt kru@ie.tusur.ru}\\
\href{mailto:pmy@fdo.tusur.ru}{\tt pmy@fdo.tusur.ru}
\end{center}

\vskip .2 in


\begin{abstract}
We consider properties of the generalized Fibonacci and Lucas numbers. We obtain identities related to the generalized Fibonacci, Lucas, Catalan, and harmonic numbers. We also derive explicit expressions for the generating functions of the Hadamard products of the generalized Fibonacci, Lucas, Catalan, and harmonic numbers.
\end{abstract}

\section{Introduction}

The Hadamard product of generating functions is a subject of study in a variety of fields including combinatorics, probability theory, and mathematical physics. 
The Hadamard product \cite{Hadamard} of the generating functions $A(x)=\sum_{n\geqslant 0} a_nx^n$ and $B(x)=\sum_{n\geqslant 0} b_nx^n$ is a generating function defined by
$$
A(x) \ast B(x)=\sum_{n\geqslant 0} a_nb_nx^n.
$$  

It is well known \cite{Woodcock,Lando} that the Hadamard product of two rational generating functions is a rational function. However, there is no general method to obtain an explicit expression for the Hadamard product of two generating functions. Some studies have explored specific
classes of generating functions. For example, Shapiro \cite{Shapiro} provided a combinatorial proof of the Hadamard product of generating
functions for Chebyshev polynomials given by
\begin{equation}
\label{Sha1}
\frac{1}{1-ax-x^2} \ast \frac{1}{1-bx-x^2}
                =  \frac{1-x^2}{1-abx-(2+a^2+b^2)x^2
                -abx^3+x^4}.
\end{equation}

Kim \cite{Kim1,Kim2} generalized \eqref{Sha1} to the case
    $$\frac{1}{1-ax-x^2}\ast \frac{x^m}{1-bx-x^n}.$$

Kar \cite{Kar2022} considered a rational generating function as a polynomial plus a sum of power series. Thus, he reduced the problem of finding the Hadamard product of two rational generating functions for the case of the Hadamard product
$$
\frac{1}{(1-ax)^{m+1}} \ast \frac{1}{(1-bx)^{n+1}}.
$$
Kar and Gessel \cite{Gessel2024}  extended  results to find the Hadamard product 
$$
\frac{x^i}{(1-ax)^{m+1}} \ast \frac{x^j}{(1-bx)^{n+1}}.
$$

Potekhina and Tolovikov \cite{Potekhina2014,Potekhina2016} developed a method to obtain the Hadamard product for the case 
$$
\frac{1}{1-d_1x+d_2x^2+\cdots +d_nx^n}\ast\frac{x^k}{1-b_1x+b_2x^2+\cdots +b_mx^n}.
$$

It is also known that if one of the generating functions is rational and the other is algebraic or $D$-finite, then their Hadamard product is, respectively, an algebraic or $D$-finite generating function \cite{Schutzenberger,Stanley_diff}. However, even in such cases, there are no general methods for finding explicit expressions for the Hadamard product. Combinatorial techniques can sometimes be used to derive explicit expressions for the Hadamard product in certain specific cases. For example, Boyadzhiev \cite[p.~378, Equation (2.27)]{Boyadzhiev} found an explicit expression for the Hadamard product 
$$
\sum_{n>0} H_nn^mx^n,
$$
where $H_n$ are harmonic numbers. 

Chen \cite{Chen} obtained remarkable generating functions for the Hadamard product of central binomial numbers and harmonic numbers
$$
\frac{1}{1-x}\log\left(\frac{1}{1-x}\right)\ast\frac{1}{\sqrt{1-4x}}=
{{\frac{2}{{\sqrt{1-4x}}}\log \left(\frac{{\sqrt{1-4x}+1}}{2\sqrt{1-4x}}\right).
 }}
$$

Barry \cite{Barry} obtained an explicit expression for the Hadamard product of  $C_nF_{n+1}$ of the Catalan numbers and the generalized Fibonacci numbers
$$
\frac{1-\sqrt{1-4x}}{2x}\ast \frac{1}{1-ax-bx^2}=\frac{1}{x}\sqrt{\frac{1-2ax-\sqrt{1-4ax-16bx^2}}{2(a^2-4b)}}.
$$

The Hadamard product has many applications. Bragg \cite{Bragg2004} applied the Hadamard product to evaluate an extensive number of trigonometric integrals in terms of sums. Prodinger and Selkirk \cite{Prodinger2019} used generating functions and the Hadamard product for getting an explicit expression of the partial sum of Tetranacci numbers. E. D. Leinartas and E. K. Leinartas \cite{Leinartas2024} applied the Hadamard composition for solving some systems of difference
equations. Zhilinskii \cite{Zhilinskii2012} applied the symmetrized Hadamard product for construction of diagonal in polyad quantum number effective resonant vibrational Hamiltonians. Potekhina and Tolovikov \cite{Potekhina2014} used the Hadamard product for calculation of distributions of statistics relating to oscillating random walks with integer steps.

