Evaluation for certain reciprocal sums of linear recurrencesequences

In this thesis, we will consider second order linear homogeneous recurrences $\lbrace U_{n}\rbrace$ and $\lbrace V_{n}\rbrace$ defined by the rules for $n\ge{2}$ \begin{equation*} U_{n}=pU_{n-1}+rU_{n-2}\text{ and }V_{n}=pV_{n-1}+rV_{n-2}, \end{equation*}% where the initial conditions $U_{0}=0$, $U_{1}=1$ and $V_{0}=2$, $V_{1}=p$, respectively. We will evaluate the following reciprocal sums including terms of these sequences \begin{equation*} \sum\limits_{k=0}^{n}(-r)^{k}\frac{V_{k+d+1}}{U_{k+d}U_{k+d+1}U_{k+d+2}}\text{ \ \ \ \ ,\ \ \ \ \ }\sum\limits_{k=0}^{n}(-r)^{k}\frac{U_{k-d}}{U_{k+d}U_{k+d+1}U_{k+d+2}} \end{equation*} and \begin{equation*} \sum\limits_{k=0}^{n}(-r)^{k}\frac{U_{k+c}U_{k+c+1}\ldots U_{k+c+m-1}}{ X_{k+d}X_{k+d+1}\ldots X_{k+d+m+1}} \end{equation*} where $X_{n}$ is $U_{n}$ or $V_{n}$.