{"id":5080,"date":"2017-05-23T16:32:20","date_gmt":"2017-05-23T14:32:20","guid":{"rendered":"http:\/\/pimedios.es\/?p=5080"},"modified":"2017-06-14T13:28:44","modified_gmt":"2017-06-14T11:28:44","slug":"un-paseo-por-los-coeficientes-binomiales","status":"publish","type":"post","link":"https:\/\/pimedios.jesussoto.es\/?p=5080","title":{"rendered":"Un paseo por los coeficientes binomiales"},"content":{"rendered":"

Cuando tratamos en probabilidad los n\u00fameros combinatorios, estamos hablando de las\u00a0 formas en que se puede extraer subconjuntos a partir de un conjunto dado. Tambi\u00e9n los llamamos coeficientes binomiales, \u00a0que nos aparecen en el famoso tri\u00e1ngulo de Pascal<\/a>.<\/p>\n

\"TrianguloPascalC.svg\"<\/a>
De Drini<\/a> – Trabajo propio<\/span>, CC BY-SA 3.0<\/a>, Enlace<\/a><\/figcaption><\/figure>\n

Esta concepci\u00f3n define el coeficiente binomial de dos n\u00fameros enteros $n,k>0$ como
\n$${\\displaystyle {n \\choose k}={\\frac {n!}{k!(n-k)!}}}$$<\/p>\n

Hasta aqu\u00ed lo que normalmente vemos. Pero, \u00bfpodemos hacer el n\u00famero binomial siguiente?:
\n$${\\displaystyle {-4 \\choose 3}}.$$
\nPara resolver el problema necesitamos herramientas nuevas. Sabemos que el n\u00famero combinatorio tambi\u00e9n puede escribirse como
\n$${\\displaystyle {n \\choose k}={\\frac {n(n-1)\\ldots (n-k+1)}{k!}}}.$$
\nDefinamos, para cualquier real, $x\\in\\mathbb{R}$, y $k\\in\\mathbb{Z}^+$,
\n$$(x)_0=1$$
\ny
\n$$(x)_k=x(x-1)\\ldots(x-k+1),\\, k>0.$$
\nEntonces,
\n$${\\displaystyle {x \\choose k}={\\frac {(x)_k}{k!}}}.$$
\nAhora,
\n$${-4 \\choose 3}=\\frac{-4(-4-1)(-4-2)}{3!}=-\\frac{4\u00b75\u00b76}{6}=-20.$$
\nDel mismo modo
\n$${1\/2 \\choose 3}=\\frac{\\tfrac{1}{2}\\left(\\tfrac{1}{2}-1\\right)\\left(\\tfrac{1}{2}-2\\right)}{3!}=\\frac{1}{16}.$$
\nParece rid\u00edculo, porque no podemos interpretarlo como al principio. Pero esto no es \u00f3bice para dar una definici\u00f3n que, alg\u00fan d\u00eda, tenga interpretaci\u00f3n.<\/p>\n

Generalizemos a\u00fan m\u00e1s. Conocemos la relaci\u00f3n de los factoriales con la funci\u00f3n gamma: $\\Gamma(n+1)=n!$ para todo $n$ natural. Entonces
\n$${\\displaystyle {n \\choose k}={\\frac {\\Gamma(n+1)}{\\Gamma(k+1)\\Gamma(n-k+1)}}}.$$
\nApliqu\u00e9moslo a n\u00fameros reales $x,y$:
\n$${\\displaystyle {x \\choose y}={\\frac {\\Gamma(x+1)}{\\Gamma(y+1)\\Gamma(x-y+1)}}}.$$
\nEs m\u00e1s, la funci\u00f3n Gamma nos lleva a un resultado curioso:
\n$${x \\choose y}\\cdot {y \\choose x}={\\frac {\\sin((x-y)\\pi)}{(x-y)\\pi}}.$$
\n\u00bfY lo podemos hacer m\u00e1s complejo? S\u00ed. Para cualesquiera $z,w\\in\\mathbb{C}$ definimos el coeficiente binomial ${z \\choose w}$ como
\n$${\\displaystyle {z \\choose w}=\\lim_{u\\to z}\\lim_{v\\to w}{\\frac {\\Gamma(u+1)}{\\Gamma(v+1)\\Gamma(u-v+1)}}}.$$
\nCon las definiciones escritas vemos que
\n$${1 \\choose \\tfrac{1}{2}}={\\frac {\\Gamma(1+1)}{\\Gamma(\\tfrac{1}{2}+1)\\Gamma(1-\\tfrac{1}{2}+1)}}=\\frac{4}{\\pi};\\quad {\\tfrac{1}{2} \\choose 1}=\\frac{1}{2}.$$<\/p>\n

Esta entrada participa en la Edici\u00f3n 8.4 \u201cMatem\u00e1ticas de todos y para todos\u201d<\/a> del Carnaval de Matem\u00e1ticas cuyo anfitri\u00f3n es, en esta ocasi\u00f3n, matematicascercanas<\/a><\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"

Hablamos sobre los coeficientes binomiales, su definici\u00f3n y la extensi\u00f3n a otros cuerpos.<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-5080","post","type-post","status-publish","format-standard","hentry","category-ocio","entry"],"_links":{"self":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/5080","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5080"}],"version-history":[{"count":7,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/5080\/revisions"}],"predecessor-version":[{"id":5088,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/5080\/revisions\/5088"}],"wp:attachment":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5080"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5080"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5080"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}