In this paper based on the identity for the generalized Fibonacci $F_n$ and Lucas $L_n$ numbers 
$$
F_{kn+2m}\sum_{i=0}^nT_{n-i,i} =\sum_{i=0}^n T_{n-i,i}L_{ki+m}F_{k(n-i)+m},
$$
where $T_{n,i}\in \mathbb{R}$ and the property $T_{n-i,i}=T_{i,n-i}$ is satisfied, we derive explicit expressions for the generating functions of the Hadamard products of the forms
$$
F(x)*H(x)=\sum_{n>0}F_nH_nx^n
$$
$$
L(x)*H(x)=\sum_{n>0}F_nH_nx^n
$$
$$
F(x)*C(x)=\sum_{n>0}F_nC_nx^n
$$
$$
L(x)*C(x)=\sum_{n\geqslant 0}L_nC_nx^n,
$$
where $H(x)$ is the generating function for harmonic numbers, 
$L(x)$ is the generating function for the generalized Lucas numbers,
and $C(x)$ is the generating function for the Catalan numbers.

\section{Identity for the generalized Fibonacci and Lucas numbers}

The Fibonacci and Lucas numbers are among the most studied objects in combinatorics. A vast number of publications and monographs are devoted to those numbers \cite{Andrica, Honsberger,Koshy}. 
Numerous generalizations of the Fibonacci and Lucas numbers have also been proposed \cite{Bicknell, Dujella, Horadam, Falcon, Edson, Kalman, Simsek}. 
Siebeck \cite{Siebeck} was one of
the first to study generalized Fibonacci numbers. He examined the recurrence
of the form 
$N_r=aN_{r-1}+cN_{r-2}$ with initial conditions $N_0=0$, $N_1=1$. He also wrote the following generating function
$$
\frac{x}{1-ax-cx^2}
$$
and the recursive formula
\begin{equation}\label{LU1}
N_{n+m}=cN_{n-1}N_m+N_{m+1}N_n.
\end{equation}

Lucas then proposed a generalization, which became known as the Lucas sequence. He studied the recurrence relation of the form
$$
X_{n+2}=PX_{n+1}-QX_n,
$$
where $P,Q$ are nonzero rational integers, and introduced two sequences $U_n(P,Q)$ with $U_0=0$ $U_1=1$ and $V_n(P,Q)$ with $V_0=2$, $V_1=P$.  He obtained the identities
\begin{equation}\label{LU2}
U_{n+m}=U_{m+1}U_n-QU_{n-1}U_m
\end{equation}
$$
2U_{n+m}=V_nU_m+U_nV_m.
$$
Here $P$ and $Q$ are present in general form. By comparing  the identities (\ref{LU1}) and (\ref{LU2}), we find
that they are completely the same.
Kalman and Mena~\cite{Kalman} made a further similar generalization and proposed a generalization for the Fibonacci numbers of the form
$$
A_{n+2}=aA_{n+1}+bA_n,
$$
where $a,b \in \mathbb{R}$. For fixed $a$ and $b$ they denoted the set of all such sequences by $\mathcal{R}(a,b)$. With initial values $A_0=0$, $A_1=1$ the resulting sequence  $(a,b)$  was referred to as the Fibonacci sequence; similarly, with $A_0=2$, $A_1=b$, they defined the Lucas sequence $(a, b)$. Then the Fibonacci sequence was written as $F_n$, and the Lucas sequence as $L_n$ when the values $a$ and $b$ were not explicitly specified $(a,b)$. They also wrote the identities for the generalized Fibonacci and Lucas numbers
$$
L_n=F_{n+1}+bF_{n-1}.
$$
In addition, they presented the following identity accomplished for the generalized Fibonacci numbers%This is Equation 19 in [Kalman Mena].
\begin{equation}\label{Equa1}
F_{n+m}=bF_{n-1}F_{m}+F_{n}F_{m+1}.
\end{equation}
This identity is completely coincides with the identities (\ref{LU1}) and (\ref{LU2}) obtained earlier. Hereafter, we adopt  the notation by Kalman and Mena~\cite{Kalman}.
The generating function for these generalized Fibonacci numbers is
$$
F(x)=\frac{x}{1-ax-bx^2},
$$
where  $a, b\in \mathbb{R}$. 

The generating function for these generalized Lucas numbers is
$$
L(x)=\frac{2-a\,x}{1-ax-bx^2},
$$
where  $a, b\in \mathbb{R}$. 

By  setting the  values $a$ and $b$, we obtain a particular sequence. For example, in the case $\mathcal{R}(1,1)$ the sequence $F_n$ corresponds to the Fibonacci numbers and $L_n$ to the Lucas numbers. Similarly, for $\mathcal{R}(2,1)$ $F_n$ represents the Pell numbers and $L_n$ the Pell-Lucas numbers.
Table~\ref{TablOeis} presents the sequences $F_n$ and $L_n$ for different values of $a$ and $b$, as they occur in the {\it On-Line Encyclopedia of
Integer Sequences} OEIS \cite{OEIS}.

\begin{table}[H]
\label{TablOeis}
\begin{center}
\begin{tabular}{llcc}\hline
$a$ & $b$ & $F_n$ & $L_n$ \\ \hline
1 &1& \seqnum{A000045} & \seqnum{A000032}\\
2 & 1& \seqnum{A000129} & \seqnum{A002203}\\
3 &1& \seqnum{A006190} & \seqnum{A006497}\\
4 &1& \seqnum{A001076} & \seqnum{A014448}\\
5 &1& \seqnum{A052918} & \seqnum{A087130}\\
6 &1& \seqnum{A005668} & \seqnum{A085447}\\
7 &1& \seqnum{A054413} & \seqnum{A086902}\\
8 &1& \seqnum{A041025} & \seqnum{A086594}\\
9 &1& \seqnum{A099371} & \seqnum{A087798}\\
1 &2& \seqnum{A001045} & \seqnum{A014551}\\
1 &3&\seqnum{A006130} & \seqnum{A075118}\\
2 &2& \seqnum{A002605} & \seqnum{A080040}\\
2 &3& \seqnum{A015518} & \seqnum{A102345}\\
3 &2& \seqnum{A007482} & \seqnum{A206776}\\
3 &3& \seqnum{A030195} & \seqnum{A172012}
\end{tabular}\caption{The generalized Fibonacci and Lucas numbers as sequences in the OEIS.}
\end{center}
\end{table}

