<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[12pt,reqno]{article}

\usepackage[usenames]{color}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amscd}

\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,amsmath,amssymb}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{latexsym}
\usepackage{epsf}

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

\newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}}
\newcommand{\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 On Two New Classes of $B$-$q$-bonacci
\vskip .1in
Polynomials}
\vskip 1cm
\large  Suchita Arolkar\\
Department of Mathematics and Statistics\\
Dnyanprassarak Mandal's College and Research Centre\\
Assagao Bardez Goa\\
403 507, India\\
\href{mailto:Suchita.golatkar@yahoo.com}{\tt Suchita.golatkar@yahoo.com}\\
\ \\
Yeshwant Shivrai Valaulikar\\
Department of Mathematics \\
Goa University \\
Taleigao Plateau Goa\\
403 206, India\\
\href{mailto:ysv@unigoa.ac.in}{\tt ysv@unigoa.ac.in }
\end{center}

\vskip .2 in

\begin{abstract}
In this paper we define two new classes of polynomials associated with generalized Fibonacci polynomials.  We call them $h(x)$-$B$-$q$-bonacci polynomials and incomplete $h(x)$-$B$-$q$-bonacci polynomials. We present some 
identities for the two classes of polynomials, and the
convolution property of $h(x)$-$B$-$q $-bonacci polynomials
and its applications.
\end{abstract}

\section{Introduction}
The Fibonacci sequence, polynomials associated with the Fibonacci sequence, and their extended forms produce interesting and fascinating properties. For details see \cite{Thomas, Vajda}. Arolkar and Valaulikar introduced the $B$-tribonacci sequence \cite{Sa1} and $B$-tribonacci polynomials \cite{Sa2}. The $B$-tribonacci sequence \cite{Sa1} and $B$-tribonacci polynomials \cite{Sa2} are further extended to $q^{th}$ order recurrence relations in \cite{Sa5} and \cite{Sa6} respectively. Arolkar and Valaulikar extended and studied the $h(x)$-Fibonacci polynomials \cite{Nalli} to $h(x)$-$B$-tribonacci polynomials \cite{Sa4}.  Filipponi \cite{Filipponi} introduced the incomplete Fibonacci and Lucas numbers. Ram\'{i}rez \cite{Jose} studied various identities related to the incomplete $k$-Fibonacci and $k$-Lucas numbers.  Ram\'{i}rez \cite{Jose1} introduced interesting classes of polynomials, namely, the incomplete $h(x)$-Fibonacci and $h(x)$-Lucas polynomials. Arolkar and Valaulikar \cite{Sa3} extended the incomplete $h(x)$-Fibonacci and $h(x)$-Lucas polynomials. Ram\'{i}rez and Sirvent \cite{victor} defined and studied identities related to the incomplete tribonacci numbers and polynomials. Yilmaz and Taskara \cite{Nazmiye} obtained identities for the incomplete tribonacci-Lucas numbers and polynomials.

The aim of this paper is to extend two classes of polynomials, namely, the $h(x)$-$B$-tribonacci polynomials \cite{Sa4} and incomplete $h(x)$-$B$-tribonacci polynomials of \cite{Sa3} to the $q^{th}$ order relations. We call them the $h(x)$-$B$-$q$-bonacci polynomials and incomplete $h(x)$-$B$-$q$-bonacci polynomials. We study some properties of these polynomials.

\section{$h(x)$-$B$-$q$-bonacci polynomials}\label{sec:2}
We first define the class of $h(x)$-$B$-$q$-bonacci polynomials.
\begin{definition}
Let $h(x)$ be a polynomial with real coefficients. The $h(x)$-$B$-$q$-bonacci polynomials, denoted by ${({}^qB)_{h,n}}(x), n \in {\N} \cup \{0\}, q \geq 2 $,
are defined by
\begin{equation}\label{eq:1.1}
({}^qB)_{h,n+q-1}(x)=
\sum_{r=0}^{q-1}\frac{(q-1)^{\underline{r}}}{r!}\ \ h^{q-1-r}(x) ({}^qB)_{h,n+q-2-r}(x),\forall n \geq 1,
\end{equation}
with $({}^qB)_{h, i}(x) = 0,\  i = 0,1,2,3,\ldots,q-2$ and $({}^qB)_{h,q-1}(x) = 1$,
where the coefficients of the terms on the right-hand side are the terms of the binomial expansion of $(h(x)+1)^{q-1}$ and $({}^qB)_{h,n}(x)$ is the $n^{th}$ polynomial.
\end{definition}
For simplicity, henceforth we denote
$({}^qB)_{h,n}(x)$ by $({}^qB)_{h,n}$ and $h(x)$ by $h$.
We have the following identities for $({}^qB)_{h,n}$.
\begin{enumerate}
\item [(1)] The $n^{th}$ term $({}^qB)_{h,n}$ of (\ref{eq:1.1}) is given by
\begin{equation}\label{eq:2.1}
({}^qB)_{h,n} =
\sum_{r=0}^{\left \lfloor \frac{(q-1)\left(n-(q-1)\right)}{q}\right \rfloor} \frac{\left((q-1)\left(n-(q-1)-r\right)\right)^{\underline{r}}}{r!}\ \ h^{(q-1)(n-(q-1)-r)-r},
\end{equation}
$n \geq q-1,$ where $\lfloor\cdot\rfloor$ denotes the floor function.
\begin{proof}
We prove the identity using induction on $n$.

