{"id":22329,"date":"2025-12-21T16:17:15","date_gmt":"2025-12-21T15:17:15","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=22329"},"modified":"2026-01-30T09:55:34","modified_gmt":"2026-01-30T08:55:34","slug":"beta-gamma-normal-and-jacobi-laguerre-hermite","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2025\/12\/21\/beta-gamma-normal-and-jacobi-laguerre-hermite\/","title":{"rendered":"Beta\/Gamma\/Normal and Jacobi\/Laguerre\/Hermite"},"content":{"rendered":"<figure id=\"attachment_22383\" aria-describedby=\"caption-attachment-22383\" style=\"width: 200px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/en.wikipedia.org\/wiki\/Anonymus_(notary_of_B%C3%A9la_III)\"><img loading=\"lazy\" src=\"http:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2025\/12\/Anonymus-200x300.jpg\" alt=\"Photo of Statue of Anonymus in City Park in Budapest\" width=\"200\" height=\"300\" class=\"size-medium wp-image-22383\" srcset=\"https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2025\/12\/Anonymus-200x300.jpg 200w, https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2025\/12\/Anonymus-687x1030.jpg 687w, https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2025\/12\/Anonymus-768x1152.jpg 768w, https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2025\/12\/Anonymus.jpg 960w\" sizes=\"(max-width: 200px) 100vw, 200px\" \/><\/a><figcaption id=\"caption-attachment-22383\" class=\"wp-caption-text\">Statue of Anonymus in the City Park of Budapest - Mikl\u00f3s Ligeti, 1903.<\/figcaption><\/figure>\n<p style=\"text-align:justify;\">This post is about the Beta\/Gamma\/Normal or Jacobi\/Laguerre\/Hermite trilogy.<\/p>\n<p style=\"text-align:justify;\"><strong>Beta.<\/strong> The Beta distribution $\\mathrm{Beta}_{[0,1]}(a,b)$ on $[0,1]$ has density proportional to \\[   x\\mapsto x^{a-1}(1-x)^{b-1}\\mathbf{1}_{x\\in [0,1]}. \\] Its image by $x\\mapsto 2x-1$ is $\\mathrm{Beta}_{[-1,1]}(a,b)$ on $[-1,1]$, with density proportional to \\[   x\\mapsto (1+x)^{a-1}(1-x)^{b-1}\\mathbf{1}_{x\\in[-1,1]}. \\] In particular $\\mathrm{Beta}_{[-1,1]}(a,a)$ on $[-1,1]$ has density proportional to \\[   x\\mapsto (1-x^2)^{a-1}\\mathbf{1}_{x\\in[-1,1]}, \\] also known as a Barenblatt profile. More generally, the $\\mathrm{Beta}_{[\\alpha,\\beta]}(a,b)$ distribution is \\[   x\\mapsto(x-\\alpha)^{a-1}(\\beta-x)^{b-1}\\mathbf{1}_{x\\in[\\alpha,\\beta]} \\] up to the normalizing constant. In particular<\/p>\n<p style=\"text-align:justify;\">\n<ul>\n<li>$\\mathrm{Beta}_{[\\alpha,\\beta]}(1,1)$ is the uniform distribution <\/li>\n<li>$\\mathrm{Beta}_{[\\alpha,\\beta]}(\\frac{3}{2},\\frac{3}{2})$ is the semi-circle distribution <\/li>\n<li>$\\mathrm{Beta}_{[\\alpha,\\beta]}(\\frac{1}{2},\\frac{1}{2})$ is the arcsine distribution. <\/li>\n<\/ul>\n<p style=\"text-align:justify;\">Beta distributions are strongly related to many other distributions.<\/p>\n<p style=\"text-align:justify;\"><strong>$\\ell^1$ geometry.