Next, we present several identities that will be applied to obtain the generating functions in the following sections.
\begin{proposition}
The following identity for the generalized Fibonacci and Lucas numbers is true
\begin{equation}\label{Prop1}
2F_{n+m}=F_mL_n+L_mF_n.
\end{equation}
\end{proposition}

\begin{proof}
Based on (\ref{Equa1}) we obtain
$$
F_{n+m}=bF_{n-1}F_m+F_{n}F_{m+1}
$$
and
$$
F_{n+m}=bF_{n}F_{m-1}+F_{n+1}F_{m}.
$$

By adding and grouping, we obtain
$$
2F_{n+m}=bF_nF_{m-1}+F_{n+1}F_{m}+bF_{n-1}F_m+F_nF_{m+1},
$$
$$
2F_{n+m}=\bigl(bF_nF_{m-1}+F_nF_{m+1}\bigr)+\bigl(F_{n+1}F_m+bF_{n-1}F_m\bigr),
$$
$$
2F_{n+m}=F_n\bigl(bF_{m-1}+F_{m+1}\bigr)+F_{m}\bigl(F_{n+1}+bF_{n-1}\bigr).
$$

Using
$$
L_n=F_{n+1}+bF_{n-1},
$$
we obtain the sought-for formula
$$
2F_{n+m}=F_mL_n+L_mF_n.
$$
\end{proof}

As a consequence of this statement, where $n=m$, we obtain the identity
\begin{equation}\label{LF2}
F_{2\,n}=F_{n}L_n.
\end{equation}

We consider an identity that connects the sequences $F_{n}$, $L_{n}$ and  $T_{n,i}$. Let $T_{n,i} \in \mathbb{R}$, $n,i \in \mathbb{N}$ and suppose that $T_{n, i}$  satisfies the property $T_{i, n-i} = T_{n-i, i}$.
Then the sum takes the form  
$$
p_n=\sum_{i=0}^n T_{n-i,i}.
$$
Next we prove the following theorem.
\begin{theorem}
For the sequences of numbers $F_n$, $L_n$ and $p_n$, $k, m\in \mathbb{N}$ the following identity holds
\begin{equation}\label{LIden20}
p_n{F}_{kn+2m}=\sum_{i=0}^n T_{n-i,i}L_{ki+m}F_{k(n-i)+m}.
\end{equation}
\end{theorem}
\begin{proof}
We substitute $n=0$ in the obtained identity (\ref{LIden20}) to get
$$
p_0F_{0+2m}=p_0L_mF_m.
$$
Based on the identity (\ref{LF2}), we obtain the equality.
Now we consider multiple pairs of members on the right side (\ref{LIden20}) such as
$$
T_{n-i,i}=T_{i,n-i}.
$$
Then for the sum
$$
T_{i,n-i}L_{ki+m}F_{k(n-i)+m}+T_{n-i,i}L_{k(n-i)+m}F_{ki+m}
$$
there holds true
$$
T_{n-i,i}\bigl(L_{ki+m}F_{k(n-i)+m}+L_{k(n-i)+m}F_{ki+m}\bigr).
$$
Based on the identity, we write
$$
L_{ki+m}F_{k(n-i)+m}+L_{k(n-i)+m}F_{ki+m}=2F_{ki+m+k(n-i)+m}=2F_{kn+2m}.
$$
For odd values of $n$, multiple pairs of members on the right side (\ref{LIden20}) are even and the resulting identity applies to each of these pairs. 
For even values of $n$ multiple pairs of members on the right side (\ref{LIden20}) are odd. 
We consider $n=i+i$, then the sum of members takes the expression
$$
T_{i,i}L_{ki+m}F_{ki+m}
$$
Based on the identity (\ref{LF2}), we obtain
$$
L_{ki+m}F_{ki+m}=F_{2ki+2m}=F_{kn+2m}.
$$
Then the right side of the identity (\ref{LIden2}) can be written as
$$
\sum_{i=0}^n T_{n-i,i}L_{ki+m}F_{k(n-i)+m}=F_{kn+2m}\sum_{i=0}^n T_{i,n-i}.
$$
Knowing that
$$
p_n=\sum_{i=0}^n T_{n-i,i},
$$
we obtain the desired identity
$$
p_nF_{kn+2m}=\sum_{i=0}^n T_{n-i,i}L_{ki+m}F_{k(n-i)+m}.
$$
\end{proof}