For $n = q-1$, $(\ref{eq:2.1})$ implies
\begin{align*}
({}^qB)_{h,q-1} =
\sum_{r=0}^{0} \frac{\left((q-1)\left(-r\right)\right)^{\underline{r}}}{r!}\ \ h^{(q-1)(-r)-r} =1.
\end{align*}
Hence  $(\ref{eq:2.1})$ is true for $n=q-1$.
Assume that $(\ref{eq:2.1})$ is true for $n \leq m$.
We divide the result into $q$ cases, namely, 
$m = qk,  qk+1,  qk+2, \ldots , qk+(q-1)$, for some $k \geq 1$.

\noindent Case (i):   Let $m=qk$ and $t=\left\lfloor\frac{(q-1)\left(qk-s-(q-1)\right)}{q}\right \rfloor$. Then
\begin{align*}
&amp;\sum_{s=0}^{q-1}\frac{(q-1)^{\underline{s}}}{s!}\ h^{q-1-s} ({}^qB)_{h,qk-s}\\
&amp;=\sum_{s=0}^{q-1}\frac{(q-1)^{\underline{s}}}{s!}\sum_{r=0}^{t} \frac{\left((q-1)\left(qk-(q-1)-(r+s)\right)\right)^{\underline{r}}}{r!} \ h^{(q-1)(qk+1-(q-1)-(r+s))-(r+s)}\\
&amp;=
\sum_{s=0}^{q-1}\frac{(q-1)^{\underline{s}}}{s!}\;\sum_{p=s}^{(q-1)k-(q-2) } \frac{\left((q-1)\left(qk-(q-1)-p\right)\right)^{\underline{p-s}}}{(p-s)!} \ h^{(q-1)(qk+1-(q-1)-p)-p}\\
&amp;=\Big( \frac{(q-1)^{\underline{0}}}{0!}
\sum_{p=0}^{(q-1)k-(q-2)} \frac{\left((q-1)\left(qk-(q-1)-p\right)\right)^{\underline{p}}}{p!}\\
&amp;+ \frac{(q-1)^{\underline{1}}}{1!} \sum_{p=1}^{(q-1)k-(q-2)} \frac{\left((q-1)\left(qk-(q-1)-p\right)\right)^{\underline{p-1}}}{(p-1)!}
+\cdots\\
&amp;+\frac{(q-1)^{\underline{(q-1)}}}{(q-1)!} \sum_{p=q-1}^{(q-1)k-(q-2)} \frac{\left((q-1)\left(qk-(q-1)-p\right)\right)^{\underline{p-(q-1)}}}{(p-(q-1))!}\Big)\ h^{(q-1)(qk+1-(q-1)-p)-p}\\
&amp;=\Bigg(\frac{\left((q-1)\left(qk-(q-1)\right)\right)^{\underline{0}}}{0!}\\
&amp;+ \left(\frac{\left((q-1)\left(qk-(q-1)-1\right)\right)^{\underline{1}}}{1!}+ \frac{(q-1)^{\underline{1}}}{1!} \frac{\left((q-1)\left(qk-(q-1)-1\right)\right)^{\underline{0}}}{0!}\right)+\cdots\\
&amp;+ \sum_{p= q-1}^{(q-1)k-(q-2) } \sum_{s=0}^{q-1}\frac{(q-1)^{\underline{s}}}{s!} \frac{\left((q-1)\left(qk-(q-1)-p\right)\right)^{\underline{p-s}}}{(p-s)!}\Bigg)\ h^{(q-1)(qk+1-(q-1)-p)-p}.
\end{align*}
Therefore, using $\sum_{s=0}^{q-1}\frac{(q-1)^{\underline{s}}}{s!} \frac{n^{\underline{p-s}}}{(p-s)!} = \frac{(n+(q-1))^{\underline{p}}}{p!},$ we have
\begin{align*}
&amp;\sum_{s=0}^{q-1}\frac{(q-1)^{\underline{s}}}{s!}\;h^{q-1-s} ({}^qB)_{h,qk-s}\\
&amp;=\sum_{p= 0}^{(q-1)k-(q-2) } \frac{\left((q-1)\left(qk+1-(q-1)-p\right)\right)^{\underline{p}}}{p!}\ h^{(q-1)(qk+1-(q-1)-p)-p}\\
&amp;=({}^qB)_{h,qk+1}.
\end{align*}
Thus, assuming the result for $m= qk $, we have proved it for $ m = qk + 1 $.
Similarly, we can prove the other cases.

