{"id":9274,"date":"2023-10-07T16:37:22","date_gmt":"2023-10-07T14:37:22","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=9274"},"modified":"2023-10-07T16:37:24","modified_gmt":"2023-10-07T14:37:24","slug":"les-polynomes-de-bernoulli","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2023\/10\/07\/les-polynomes-de-bernoulli\/","title":{"rendered":"Les polyn\u00f4mes de Bernoulli"},"content":{"rendered":"\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Les polyn\u00f4mes de Bernoulli m&rsquo;ont \u00e9t\u00e9 enseign\u00e9s alors que je n&rsquo;\u00e9tais qu&rsquo;en Premi\u00e8re, il y a moultes ann\u00e9es&#8230; Retour en arri\u00e8re&#8230;<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/10\/07\/les-polynomes-de-bernoulli\/#Polynomes_de_Bernoulli_introduction\" >Polyn\u00f4mes de Bernoulli: introduction<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/10\/07\/les-polynomes-de-bernoulli\/#Introduction_de_lintroduction\" >Introduction de l&rsquo;introduction<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/10\/07\/les-polynomes-de-bernoulli\/#Application_a_lanalyse_numerique\" >Application \u00e0 l&rsquo;analyse num\u00e9rique<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/10\/07\/les-polynomes-de-bernoulli\/#Polynomes_de_Bernoulli_etape_suivante\" >Polyn\u00f4mes de Bernoulli: \u00e9tape suivante<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2023\/10\/07\/les-polynomes-de-bernoulli\/#Polynome_de_Bernoulli_generalisation\" >Polyn\u00f4me de Bernoulli: g\u00e9n\u00e9ralisation<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Polynomes_de_Bernoulli_introduction\"><\/span>Polyn\u00f4mes de Bernoulli: introduction<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><img loading=\"lazy\" decoding=\"async\" width=\"268\" height=\"300\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/10\/Jakob_Bernoulli-268x300.jpg\" alt=\"polyn\u00f4mes de Bernoulli: Jakob Bernoulli\" class=\"wp-image-9285\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/10\/Jakob_Bernoulli-268x300.jpg 268w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/10\/Jakob_Bernoulli.jpg 414w\" sizes=\"auto, (max-width: 268px) 100vw, 268px\" \/><figcaption class=\"wp-element-caption\">Jakob Bernoulli, le bogoss des polyn\u00f4mes<\/figcaption><\/figure>\n<\/div>\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Introduction_de_lintroduction\"><\/span>Introduction de l&rsquo;introduction<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Cherchons un polyn\u00f4me du second degr\u00e9 \\(P_2(x)=ax^2+bx+c\\) telle que:$$\\forall x\\in\\mathbb{R},\\quad P_2(x+1)-P_2(x)=x.$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">\u00ab\u00a0&#8211; Mais pourquoi ???\u00a0\u00bb me direz-vous, tel Sheldon lisant la th\u00e8se de Bert apr\u00e8s que ce dernier ait re\u00e7u un prestigieux prix (on est fan ou on ne l&rsquo;est pas&#8230; si vous avez la ref!)<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Parce que j&rsquo;ai ai envie, et puis c&rsquo;est tout! Patience petit scarab\u00e9e&#8230; Tu vs comprendre!<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Regardons comment trouver les coefficients <em>a<\/em>, <em>b<\/em> et <em>c<\/em>:$$\\begin{array}{ll}\\forall x\\in\\mathbb{R},\\ P_2(x+1)-P_2(x)=x &amp; \\iff a(x+1)^2+b(x+1)+c-(ax^2+bx+c)=x\\\\ &amp; \\iff 2ax+a+b=x\\\\ &amp; \\iff a=\\frac{1}{2},\\ b=-\\frac{1}{2}\\end{array}$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Comme la valeur de <em>c<\/em> n&rsquo;a aucune importance, on va la prendre nulle. On va se la jouer physicien&#8230; (beurk! un physicien&#8230; !)<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On a donc:$$P_2(x)=\\frac{1}{2}x^2-\\frac{1}{2}x.$$Super! Bon, on fait quoi avec \u00e7a&#8230; ?<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Application_a_lanalyse_numerique\"><\/span>Application \u00e0 l&rsquo;analyse num\u00e9rique<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Consid\u00e9rons le polyn\u00f4me \\(P_2\\) non pas sur l&rsquo;ensemble des r\u00e9els mais sur celui des entiers naturels. On a alors:$$\\forall n\\in\\mathbb{N},\\quad P_2(n+1)-P_2(n)=n.