Let us consider some examples of applying the identity (\ref{LIden20}). 
If we let
$$
T_{n,i}=r_{n}r_{i},
$$
where  $r_n\in \mathbb{R}$ and
$$
p_{n}=\sum_{i=0}^n r_{i}r_{n-i},
$$
then the identity (\ref{LIden20}) is
\begin{equation}\label{LIden2}
p_nF_{kn+2m}=\sum_{i=0}^n r_iL_{ki+m}r_{n-i}F_{k(n-i)+m}.
\end{equation}
If $r_n=1$, then the identity (\ref{LIden2}) is
$$
(n+1)F_{kn+2m}=\sum_{i=0}^n L_{ki+m}F_{k(n-i)+m}.
$$
For $m=n$ we obtain
$$
(n+1)F_{(k+1)n}=\sum_{i=0}^n L_{ki+n}F_{k(n-i)+n}.
$$
Let us consider the case of $r_n=F_n$. We substitute this into the identity (\ref{LIden2})
$$
p_nF_{n+2m}=\sum_{i=0}^n F_iL_{i+m}F_{n-i}F_{n-i+m}.
$$
Here
$$
p_n=\sum_{i=0}^n F_iF_{n-i}.
$$

Hoggatt \cite{HB} obtained the following explicit expression for the convolution of Fibonacci numbers $\mathcal{R}(1,1)$ 
$$
\sum_{i=0}^n F_iF_{n-i}=\frac{1}{5}\left((n-1)F_n+(n+1)F_{n-1}\right).
$$
Using his method, we can obtain the convolution formula for the generalized Fibonacci numbers $\mathcal{R}(a,b)$ 
\begin{equation}\label{LComvo}
\sum_{i=0}^n F_iF_{n-i}=\frac{(n-1)F_{n+1}+b(n+1)F_{n-1}}{a^2+4b}.
\end{equation}
When $m=0$, $k=1$, $\mathcal{R}(1,1)$, we obtain the identity
$$
\sum_{i=0}^n F_i^2L_{n-i}F_{n-i}=F_{n}\frac{(n-1)F_{n+1}+(n+1)F_{n-1}}{5}.
$$
For Pell numbers we obtain $m=0$, $k=1$, $\mathcal{R}(2,1)$
$$
\sum_{i=0}^n F_i^2L_{n-i}F_{n-i}=F_{n}\frac{(n-1)F_{n+1}+(n+1)F_{n-1}}{8}.
$$
For Jacobsthal numbers, we obtain $m=0$, $k=1$, $\mathcal{R}(1,2)$
$$
\sum_{i=0}^n F_i^2L_{n-i}F_{n-i}=F_{n}\frac{(n-1)F_{n+1}+(n+1)F_{n-1}}{9}.
$$

Note that the property $T_{i,n-i}=T_{n-i,i}$ can be replaced by the property $T_{n,i}=T_{n,n-i}$ for some triangle $T_{n,k}$ (for $n<k$ $T_{n,k}=0$), provided that the triangle is symmetrical with respect to the main diagonal. In this case, the identity (\ref{LIden20}) takes the form
$$
F_{kn+2m}\sum_{i=0}^n T_{n,i}=\sum_{i=0}^n T_{n,i}F_{ki+m}L_{k(n-i)+m}.
$$
We write the identity for Pascal's triangle
$$
2^nF_{kn+2m}=\sum_{k=0}^n \binom{n}{i}F_{ki+m}L_{k(n-i)+m}.
$$
Similarly, for the entries of Pascal's triangle (see \seqnum{A008459})
$$
\binom{2n}{n}F_{kn+2m}=\sum_{k=0}^n \binom{n}{i}^2F_{ki+m}L_{k(n-i)+m}.
$$
For Eulerian numbers $E(n,k)$ \cite[par.~6.2, pp.~267--272]{Graham} the following identity can be written 
$$
(n+1)!F_{kn+2m}=\sum_{k=0}^n E(n,k)F_{ki+m}L_{k(n-i)+m}.
$$
Leibniz's harmonic triangle \cite{Leibnic}, Lozanic's triangle
\cite{Losanitsch}, and others \cite{OEIS} have such properties.
The resulting identity (\ref{LIden20}) allows us to write a functional equation for the Hadamard product in the form
\begin{equation}\label{hadam_eq}
[F(x)*R(x)^2]=[F(x)*R(x)][L(x)*R(x)],
\end{equation}
where $F(x)$ is the generating function for the generalized Fibonacci numbers, $L(x)$ is the generating function for the generalized Lucas numbers and
$$ 
R(x)=\sum_{n=0}^{\infty} r_nx^n.
$$
Knowing two of these three functions, one can find the third function.
Further we consider obtaining the generating functions based on the equation (\ref{hadam_eq}).

\section{Hadamard product of the generalized Fibonacci and harmonic numbers}

Harmonic numbers are important in various fields of number theory and 
algorithms. The harmonic number $H_n$ is defined as
$$
H_n=\sum_{k=1}^n \frac{1}{k}.
$$
The generating function has a form
$$
H(x)=\frac{1}{1-x}\log\left(\frac{1}{1-x}\right).
$$
We know the identity the for harmonic numbers 
$$
\sum_{i=1}^{n-1} \frac{1}{i(n-i)}=\frac{2}{n}{{{H}\left(n-1\right)}}.
$$

Let us find the generating function for the Hadamard product of the form
$$
\sum_{n>0} F_nH_nx^n.
$$
To do this, we prove the following theorem.

\begin{theorem}
The Hadamard product of the generating function of the generalized Fibonacci numbers and harmonic numbers has the form
\begin{equation}\label{hadamar1}
\frac{x}{1-ax-bx^2}*\frac{\log(x-1)}{1-x}=\frac{{{\frac{(2-ax)}{{\sqrt{4b+a^2}}}\log \left(\frac{2-\sqrt{a^2+4b}x-ax}{2+\sqrt{a^2+4b}x-ax}\right)}}+x
 \log \left(1-ax-bx^2\right)}{2\left(1-ax-bx^2\right)}.
\end{equation}
\end{theorem}
\begin{proof}

We consider the case $r(n)=\frac{1}{n}$ with $r(0)=0$, $m=0$, $k=1$. Then the identity (\ref{LIden2}) is

$$
p_nF_n=\sum_{i=1}^{n-1} \frac{L_i}{i}\frac{F_{n-i}}{n-i},
$$
where
$F_n$ are the generalized Fibonacci numbers, $L_n$ are the generalized Lucas numbers.