We conclude that
$\sum_{s=0}^{q-1}\frac{(q-1)^{\underline{s}}}{s!}\;h^{q-1-s} ({}^qB)_{h,m-s}
=({}^qB)_{h,m+1}$.

Hence, by induction, the result follows.
\end{proof}

\item [(2)] The sum of the first $n+1$ terms of (\ref{eq:1.1}) is given by
\begin{equation}\label{eq:2.2}
\sum_{r=0}^{n} ({}^qB)_{h,r} = \frac {({}^qB)_{h,n+1} +\sum_{i=0}^{q-2}\sum_{r=1+i}^{q-1} \frac{(q-1)^{\underline{r}}}{r!}\; h^{q-1-r}\ \ ({}^qB)_{h,n-i}  -1}{(h+1)^{q-1}-1},\ \
\end{equation}
provided
$
\begin{cases}
h \neq 0, &amp; \text{if $q$ is even;}\\
h \neq 0,-2, &amp; \text{if $q$ is odd.}
\end{cases}
$
\begin{proof}
We obtain the result by induction on $n$.
For $n=q-1,$ $\sum_{r=0}^{q-1} ({}^qB)_{h,r} = ({}^qB)_{h,q-1} =1$.
Also,
\begin{align*}
&amp;\frac {({}^qB)_{h,q} +\sum_{i=0}^{q-2}\sum_{r=1+i}^{q-1} \frac{(q-1)^{\underline{r}}}{r!}\ h^{q-1-r}\ \ ({}^qB)_{h,q-1-i}  -1}{(h+1)^{q-1}-1}\\
&amp; = \frac {h^{q-1} + \sum_{r=1}^{q-1} \frac{(q-1)^{\underline{r}}}{r!}\ h^{q-1-r}\;({}^qB)_{h,q-1}  -1}{(h+1)^{q-1}-1}=1.
\end{align*}
Thus, (\ref{eq:2.2}) is true for $n=q-1$.

Assume that (\ref{eq:2.2}) is true for $n\leq m$. Then
\begin{align*}
&amp;\sum_{r=0}^{m+1} ({}^qB)_{h,r} = \sum_{r=0}^{m} ({}^qB)_{h,r} + ({}^qB)_{h,m+1} \\
&amp;=\frac {({}^qB)_{h,m+1} +\sum_{i=0}^{q-2}\sum_{r=1+i}^{q-1} \frac{(q-1)^{\underline{r}}}{r!}\ h^{q-1-r}\ ({}^qB)_{h,m-i}  -1}{(h+1)^{q-1}-1} + ({}^qB)_{h,m+1}\\
&amp;=\frac {({}^qB)_{h,m+2} +\sum_{i=0}^{q-2}\sum_{r=1+i}^{q-1} \frac{(q-1)^{\underline{r}}}{r!}\ h^{q-1-r}\ \ ({}^qB)_{h,m+1-i}  -1}{(h+1)^{q-1}-1}.
\end{align*}
Thus, the result is true for $ n= m+1$.
Hence, by induction, the result follows.
\end{proof}
\item [(3)] The generating  function  for (\ref{eq:1.1}) is given by
\begin{equation}\label{eq:2.3}
 ({}^qG_{(B)})_{h}(z) = \frac{1}{1-z(h+z)^{q-1}},
 \end{equation}
 provided $|(z(h+z)^{q-1})|&lt;1$.

