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\title[Scott Christian College]{Workshop on \LaTeX}
\author[Dr.D.S.Dilip]{Dr.D.S.Dilip}
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\institute[St.Johns College]{
{Assistant Professor \\Department of Mathematics\\
St.Johns College\\
Anchal - 691 306, Kollam District.\\
 $e-mail:mr.dilipmaths@rediffmail.com$}}
\date[13 - 02 - 18]{February 13, 2018}
\begin{document}
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\begin{block}{Definition}
If $\{y(n)\}$ is a real sequence then \[\Delta y(n)=y(n+1)-y(n),\ n\in\mathbb{N}\]
\[\Delta y(n)=y(n+\ell)-y(n),\ \ell\in\mathbb{N}\]
\[\Delta_\ell y(n)=y(n+\ell)-y(n)\]
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\section{}
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\begin{block}{Relation and Basic theory}
\begin{enumerate}
	\item $E^\ell=1+\Delta_\ell=(1+\Delta)^\ell$.
	\item $\Delta_\ell^r=\sum\limits_{n=0}^r(-1)^nrC_nE^{\ell(r-n)}$.
	\item $1+\Delta_{(\ell_1+\ell_2+\dotsb+\ell_n)}=\prod\limits_{i=1}^{n}(1+\Delta_{\ell_i})$.
	\item $\Delta_\ell^n=\Delta_{n\ell}-nC_1\Delta_{(n-1)\ell}+nC_2\Delta_{(n-2)\ell}+\dotsb+(-1)^{n-1}nC_{n-1}\Delta_\ell$.
\end{enumerate}
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  \begin{frame}\frametitle{Preliminaries}
 \begin{theorem}[Generalized Leibnitz's theorem]
\[\Delta_\ell^n(x_ky_k)=x_k\Delta_\ell^n(y_k)+nC_1\Delta_\ell(x_k)\Delta_\ell^{n-1}(y_{k+\ell})\qquad\qquad\qquad\qquad\]\[\qquad\qquad\qquad\qquad+nC_2\Delta_\ell^2(x_k)\Delta_\ell^{n-2}(y_{k+2\ell})+\dotsb+nC_n\Delta_\ell^n(x_k)y_{k+n\ell}.\]
\end{theorem}
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%\section{references}
 \begin{frame}[allowframebreaks]{References}
\begin{thebibliography}{9}
\beamertemplatebookbibitems
\bibitem{agar00}R.P Agarwal, \emph{Difference  Equations and Inequalities}, Marcel Dekker, New York, 2000.
\beamertemplatearticlebibitems
  \bibitem{bri.ge06} M.Maria Susai Manuel, G.Britto Antony Xavier and E.Thandapani, \emph{Theory of Generalized Difference Operator and Its Applications}, Far East Journal of Mathematical Sciences, 20(2) (2006), 163 - 171.
\bibitem{brit415}M.Maria Susai Manuel, G.Britto Antony Xavier and E.Thandapani, \emph{Qualitative Properties of Solutions of Certain Class of Difference Equations }, Far East Journal of Mathematical Sciences, 23(3) (2006), 295-304.
\bibitem{brit904}M.Maria Susai Manuel, G.Britto Antony Xavier and E.Thandapani, \emph{Generalized Bernoulli Polynomials Through Weighted Pochhammer Symbols}, Far East Journal of Applied Mathematics, 26(3) (2007), 321-333.
	\bibitem{bri.on07} M.Maria Susai Manuel, A.George Maria Selvam and G.Britto Antony Xavier, \emph{On the Solutions and applications of  Some Class of Generalized Difference Equations}, Far East   Journal of  Applied Mathematics, 28(2) (2007), 223 - 241.
	\bibitem{brit.ro06}M.Maria Susai Manuel, A.George Maria Selvam and G.Britto Antony Xavier, \emph{Rotatory and Boundedness of  Solutions of Certain Class of Difference Equations}, International  Journal of Pure and Applied Mathematics, 33(3) (2006), 333-343.


\beamertemplatebookbibitems
\bibitem{ronal90}Ronald E.Mickens, \emph{Difference Equations}, Van Nostrand Reinhold Company, New York, 1990.
\bibitem{sab99} Saber N.Elaydi, \emph{An Introduction To Difference Equations}, Second Edition,  Springer, 1999. 
\bibitem{wal91} Walter G.Kelley, Allan C. Peterson, \emph{Difference Equations, An Introduction with Applications}, Academic Press, inc 1991.
\end{thebibliography}
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\Huge \texttt{THANK YOU}
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