Then
$$
p_n=\sum_{i=1}^{n-1} \frac{1}{i(n-i)}=\frac{2}{n}H_{n-1}.
$$

Next, we find the generating function $U_1(x,a,b)$ for $p(n)F(n)$. We note that
$$
p_nF_n=[x^n]U_1(x)=[x^n]U_2(x) U_3(x),
$$
where 
$$
U_2(x)=\sum_{n=1}^{\infty}\frac{F_n}{n}x^n,
$$
$$
U_3(x)=\sum_{n=1}^{\infty}\frac{L_n}{n}x^n.
$$

Then
%fixed #1 
$$U_2(x,a,b)=\int\cfrac{1}{(1-ax-bx^2)}dx=\frac{1}{{\sqrt{4b+a^2}}}{{\log \left(\frac{-\sqrt{4 b+a^2}x+a x-2}{\sqrt{4 b+a^2}x+
 ax-2}\right)}},$$
$$U_3(x,a,b)=\int\left( \frac{2-ax}{(1-ax-bx^2)x}-\frac{2}{x}\right)dx= -\log \left(1-ax-bx^2\right).
$$

Therefore the sought-for generating function is
$$
U_1(x,a,b)=\frac{1}{{\sqrt{a^2+4 b}}}{{\log\left(\frac{2-\sqrt{a^2+4b} x-a x}{2+\sqrt{a^2+4b} x-a x}\right)}}\log \left(1-a x-b x^2\right)
$$
and
$$
\frac{2}{n}H_{n-1} F_n=[x^n]U_1(x,a,b).
$$

Therefore we can write 
$$
H_{n-1}F_n=[x^n]{{{\frac{x d U_1(x,a,b)}{2 d x}}}}.
$$

Then the generating function for the Hadamard product $H_{n}F_{n+1}$ is
$$U_4(x,a,b)={\frac{(a+2bx)\log \left(\frac{2-\sqrt{a^2+4b}x-ax}{2+\sqrt{a^2+4b}x-ax}\right)+\sqrt{a^2+4 b}
 \log \left(1-ax-bx^2\right)}{2\sqrt{a^2+4b}
 \left(1-ax-bx^2\right)}}.
$$

Next, we find the generating function $U_5(x,a,b)$ for $H_nF_n$. 
We write  
$$
H_nF_n=H_{n-1}F_n+\frac{1}{n}F_n,
$$
where
$$
H_{n}F_n=[x^n]\left({{{\frac{x d U_1(x,a,b)}{2 d x}}}}+U_2(x,a,b)\right).
$$

Then the generating function for the Hadamard product $H_nF_n$ is
%fixed #2
$$U_5(x,a,b)={\frac{{{\frac{(2-ax)}{{\sqrt{4b+a^2}}}\log \left({\frac{2-\sqrt{a^2+4b}x-ax}{2+\sqrt{a^2+4b} x-a x}}\right)}}+x
 \log \left(1-ax-bx^2\right)}{2\left(1-ax-bx^2\right)}}.
$$

\end{proof}

%$${{{{\left(a x-2\right) \log \left({{-\sqrt{4 b+a^2} x+a x-2
% }\over{\sqrt{4 b+a^2} x+a x-2}}\right)}\over{\sqrt{4 b+a^2}}}+x
%  \log \left(-b x^2-a x+1\right)}\over{2 \left(b x^2+a x-1
% \right)}}$$


Next we consider some examples. If we let $a=1$, $b=1$, and $\mathcal{R}(1,1)$, then the Hadamard product for the Fibonacci numbers and the harmonic numbers is
$$H_nF_n=[x^n]U_5(x,1,1)=[x^n]\left({\frac{\sqrt{5} x \log \left(1-x-x^2\right)+(2-x) \log \left({\frac{\left(2-\sqrt{5}+1\right) x}{\left(
 2+\sqrt{5}-1\right) x}}\right)}{2\sqrt{5}(x^2+x-1)}}\right).
$$
The presented series converges at $x=\frac{1}{2}$.
Therefore it follows that
$$\sum_{n=1}^{\infty }{{\frac{H_n F_n}{2^{n}}}}={{\frac{3}{\sqrt{5}}\log \left({\frac{3+\sqrt{5}}{3-\sqrt{5}}}\right)+
\log 4}}={\frac{6\log (1+\phi)}{\sqrt{5}}}+2\log 2,
$$
where $\phi$ is the golden ratio.