 \begin{proof} Let $t = \left \lfloor\frac{(q-1)\left(n-(q-1)\right)}{q}\right \rfloor$.  Then
\begin{align*}
({}^qG_{(B)})_{h}(z)&amp;= \sum_{n=0}^{\infty}({}^qB)_{h,n}\ z^{n-(q-1)}\\
&amp;=\sum_{n=0}^{q-2}({}^qB)_{h,n}\ z^{n-(q-1)}+\sum_{n=q-1}^{\infty}({}^qB)_{h,n}\ z^{n-(q-1)}\\
&amp;=\sum_{n=q-1}^{\infty}({}^qB)_{h,n}\;z^{n-(q-1)},\ \text{since}\ ({}^qB)_{h, i} = 0,\ i = 0,1,2,3,\ldots,q-2\\
&amp;=\sum_{n=q-1}^{\infty} \sum_{r=0}^{t} \frac{\left((q-1)\left(n-(q-1)-r\right)\right)^{\underline{r}}}{r!}\ \ h^{(q-1)(n-(q-1)-r)-r}\ z^{n-(q-1)}\\
&amp;=1+ h^{q-1} z+ \left(h^{2(q-1)}+ \frac{(q-1)^{\underline{1}}}{1!} h^{q-2}\right) z^{2}+\dots\\
&amp;=1+z(h+z)^{q-1}+z^{2}(h+z)^{2(q-1)}+\dots\\
&amp;= \frac{1}{1-z(h+z)^{q-1}},\ \text{provided}\ |(z(h+z)^{q-1})|&lt;1.
\end{align*}
\end{proof}
\end{enumerate}
We now obtain the following property.
\begin{theorem}\textbf{$($Convolution property for  $({}^qB)_{h,n}$$)$}

