Terminados apuntes de Álgebra Conmutativa

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+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input{../defs}
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
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+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
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+\tracking_changes false
+\output_changes false
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+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+kern-1em
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{align*}
+\binom{n}{k} & =\binom{n}{n-k}; &  &  & \binom{r}{k} & =(-1)^{k}\binom{k-r-1}{k};\\
+\binom{r}{k} & =\frac{r}{k}\binom{r-1}{k-1}, & k & \neq0; & \binom{n}{m} & =(-1)^{n-m}\binom{-(m+1)}{n-m}, & n & \geq0;\\
+\binom{r}{k} & =\frac{r}{r-k}\binom{r-1}{k}, & k & \neq r; & \sum_{k=0}^{n}\binom{r+k}{k} & =\binom{r+n+1}{n}, & n & \geq0;\\
+\binom{r}{k} & =\binom{r-1}{k}+\binom{r-1}{k-1}; &  &  & \sum_{k=0}^{n}\binom{k}{m} & =\binom{n+1}{m+1}, & m,n & \geq0;
+\end{align*}
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{align*}
+\binom{r}{m}\binom{m}{k} & =\binom{r}{k}\binom{r-k}{m-k}, & \sum_{k}\binom{r}{k}\binom{s}{n-k} & =\binom{r+s}{n};\\
+\sum_{k}\binom{r}{m+k}\binom{s}{n+k} & =\binom{r+s}{r-m+n}, & \sum_{k}\binom{r}{k}\binom{s+k}{n}(-1)^{r-k} & =\binom{s}{n-r}, & r & \geq0;
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{align*}
+\sum_{k=0}^{r}\binom{r-k}{m}\binom{s}{k-t}(-1)^{k-t} & =\binom{r-t-s}{r-t-m}, & t,r,m & \geq0;\\
+\sum_{k=0}^{r}\binom{r-k}{m}\binom{s+k}{n} & =\binom{r+s+1}{m+n+1}, & n\geq s & \geq0,\ m,r\geq0;\\
+\sum_{k\geq0}\binom{r-tk}{k}\binom{s-t(n-k)}{n-k}\frac{r}{r-tk} & =\binom{r+s-tn}{n};
+\end{align*}
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{align*}
+\sum_{k}\binom{n}{k}x(x-kz)^{k-1}(y+kz)^{n-k} & =(x+y)^{n}, & x & \neq0;
+\end{align*}
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{align*}
+\sum_{k}\binom{r}{k}x^{k}y^{r-k} & =(x+y)^{r}, & r & \geq0; & \sum_{k}\binom{r}{k}x^{k} & =(1+x)^{r}, & r & \geq0\text{ o }|x|<1;
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document