If we let $a=1$, $b=2$, and $\mathcal{R}(1,2)$, then the Hadamard product for the Jacobsthal numbers $J_n$ and the harmonic numbers is
$$H_nJ_n=[x^n]U_5(x,1,2)=[x^n]\left({\frac{(2-x)\log({\frac{1+x}{1-2x}})-3x
 \log (1-x-2\,x^2)}{6(1-x-2x^2)}}\right).
$$

The presented series converges at $x=\frac{1}{3}$.
Therefore it follows that
$$\sum_{n=1}^{\infty }\frac{1}{3^n}H_nJ_n={\frac{3\log 3+2\log 2}{4}}$$.
 
The obtained generating functions $U_4(x,a,b)$ and $U_5(x,a,b)$ allow us to write the generating function for $H(n)F(n+j+1)$ with $j\ge 0$. For this purpose we use the identity (\ref{Equa1}) and obtain
$$
H(n)F(n+j+1)=F(j+1)H(n)F(n+1)+b F(j)H(n)F(n).
$$

Therefore the generating function has the following expression
\begin{equation}
U_6(x,a,b)=F(j+1)U_5(x,a,b)+b F(j)U_4(x,a,b).
\end{equation}               
                                                           
\section{Hadamard product of Fibonacci and Catalan numbers}

Catalan numbers, along with Fibonacci numbers and harmonic numbers are the subject of  extensive research by many mathematicians. For instance, Stanley \cite{Stanley} gives over 300 combinatorial objects that have been described by Catalan numbers. We now provide an explicit expression for the Catalan numbers
$$
C_n=\frac{1}{n+1}\binom{2 n}{n},
$$
a generating function 
$$
C(x)=\frac{1-\sqrt{1-4x}}{2x}.
$$
and the well-known recurrence relation
$$
C_{n+1}=\sum_{i=0}^n C_iC_{n-i},
$$
where $C_0=1$.

Let us find the Hadamard product 
$$
[C(x)*F(x)]=\sum_{n>0} C_nF_nx^n
$$

For that we consider the application of the obtained identity (\ref{LIden2}) for $r(n)=C_n$, where $k=1$ and $C_n$ are the Catalan numbers.

Then the identity is 
\begin{equation}\label{LCatId}
C_{n+1} F_{n+m}=\sum_{i=0}^n C_iL_{i+m}C_{n-i}F_{n-i+m}.
\end{equation}

Applying the result obtained by Barry and Mwafise \cite[p.~22]{Barry}
\begin{equation}\label{FBarry}
C_np_{n+1}=[x^n]V_1(x,a,b)=[x^n]\left(\frac{1}{x}\sqrt{\frac{1-2ax-\sqrt{1-4ax-16bx^2}}{(2(a^2+4b))}}\right),
\end{equation}
where
$$
p_n=[x^n]\left(\frac{1}{1-ax-bx^2}\right),
$$
we obtain the following generating function
$$
V_2(x,a,b)=\sum_{n=0}^{\infty} C_n\,L_{n+1}x^n.
$$

\begin{corollary}
The generating function for the product of the generalized Lucas and the Catalan numbers $C_n L_{n+1}$
has the following expression
\begin{equation}\label{LGF1}
V_2(x,a,b)=\frac{1}{x}\left({{1-{{\sqrt{\frac{1-2 a x+\sqrt{1-4 a x-16 b x^2}}{{2}
 }}}}}}\right).
\end{equation}
\end{corollary}

\begin{proof}
If we let 
$$
V_1(x)=\sum_{n=0} C_nF_{n+1}x^n,
$$
then we use the recurrence equation based on the identity (\ref{LCatId}) for $m=1$
$$
C_{n+1} F_{n+2}=\sum_{i=0}^n C_iL_{i+1}C_{n-i}F_{n-i+1}.
$$
Then
$$
C_nF_{n+1}=\left\{
\begin{array}{ll}
1, & \text{if $n=0$;}\\
\sum\limits_{i=0}^{n-1} C_iL_{i+1}C_{n-i-1}F_{n-i},& \text{if $n>0$.}\\
\end{array}
\right.
$$
Then we write the following functional equation related to the generating functions $V_1(x,a,b)$ and $V_2(x,a,b)$
$$
V_1(x,a,b)=1+x V_1(x,a,b) V_2(x,a,b).
$$

Therefore we find the explicit expression for the sought-for generating function
$$
V_2(x,a,b)=\frac{1}{x}\left(1-\frac{1}{V_1(x,a,b)}\right).
$$
\end{proof}

\begin{corollary}
The generating function for the product of the generalized Fibonacci and the Catalan numbers $C_nF_n$ has the following expression
\begin{equation}\label{L_V3}
V_3(x,a,b)=\frac{1}{2bx}\left({{1-\sqrt{{\frac{2 b \sqrt{-16 b x^2-4 a x+1}+4 a b x+2 b+a^2
 }{4 b+a^2}}}}}\right).
\end{equation}
\end{corollary}
 
\begin{proof}
By definition we obtain
$$
L_{n+1}=F_{n+2}+b F_{n}=a F_{n+1}+2 b F_n.
$$

Then
$$
C_nL_{n+1}=a C_nF_{n+1}+2 b F_nC_n,
$$
$$
V_2(x,a,b)=a V_1(x,a,b)+2 b V_3(x,a,b),
$$
$$
V_3(x,a,b)=\frac{1}{2 b}\left(V_2(x,a,b)-a V_1(x,a,b)\right).
$$