For all $n \geq q-1,$ we have
\begin{equation}\label{eq:2.5}
\frac{d}{dx}\left(({}^qB)_{h,n}\right)= (q-1) \frac{dh}{dx}\left( \sum_{r=0}^{q-2} \frac{(q-2)^{\underline{r}}}{r!}\ h^{(q-2)-r}\ \sum_{i=0}^{n+q-2-r}({}^qB)_{h,i}\ ({}^qB)_{h,n+q-2-r-i}\right).
\end{equation}
\end{theorem}
\begin{proof}
Equation (\ref{eq:2.3}) implies
\begin{align*}
\sum_{n=0}^{\infty}({}^qB)_{h,n}\ z^{n-(q-1)} = \frac{1}{1-z(h+z)^{q-1}}.
\end{align*}
Differentiating both sides with respect to $x,$ we get
\begin{align*}
&amp;\sum_{n=0}^{\infty}\frac{d}{dx}\left(({}^qB)_{h,n}\right)z^{n-(q-1)}
=z(q-1)(h+z)^{q-2}\ \frac{1}{\left(1-z(h+z)^{q-1}\right)^{2}}\ \frac{dh}{dx}\\
&amp;=\left((q-1)\ \sum_{r=0}^{q-2} \frac{(q-2)^{\underline{r}}}{r!}\ h^{(q-2)-r}z^{r+1}\ \left(\sum_{n=0}^{\infty}({}^qB)_{h,n}\ z^{n-(q-1)}\right)^{2}\right)\ \frac{dh}{dx}\\
&amp;= \left((q-1)\ \sum_{r=0}^{q-2} \frac{(q-2)^{\underline{r}}}{r!}\ h^{(q-2)-r}\ z^{-2(q-1)+r+1}\ \left( \sum_{n=0}^{\infty}({}^qB)_{h,n}\ z^{n}\right)^{2}\right)\ \frac{dh}{dx}\\
&amp;= (q-1)\frac{dh}{dx}\ \sum_{r=0}^{q-2} \frac{(q-2)^{\underline{r}}}{r!}\ h^{(q-2)-r}\ \sum_{n=0}^{\infty}\left( \sum_{i=0}^{n}({}^qB)_{h,i}({}^qB)_{h,n-i}\ z^{n-2(q-1)+r+1}\right).
\end{align*}
Comparing the coefficients of $z^{n-(q-1)},$ we get
\begin{eqnarray*}
\frac{d}{dx}\left(({}^qB)_{h,n}\right) = (q-1)\frac{dh}{dx} \left(\;\sum_{r=0}^{q-2} \frac{(q-2)^{\underline{r}}}{r!}\; h^{(q-2)-r}\; \sum_{i=0}^{n+q-2-r}({}^qB)_{h,i}({}^qB)_{h,n+q-2-r-i}\right).
\end{eqnarray*}
\end{proof}
We now give an application of the convolution property.  It is required to prove an identity in the next section.
\begin{theorem}\label{thm:3}
For $n\geq q-1$,
\begin{align*}
\sum_{r=0}^{\left \lfloor\frac{(q-1)(n-(q-1))}{q}\right \rfloor} r\;\frac{\left((q-1)(n-(q-1)-r)\right)^{\underline{r}}}{r!}\;\;h^{(q-1)(n-(q-1))-qr}
\end{align*}
\begin{align*}
=\frac{(q-1)\left(n-(q-1)\right)}{q}\ ({}^qB)_{h,n}
\end{align*}
\begin{equation}\label{eq:2.6}
-\frac{h}{q}\ (q-1) \left(\sum_{r=0}^{q-2} \frac{(q-2)^{\underline{r}}}{r!}\ h^{(q-2)-r}\ \sum_{i=0}^{n+q-2-r}({}^qB)_{h,i}\ ({}^qB)_{h,n+q-2-r-i}\right).
\end{equation}
\end{theorem}
\begin{proof} Equation (\ref{eq:2.1}) implies
\begin{align*}
({}^qB)_{h,n} =
\sum_{r=0}^{\left \lfloor\frac{(q-1)(n-(q-1))}{q} \right \rfloor} \frac{\left((q-1)(n-(q-1)-r)\right)^{\underline{r}}}{r!}\ h^{(q-1)(n-(q-1))-qr}
.
\end{align*}
Differentiating both sides with respect to $x$ and simplifying, we get
\begin{align*}
&amp;\frac{d}{dx}\big(({}^qB)_{h,n}\big)h =  ((q-1)(n-(q-1)))\ ({}^qB)_{h,n}\ \frac{dh}{dx}\\
&amp;-q \ \frac{dh}{dx}\sum_{r=0}^{\left \lfloor\frac{(q-1)(n-(q-1))}{q}\right \rfloor}r\ \frac{((q-1)(n-(q-1)-r))^{\underline{r}}}{r!}\ h^{(q-1)(n-(q-1))-qr}.
\end{align*}
Thus,
\begin{align*}
&amp;\frac{dh}{dx}\ \sum_{r=0}^{\left \lfloor\frac{(q-1)(n-(q-1))}{q}\right \rfloor} r\ \frac{((q-1)(n-(q-1)-r))^{\underline{r}}}{r!}\ h^{(q-1)(n-(q-1))-qr}\\
&amp;= \frac{(q-1)(n-(q-1))}{q}\ ({}^qB)_{h,n}\ \frac{dh}{dx}-\frac{h}{q}\ \frac{d}{dx}\left(({}^qB)_{h,n}\right).
\end{align*}
Hence (\ref{eq:2.5}) implies
\begin{align*}
&amp;\sum_{r=0}^{\left \lfloor\frac{(q-1)(n-(q-1))}{q}\right \rfloor} r \ \frac{((q-1)(n-(q-1)-r))^{\underline{r}}}{r!}\ \ h^{(q-1)(n-(q-1))-qr}\\
&amp;= \frac{(q-1)(n-(q-1))}{q}\ ({}^qB)_{h,n}\\
&amp;-\frac{h}{q}\ (q-1) \left(\ \sum_{r=0}^{q-2} \frac{(q-2)^{\underline{r}}}{r!}\ h^{(q-2)-r}\ \sum_{i=0}^{n+q-2-r}({}^qB)_{h,i}\ ({}^qB)_{h,n+q-2-r-i}\right).
\end{align*}
\end{proof}

\section{Incomplete $h(x)$-$B$-$q$-bonacci Polynomials}
In this section we define the class of incomplete $h(x)$-$B$-$q$-bonacci polynomials and discuss some of its properties.

\begin{definition}
The incomplete $h(x)$-$B$-$q$-bonacci polynomials are defined by
\begin{equation}\label{eq:3.1}
({}^qB)^{l}_{h,n}(x) =
\sum_{r=0}^{l} \frac{\left((q-1)(n-(q-1)-r)\right)^{\underline{r}}}{r!}\ \ h^{(q-1)(n-(q-1)-r)-r}(x),\\
\end{equation}
\begin{align*}
\forall \ 0 \leq l \leq \left \lfloor\frac{(q-1)(n-(q-1))}{q}\right \rfloor \ \text{and}\ n \geq q-1.\\
\end{align*}
\end{definition}

Note that
$({}^qB)^{\left \lfloor\frac{(q-1)(n-(q-1))}{q}\right \rfloor}_{h,n} (x)= ({}^qB)_{h,n}(x)$.

