Lineer indirgeme dizilerinin bazı ters toplamlarının hesaplanması

Abstract

Bu tezde, $U_{0}=0$, $U_{1}=1$ ve $V_{0}=2$, $V_{1}=p$ başlangıç koşulları olmak üzere her $n/ge{2}$ için /begin{equation*}U_{n}=pU_{n-1}+rU_{n-2}/text{ ve }V_{n}=pV_{n-1}+rV_{n-2},/end{equation*}%kuralları ile tanımlanan ikinci basamaktan lineer homojen indirgeme dizileri $/lbrace U_{n}/rbrace$ ve $/lbrace V_{n}/rbrace$ ile çalışacağız. Bu dizilerin terimlerini ihtiva eden aşağıdaki ters toplamları hesaplayacağız:/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*}ve $X_{n}$, $U_{n}$ ya da $V_{n}$ olmak üzere/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*}

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 $V_{n}$.