Knowing the generating functions for the right side terms of the expression, we obtain the sought-for expression.
\end{proof}
Let us consider some examples. If we let $a=1$, $b=1$, and $\mathcal{R}(1,1)$, then the Hadamard product for the Fibonacci and the Catalan numbers will have the expression
$$
F_nC_n=[x^n]\left(\frac{1}{2 x}\left({{1-{\frac{\sqrt{2 \sqrt{1-4 x-16 x^2}+4 x+3}}{
 \sqrt{5}}}}}\right)\right).
$$

We add this generating function to the sequence \seqnum{A119694}. The presented series converges at $x=\frac{1}{8}$. 
Therefore $$ \sum_{n=1}^{\infty} \frac{1}{2^{3n}}C_nF_n=4 \left(1-{\frac{3}{\sqrt{10}}}\right).$$

If we let $a=2$, $b=1$, and $\mathcal{R}(2,1)$, then the Hadamard product for the Pell numbers $P_n$ and the Catalan numbers will have the expression
$$
P_nC_n=[x^n]\left(\frac{1}{2 x}\left({{1-{\frac{\sqrt{2 \sqrt{1-8 x-16 x^2}+8 x+6}}{2\sqrt{2}}}}}\right)\right).
$$

We add this generating function to the sequence \seqnum{A372216}. The presented series converges at $x=\frac{1}{16}$. 
Therefore $$\sum_{n=1}^{\infty} \frac{1}{2^{4n}}C_nF_n=8-2 \sqrt{13+\sqrt{7}}.$$

Based on the identity (\ref{Equa1}) and using the generating functions $V_1(x,a,b)$ and $V_3(x,a,b)$ we obtain a generating function for the product $C_nF_{n+j+1}$, where $j\ge 0$ 
\begin{equation}
V_4(x,a,b)=F_{j+1}V_1(x,a,b)+b F_jV_3(x,a,b).
\end{equation}


\section{Acknowledgment}
We would like to thank to thank the anonymous referee for many helpful comments. This research was funded by the Russian Science Foundation, grant number
22-71-10052.

\begin{thebibliography}{99}


\bibitem{Andrica} D. B. Andrica and O. Bagdasar, {\it Recurrent Sequences: Key Results, Applications, and Problems}, Springer Nature, 2020. 

\bibitem{Barry} P. Barry and A. M. Mwafise, Classical and semi-classical orthogonal polynomials defined by Riordan arrays and their moment sequences, {\it J. Integer Sequences} {\bf 21} (2018), 
\href{https://cs.uwaterloo.ca/journals/JIS/VOL21/Barry/barry362.html}{Article 18.1.5}.

\bibitem{Bicknell} M. Bicknell-Johnson and C. P. Spears, Classes of identities for the generalized Fibonacci numbers $G_n=G_{n-1}+G_{n-c}$ for matrices with constant valued determinants, {\it Fib. Quart.} {\bf 34} (1996), 121--128.

\bibitem{Boyadzhiev} K. N. Boyadzhiev, Series transformation formulas of Euler type, Hadamard product of series, and harmonic number identities, {\it Indian J. Pure Appl. Math.} {\bf 42} (2011), 371--386. 

\bibitem{Bragg2004} L. R. Bragg, Extended Hadamard products, trigonometric integrals and associated sums, {\it Rocky Mountain J. Math.} {\bf 34} (2004), 29--41.

\bibitem{Chen} H. Chen, Interesting series associated with central binomial coefficients, Catalan numbers and harmonic numbers, {\it J. Integer Sequences}  {\bf 19} (2015), 
\href{https://cs.uwaterloo.ca/journals/JIS/VOL19/Chen/chen21.html}{Article 16.1.5}.

\bibitem{Dujella} A. Dujella, Generalized Fibonacci numbers and the problem of diophantus, {\it Fib. Quart.} {\bf 34} (1996), 164--175.

\bibitem{Edson} M. Edson and O. Yayenie, A new generalization of Fibonacci sequences and extended Binet's formula, {\it Integers} {\bf(9)} (2009), 639--654.

\bibitem{Falcon}  S. Falc\'on and A. Plaza, On the Fibonacci $k$-numbers, {\it Chaos, Solitons, Fractals} {\bf 32} (2007), 1615--1624. 

\bibitem{Gessel2024} I. M. Gessel and I. Kar, Binomial convolutions for rational power series, {\it J. Integer Sequences}  {\bf 27} (2024), 
\href{https://cs.uwaterloo.ca/journals/JIS/VOL27/Gessel/gessel7.html}{Article 24.1.3}.

\bibitem{Graham}  R. L. Graham, D. E. Knuth, and O. Patashnik, {\it Concrete Mathematics: A Foundation for Computer Science}, Addison-Wesley, 1994.

\bibitem{Hadamard} J. Hadamard, Th\'eor\`eme sur les s\'eries enti\`eres, {\it Acta Math.} {\bf 22} (1899), 55--63.

\bibitem{HB} V. E. Hoggatt, Jr. and M. Bicknell-Johnson, Fibonacci convolution sequences, {\it Fib. Quart.} {\bf 15} (1977), 117--122.

\bibitem{Honsberger} R. A. Honsberger, {\it Second Look at the Fibonacci and Lucas Numbers}, Ch. 8 in Mathematical Gems III, Math. Assoc. America, 1985.

\bibitem{Horadam} A. F. Horadam, Generating functions for powers of a certain generalized sequence of numbers, {\it Duke Math. J.} {\bf 32} (1965), 437--446.