For simplicity, we use
$({}^qB)^{l}_{h,n}(x) = ({}^qB)^{l}_{h,n}, ({}^qB)_{h,n}(x) = ({}^qB)_{h,n}$ and $h(x)=h$. 
We prove identities related to the recurrence relation for $({}^qB)^{l}_{h,n}$.
\begin{theorem}
The recurrence relation for $({}^qB)^{l}_{h,n}$ is given by
\begin{equation}\label{eq:3.2}
 ({}^qB)^{l+q-1}_{h,n+q} = \sum_{r=0}^{q-1}\frac{(q-1)^{\underline{r}}}{r!}\;\;h^{q-1-r} ({}^qB)^{l+q-1-r}_{h,n+q-1-r},\; 0 \leq l\leq \left \lfloor \frac{(q-1)(n-q)}{q} \right \rfloor, \forall n \geq q.
\end{equation}
\end{theorem}
\begin{proof}
Consider,
\begin{align*}
&amp;\sum_{r=0}^{q-1}\frac{(q-1)^{\underline{r}}}{r!}\;h^{q-1-r}\; ({}^qB)^{l+q-1-r}_{h,n+q-1-r}\\
&amp;=\sum_{r=0}^{q-1}\frac{(q-1)^{\underline{r}}}{r!}\;h^{q-1-r}\\
&amp;\sum_{i=0}^{l+q-1-r} \frac{\left((q-1)(n+q-1-r-(q-1)-i)\right)^{\underline{i}}}{i!}\;\;h^{(q-1)(n+q-1-r-(q-1)-i)-i}\\
&amp;=\sum_{r=0}^{q-1}\frac{(q-1)^{\underline{r}}}{r!}\;h^{q-1-r}\;\sum_{i=0}^{l+q-1-r} \frac{\left((q-1)(n-r-i)\right)^{\underline{i}}}{i!}\ h^{(q-1)(n-r)-qi}\\
&amp;=\sum_{r=0}^{q-1}\frac{(q-1)^{\underline{r}}}{r!}\ \sum_{i=0}^{l+q-1-r} \frac{\left((q-1)(n-r-i)\right)^{\underline{i}}}{i!}\ h^{(q-1)(n+1)-qr-qi}\\
&amp;=\sum_{r=0}^{q-1}\frac{(q-1)^{\underline{r}}}{r!}\ \sum_{i=0}^{l+q-1-r} \frac{\left((q-1)(n-(r+i))\right)^{\underline{i}}}{i!}\ h^{(q-1)(n+1)-q(r+i)}.
\end{align*}
Taking $j = i+r,$ we get
\begin{align*}
&amp;\sum_{r=0}^{q-1}\frac{(q-1)^{\underline{r}}}{r!}\ ({}^qB)^{l+q-1-r}_{h,n+q-1-r}\ h^{q-1-r}\\
&amp;=\sum_{r=0}^{q-1}\frac{(q-1)^{\underline{r}}}{r!}\ \sum_{j=r}^{l+q-1} \frac{\left((q-1)(n-j)\right)^{\underline{j-r}}}{(j-r)!}\ h^{(q-1)(n+1)-qj}\\
&amp;=\sum_{j=0}^{l+q-1} \frac{\left((q-1)(n+1-j)\right)^{\underline{j}}}{j!}\ h^{(q-1)(n+1)-qj}\\
&amp;=({}^qB)^{l+q-1}_{h,n+q}.
\end{align*}
\end{proof}
\begin{theorem} For $s \geq 1,$
\begin{equation}\label{eq:3.3}
({}^qB)_{h,n+qs}^{l+(q-1)s}=\sum_{i=0}^{(q-1)s}\frac{((q-1)s)^{\underline{i}}}{i!}\; ({}^qB)_{h,n+i}^{l+i}\;h^{i},
\end{equation}
$0 \leq l\leq \left \lfloor \frac{(q-1)(n-s-(q-1))}{q} \right \rfloor.$
\end{theorem}
\begin{proof}
Follows using induction.
\end{proof}
\begin{theorem}
For $n\geq \left \lfloor \frac{ql+2(q-1)}{q-1} \right \rfloor,$
$({}^qB)_{h,n+(q-1)+s}^{l+(q-1)}-h^{(q-1)s}({}^qB)_{h,n+q-1}^{l+q-1}$
\begin{equation}\label{eq:3.4}
=\sum_{i=0}^{s-1}\sum_{r=1}^{q-1} \frac{(q-1)^{\underline{r}}}{r!} h^{(q-1)s-(q-1)i-r}\ ({}^qB)_{h,n+(q-1)+i-r}^{l+(q-1)-r}.
\end{equation}
\end{theorem}
\begin{proof}
Follows using induction.
\end{proof}
The next theorem is related to the sum of incomplete $h(x)$-$B$-$q$-bonacci polynomials $({}^qB)^{l}_{h,n}.$