$$En particulier, nous avons:$$\\begin{array}{ll}P_2(2)+P_2(1) &amp; = 1\\\\P_2(3)+P_2(2) &amp; = 2\\\\P_2(4)+P_2(3) &amp; = 3\\\\ \\vdots &amp; \\vdots\\\\P_2(n)+P_2(n-1) &amp; = n-1\\\\P_2(n+1)+P_2(n) &amp; = n \\end{array}$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">En effectuant la somme de ces \u00e9galit\u00e9s, on obtient:$$ \\sum_{k=1}^{n} P_2(k+1)-P_2(k) = \\sum_{k=1}^n k$$On reconna\u00eet \u00e0 gauche une <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Somme_t%C3%A9lescopique\" target=\"_blank\" rel=\"noreferrer noopener\">somme t\u00e9lescopique<\/a> (qui a \u00e9t\u00e9 photographi\u00e9e \u00e0 de multiples reprises par les diff\u00e9rents magazines <em>people<\/em> math\u00e9matiques). Tout ceci donne:$$P_2(n+1)-P_2(1)=1+2+3+\\cdots+n$$ soit : $$\\frac{1}{2}\\big[(n+1)^2-1\\big]=1+2+\\cdots+n$$et donc:$$\\boxed{\\frac{n(n+1)}{2}=1+2+3+\\cdots+n}$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Trop cool tout \u00e7a! Le polyn\u00f4me nous permet de calculer la somme des <em>n<\/em> premiers nombres entiers&#8230;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Polynomes_de_Bernoulli_etape_suivante\"><\/span>Polyn\u00f4mes de Bernoulli: \u00e9tape suivante<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">\u00c0 ce stade, on n&rsquo;a qu&rsquo;une seule envie: savoir si on peut faire pareil  \u00ab\u00a0un cran au dessus\u00a0\u00bb: posons \\(P_3\\) le polyn\u00f4me de degr\u00e9 3 tel que:$$\\forall x \\in \\mathbb{R},\\quad P_3(x+1)-P_3(x)=x^2.$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Cette derni\u00e8re condition nous m\u00e8ne, en posant \\(P_3(x)=ax^3+bx^2+cx+d\\) \u00e0:$$\\begin{array}{ll}\\forall x\\in\\mathbb{R},\\quad &amp;  a(x+1)^3+b(x+1)^2+c(x+1)-(ax^3+bx^2+cx+d)=x^2\\\\ \\iff &amp; 3ax^2+(3a+2b)x + a+b+c = x^2\\\\ \\iff &amp; \\begin{cases}3a=1\\\\3a+2b=0\\\\a+b+c=0\\end{cases}\\\\ \\iff &amp; a=\\frac{1}{3},\\ b=-\\frac{1}{2},\\ c=\\frac{1}{6}  \\end{array}$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On a alors, en prenant <em>d<\/em> = 0:$$P_3(x)=\\frac{1}{3}x^3-\\frac{1}{2}x^2+\\frac{1}{6}x$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Comme pr\u00e9c\u00e9demment, on peut alors \u00e9crire:$$\\forall n\\in\\mathbb{N},\\quad P_3(n+1)-P_3(1)=1^2+2^2+3^2+\\cdots+n^2$$ soit:$$\\boxed{\\frac{n(n+1)(2n+1)}{6}=1^2+2^2+3^2+\\cdots+n^2}$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Non mais s\u00e9rieux, c&rsquo;est pas g\u00e9nial ce polyn\u00f4me ?<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Polynome_de_Bernoulli_generalisation\"><\/span>Polyn\u00f4me de Bernoulli: g\u00e9n\u00e9ralisation<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Grands math\u00e9maticiens que nous sommes, nous pouvons nous dire que nous allons g\u00e9n\u00e9raliser tout \u00e7a en posant:$$P_n(x)=\\sum_{k=0}^n a_kx^k$$ tel que:$$P_n(x+1)-P_n(x)=x^{n-1}.$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Ces polyn\u00f4mes ne sont pas les polyn\u00f4mes de Bernoulli&#8230; Non, pas encore! Les polyn\u00f4mes de Bernoulli sont les polyn\u00f4mes \\(B_n\\) tels que:$$B_n(x)=P_{n+1}^\\prime(x)$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Alors, pourquoi d\u00e9finir les polyn\u00f4mes de Bernoulli ? En fait, je n&rsquo;en sais rien&#8230; Je trouve que mes polyn\u00f4mes sont bien plus pratiques&#8230; mais bon! Jakob (Bernoulli) n&rsquo;a pas pens\u00e9 \u00e0 mes polyn\u00f4me a priori&#8230; \ud83d\ude42 <\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Parce qu&rsquo;il faut bien le dire, mes polyn\u00f4mes sont super sympas; en effet,$$P_k(n+1)=1^k + 2^k + 3^k + \\cdots + n^k.$$ Mais bon! Les polyn\u00f4mes de Bernoulli permettent eux aussi de calculer ces sommes; on ne va donc pas s&#8217;emb\u00eater avec d&rsquo;autres polyn\u00f4mes qui leurs ressemblent fortement&#8230; <\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On retiendra donc que les polyn\u00f4mes de Bernoulli permettent de calculer les \\(\\zeta(s)\\) pour <em>s<\/em> entier strictement n\u00e9gatif (<a href=\"https:\/\/fr.wikipedia.org\/wiki\/Fonction_z%C3%AAta_de_Riemann\" target=\"_blank\" rel=\"noreferrer noopener\">fonction de Riemann<\/a>), mais pas exactement de la m\u00eame fa\u00e7on que mes polyn\u00f4mes&#8230; \ud83d\ude42<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Les polyn\u00f4mes de Bernoulli m&rsquo;ont \u00e9t\u00e9 enseign\u00e9s alors que je n&rsquo;\u00e9tais qu&rsquo;en Premi\u00e8re, il y a moultes ann\u00e9es&#8230; Retour en arri\u00e8re&#8230;<\/p>\n","protected":false},"author":1,"featured_media":9286,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[355,401,400,402],"class_list":["post-9274","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","tag-bernoulli","tag-fonction-zeta","tag-polynomes","tag-somme-des-entiers"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Les polyn\u00f4mes de Bernoulli - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Les polyn\u00f4mes de Bernoulli m&#039;ont \u00e9t\u00e9 enseign\u00e9s alors que je n&#039;\u00e9tais qu&#039;en Premi\u00e8re, il y a moultes ann\u00e9es... 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