\bibitem{Kalman} D. Kalman and R. Mena, The Fibonacci numbers-exposed, {\it Math. Mag.} {\bf 76} (2003), 167--181.

\bibitem{Kar2022} I. Kar, A new method to compute the Hadamard product of two rational functions, {\it Rose-Hulman Undergrad. Math J.} {\bf 23} (2022), Article 3.

\bibitem{Kim1} J. H. Kim, Hadamard products and tilings {\it J. Integer
Sequences} {\bf 12} (2009), 
\href{https://cs.uwaterloo.ca/journals/JIS/VOL12/Kim/kim18.html}{Article 09.7.4}.

\bibitem{Kim2} J. H. Kim, {\it Hadamard Products, Lattice Paths, and Skew Tableaux}, Brandeis Univ., 2011.

\bibitem{Koshy} T. Koshy, {\it Fibonacci and Lucas Numbers with Applications}, Wiley, 2001. 

\bibitem{Lando} S. K. Lando, {\it Lectures on Generating Functions}, 
American Mathematical Society, 2003. 

\bibitem{Leibnic} Y. Simsek, Construction of a generalization of the Leibnitz numbers and their properties, {\it Adv. Stud. Contemp. Math.} {\bf 31} (2021), 311--323.

\bibitem{Leinartas2024} E. D. Leinartas and E. K. Leinartas, Difference equations and Hadamard composition of multiple power series, {\it J. Sib. Fed. Univ. Math. Phys.} {\bf 17} (2024),  169--174.

\bibitem{Losanitsch} S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, {\it Chem. Ber.} {\bf 30} (1897), 1917--1926.

\bibitem{Potekhina2014} E. A. Potekhina and M. I. Tolovikov, Oscillating random walk and the Hadamard product of rational
functions, {\it Discrete Math. Appl.} {\bf 24} (2014), 29--44.

\bibitem{Potekhina2016} E. A. Potekhina, Application of Hadamard product to some combinatorial and probabilistic problems, {\it Diskr. Mat.} {\bf 28} (2016), 101--112.

\bibitem{Prodinger2019} H. Prodinger and S. J. Selkirk, Sums of squares of Tetranacci numbers: a generating
function approach, {\it Fibonacci Quart.} {\bf 57} (2019), 313--317.

\bibitem{Schutzenberger} M. P. Sch\"utzenberger, On a theorem of R. Jungen, {\it Proc. Amer. Math. Soc.} {\bf 13} (1962), 885--890. 

\bibitem{Shapiro} L. Shapiro, A combinatorial proof of a Chebyshev polynomial identity, {\it Diskr. Mat.} {\bf 34} (1981), 203--206.

\bibitem{Siebeck} H. Siebeck, Die recurrenten Reihen, vom Standpuncte
der Zahlentheorie aus betrachtet, {\it J. Reine Angew. Math}  {\bf 33} (1846),
71--77.
 
\bibitem{Simsek} Y. Simsek, Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials, {\it Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.} {\bf 117} (2023), Article 130.

\bibitem{OEIS} OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences. Published electronically at \url{https://oeis.org}, 2024.

\bibitem{Stanley} R. Stanley, {\it Catalan Numbers}, Cambridge Univ. Press, 2015. 

\bibitem{Stanley_diff} R. P. Stanley, Differentiably finite power series, {\it Eur. J. Comb.} {\bf 1} (1980), 175--188.

\bibitem{Woodcock} C. F. Woodcock and H. Sharif, Hadamard products of rational formal power series, {\it J. Algebra} {\bf 128} (1990), 517--527.

\bibitem{Zhilinskii2012} B. I. Zhilinskii, Generating functions for effective hamiltonians. Symmetry, topology, combinatorics, {\it Phys. Atom. Nuclei} {\bf 75} (2012), 109--117.


\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2024 {\it Mathematics Subject Classification}:
Primary 05A15; Secondary 11B39, 05A10, 11Y55.

\noindent \emph{Keywords: } 
generating function, Fibonacci number, Lucas number, Pell number, Catalan number, harmonic number, Pascal's  triangle.

\bigskip
\hrule
\bigskip
\noindent (Concerned with sequences
\seqnum{A000032},
\seqnum{A000045},
\seqnum{A000129},
\seqnum{A001045},
\seqnum{A001076},
\seqnum{A002203},
\seqnum{A002605},
\seqnum{A005668},
\seqnum{A006130},
\seqnum{A006190},
\seqnum{A006497},
\seqnum{A007482},
\seqnum{A008459},
\seqnum{A014448},
\seqnum{A014551},
\seqnum{A015518},
\seqnum{A030195},
\seqnum{A041025},
\seqnum{A052918},
\seqnum{A054413},
\seqnum{A075118},
\seqnum{A080040},
\seqnum{A085447},
\seqnum{A086594},
\seqnum{A086902},
\seqnum{A087130},
\seqnum{A087798},
\seqnum{A099371},
\seqnum{A102345},
\seqnum{A119694},
\seqnum{A172012},
\seqnum{A206776}, and
\seqnum{A372216}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received June 5 2024;
revised versions received June 10 2024; April 30 2025.
Published in {\it Journal of Integer Sequences}, December 15 2025.

\bigskip
\hrule
\bigskip

\noindent
Return to \href{https://cs.uwaterloo.ca/journals/JIS/}{Journal of Integer Sequences home page}.
\vskip .1in

\end{document}

                                                                                

                                                                                