\begin{theorem} For all $n \geq q-1,$
\begin{align*}
\sum_{l=0}^{\left \lfloor \frac{(q-1)(n-(q-1))}{q} \right \rfloor} ({}^qB)^{l}_{h,n}=\left(\left \lfloor \frac{(q-1)\left(n-(q-1)\right)}{q}\right \rfloor+\frac{q-(q-1)\left(n-(q-1)\right)}{q}\right)({}^qB)_{h,n}
\end{align*}
\begin{equation}\label{eq:5.65}
+\frac{h}{q}\ (q-1) \left(\sum_{r=0}^{q-2} \frac{(q-2)^{\underline{r}}}{r!}\ h^{(q-2)-r}\ \sum_{i=0}^{n+q-2-r}({}^qB)_{h,i}\ ({}^qB)_{h,n+q-2-r-i}\right).
\end{equation}
\end{theorem}
\begin{proof}
$\sum_{l=0}^{\left \lfloor \frac{(q-1)(n-(q-1))}{q} \right \rfloor} ({}^qB)^{l}_{h,n}$
\begin{align*}
&amp;= ({}^qB)^{0}_{h,n}+({}^qB)^{1}_{h,n}+\cdots+({}^qB)^{r}_{h,n}+\cdots+({}^qB)^{\left \lfloor \frac{(q-1)(n-(q-1))}{q}\right \rfloor}_{h,n}\\
&amp;=\frac{((q-1)(n-(q-1)))^{\underline{0}}}{0!}\ h^{(q-1)(n-(q-1))}\\
&amp;+\left(\frac{((q-1)(n-(q-1)))^{\underline{0}}}{0!}h^{(q-1)(n-(q-1))}
+\frac{(q-1)(n-(q-1)-1))^{\underline{1}}}{1!}\ h^{(q-1)(n-(q-1))-q}\right)\\
&amp;+\cdots+\\
&amp;\left(\frac{((q-1)(n-(q-1)))^{\underline{0}}}{0!}\ h^{(q-1)(n-(q-1))}+\cdots
+\frac{((q-1)(n-(q-1)-r))^{\underline{r}}}{r!}\ h^{(q-1)(n-(q-1))-qr}\right)\\
&amp;+\cdots\\
\end{align*}
\begin{align*}
&amp;+\bigg(\frac{((q-1)(n-(q-1)))^{\underline{0}}}{0!}\ h^{(q-1)(n-(q-1))}+\cdots\\
&amp;+\frac{((q-1)(n-(q-1)-r))^{\underline{r}}}{r!}\ h^{(q-1)(n-(q-1))-qr}+\cdots\\
&amp;+\frac{\left((q-1)\left(n-(q-1)-\lfloor \frac{(q-1)(n-(q-1))}{q}\rfloor\right)\right)^{\underline{\lfloor \frac{(q-1)(n-(q-1))}{q}\rfloor}}}{(\lfloor \frac{(q-1)(n-(q-1))}{q}\rfloor)!}\ h^{(q-1)(n-(q-1))- q\lfloor  \frac{(q-1)(n-(q-1))}{q}\rfloor}\bigg)\\
&amp;=\sum_{r=0}^{\left \lfloor \frac{(q-1)(n-(q-1))}{q}\right \rfloor}\left(\lfloor \frac{(q-1)(n-(q-1))}{q}\rfloor+1-r\right)\frac{((q-1)(n-(q-1)-r))^{\underline{r}}}{r!}\ h^{(q-1)(n-(q-1))-qr}\\
&amp;=\sum_{r=0}^{\left \lfloor \frac{(q-1)(n-(q-1))}{q}\right \rfloor}\left(\left \lfloor \frac{(q-1)(n-(q-1))}{q}\right \rfloor+1\right)\frac{((q-1)(n-(q-1)-r))^{\underline{r}}}{r!}\ h^{(q-1)(n-(q-1))-qr}\\
&amp;- \sum_{r=0}^{\left \lfloor \frac{(q-1)(n-(q-1))}{q}\right \rfloor} r\frac{((q-1)(n-(q-1)-r))^{\underline{r}}}{r!}\ h^{(q-1)(n-(q-1))-qr}\\
&amp;=\left(\left \lfloor \frac{(q-1)(n-(q-1))}{q}\right \rfloor+1-\frac{(q-1)(n-(q-1))}{q}\right)({}^qB)_{h,n}\\
&amp;+\frac{h}{q}(q-1) \left(\sum_{r=0}^{q-2} \frac{(q-2)^{\underline{r}}}{r!}\ h^{(q-2)-r}\ \sum_{i=0}^{n+q-2-r}({}^qB)_{h,i}\ ({}^qB)_{h,n+q-2-r-i}\right).
\end{align*}
Using (\ref{eq:2.6}) of Theorem \ref{thm:3} in Section \ref{sec:2}, the result follows.
\end{proof}

\section{Acknowledgments}
The authors thank the reviewers/referees for their comments and suggestions that helped to improve the article.

\begin{thebibliography}{99}
	\bibitem{Sa2}
	S. Arolkar and Y. S. Valaulikar, Generalized bivariate B-tribonacci and B-tri-Lucas polynomials, {\it Proc. CPMSED-2015,} Krishi sanskriti publications, 2015, pp.~10--13.
	
	\bibitem{Sa1}
	S. Arolkar and Y. S. Valaulikar, On an extension of Fibonacci
	sequence, {\it Bull. Marathwada Math. Soc.} \textbf{17} (2016), 1--8.
	
	\bibitem{Sa4}
	S. Arolkar and Y. S. Valaulikar, $h(x)$-$B$-tribonacci and $h(x)$-$B$-tri Lucas polynomials, {\it Kyungpook Math. J.} \textbf{56} (2016), 1125--1133.
	
	\bibitem{Sa3}
	S. Arolkar and Y. S. Valaulikar, Incomplete $h(x)$-$B$-tribonacci polynomials, {\it Turkish
		Journal of Analysis and Number Theory} \textbf{4} (2016), 155--158.
	
	\bibitem{Sa5}
	S. Arolkar and Y. S. Valaulikar, On a B-$\mathrm{q}$ bonacci sequence, {\it International Journal of Advances in Mathematics} \textbf{1} (2017), 1--8.
	
	
	\bibitem{Sa6}
	S. Arolkar and Y. S. Valaulikar, Identities involving partial derivatives of bivariate $B$-$q$ bonacci and $B$-$q$ Lucas polynomials, preprint.
	
	\bibitem{Filipponi}
	P. Filipponi, Incomplete Fibonacci and Lucas numbers, {\it Rend. Circ. Mat. Palermo  } \textbf{45} (1996), 37--56.

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

	\bibitem{Nalli}
	A. Nalli and P. Haukkane, On generalized Fibonacci and Lucas polynomials, {\it Chaos, Solitons $\&amp;$ Fractals} \textbf{42} (2009), 3179--3186.

	\bibitem{Nazmiye}
N. Yilmaz  and N. Taskara, Incomplete Tribonacci-Lucas numbers and
polynomials, {\it Adv. Appl. Clifford Algebr} \textbf{25} (2015),
741--753.

	\bibitem{Jose}
	J. L. Ram\'{i}rez, Incomplete $k$-Fibonacci and $k$-Lucas numbers, {\it Chinese J. of Math.} (2013),  \href{http://dx.doi.org/10.1155/2013/107145}{Article ID 107145}.

	\bibitem{victor}
	J. L. Ram\'{i}rez and V. F. Sirvent, Incomplete tribonacci numbers and polynomials, {\it J. Integer Sequences} \textbf{17} (2014), \href{https://cs.uwaterloo.ca/journals/JIS/VOL18/Mangontarum/mango2.html}{Article 14.4.2}.

	\bibitem{Jose1}
	J. L. Ram\'{i}rez,  Incomplete generalized Fibonacci and Lucas polynomials, {\it Hacet. J. Math. Stat.} \textbf{42} (2015), 363--373.

	\bibitem{Vajda}
	S. Vajda, {\it Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications}, Dover, 2008.
\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B39; Secondary 11B83.

\noindent \emph{Keywords: }
Fibonacci polynomial,
$h(x)$-$B$-$q$-bonacci polynomial, incomplete $h(x)$-$B$-$q$-bonacci polynomial.

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received October 3 2017;
revised versions received   October 5 2017; March 16 2018; April 4 2018;
April 6 2018.
Published in {\it Journal of Integer Sequences}, May 7 2018.

\bigskip
\hrule
\bigskip

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


\end{document}

                                                                                


</pre></body></html>