<\/strong> If $X_1,\\ldots,X_n$ are independent with $X_i\\sim\\mathrm{Gamma}(a_i,\\lambda)$ then \\[   V=\\frac{X}{|X|_1}=\\frac{(X_1,\\ldots,X_n)}{X_1+\\cdots+X_n}   \\sim\\mathrm{Dirichlet}(a_1,\\ldots,a_n) \\] which is a probability distribution on the simplex $\\{(v_1,\\ldots,v_n)\\in\\mathbb{R}_+:|v|_1=1\\}$, uniform when $a_1=\\cdots=a_n=1$ and in this case $X_i\\sim\\mathrm{Exp}(\\lambda)$. For all $1\\leq i\\leq n$, \\[   V_i=\\frac{X_i}{|X|_1}=\\frac{X_i}{X_1+\\cdots+X_n}   \\sim\\mathrm{Beta}_{[0,1]}\\Bigr(a_i,\\sum_{j\\neq i}a_j\\Bigr). \\] Indeed, if $A\\sim\\mathrm{Gamma}(a,\\lambda)$ and $B\\sim\\mathrm{Gamma}(b,\\lambda)$ are independent then $A+B\\sim\\mathrm{Gamma}(a+b,\\lambda)$ and $A\/(A+B)\\sim\\mathrm{Beta}_{[0,1]}(a,b)$. In particular \\[   X_1+\\cdots+X_n\\sim\\mathrm{Gamma}(a_1+\\cdots+a_n,\\lambda)   \\quad\\text{and}\\quad   V_i=\\frac{X_i}{X_i+Y_i} \\] where $Y_i:=\\sum_{j\\neq i}X_j\\sim\\mathrm{Gamma}(\\sum_{j\\neq i}a_j,\\lambda)$ is independent of $X_i\\sim\\mathrm{Gamma}(a_i,\\lambda)$.<\/p>\n<p style=\"text-align:justify;\"><strong>$\\ell^2$ geometry.<\/strong> If $X_1,\\ldots,X_n$ are iid with $X_i\\sim\\mathcal{N}(0,1)$ then \\[   U=\\frac{X}{|X|_2}=\\frac{(X_1,\\ldots,X_n)}{\\sqrt{X_1^2+\\cdots+X_n^2}} \\] is uniformly distributed on the sphere $\\mathbb{S}^{n-1}=\\{u\\in\\mathbb{R}^n:|u|_2^2=1\\}$. For all $1\\leq k\\leq n$, \\[   |\\mathrm{proj}_{\\mathbb{R}^k}(U)|_2^2   =U_1^2+\\cdots+U_k^2\\sim\\mathrm{Beta}_{[0,1]}\\Bigr(\\frac{k}{2},\\frac{n-k}{2}\\Bigr). \\] Indeed, we have \\[   U_1^2+\\cdots+U_k^2=\\frac{X_1^2+\\cdots+X_k^2}{(X_1^2+\\cdots+X_k^2)+(X_{k+1}^2+\\cdots+X_n^2)}   =\\frac{A}{A+B} \\] and $\\begin{cases}A\\sim\\chi^2(k)=\\mathrm{Gamma}(\\frac{k}{2},\\frac{1}{2})\\\\B\\sim\\chi^2(n-k)=\\mathrm{Gamma}(\\frac{n-k}{2},\\frac{1}{2})\\end{cases}$ are independent, hence the result.<\/p>\n<p style=\"text-align:justify;\"><strong>Funk-Hecke and Barenblatt.<\/strong> In particular, for all $1\\leq i\\leq n$, $U_i^2\\sim\\mathrm{Beta}_{[0,1]}(\\frac{1}{2},\\frac{n-1}{2})$, hence \\[   U_i=\\varepsilon|U_i|\\sim\\mathrm{Beta}_{[-1,1]}\\Bigr(\\frac{n-1}{2},\\frac{n-1}{2}\\Bigr). \\] This is nothing else but the Funk-Hecke formula in harmonic analysis. More generally, we get, for all $1\\leq k\\leq n-1$, that $\\mathrm{proj}_{\\mathbb{R}^k}(U)$ follows a Barenblatt distribution with density proportional to \\[   x\\mapsto(1-|x|_2^2)^{\\frac{n-k-2}{2}}\\mathbf{1}_{|x|_2\\leq1}. \\] This can be seen as a multivariate Beta distribution.<\/p>\n<p style=\"text-align:justify;\"><strong>Beta, Gamma, Normal.<\/strong> The Euler formula $\\lim_{n\\to\\infty}(1+\\frac{x}{n})^n=\\mathrm{e}^x$ gives \\begin{align*}   x^{a-1}(1-\\tfrac{x}{n})^{n-1}&\\to x^{a-1}\\mathrm{e}^{-x}\\\\     (1-\\tfrac{x^2}{2n})^{n-1}&\\to\\mathrm{e}^{-\\frac{x^2}{2}} \\end{align*} hence, for the narrow convergence, \\begin{align*}   \\lim_{n\\to\\infty}   \\mathrm{Beta}_{[0,n]}(a,n)   &=\\mathrm{Gamma}(a,1)\\\\   \\lim_{n\\to\\infty}   \\mathrm{Beta}_{[-\\sqrt{2n},\\sqrt{2n}]}(n,n)   &=\\mathcal{N}(0,1). \\end{align*} In terms of random variables and using Beta distributions on $[0,1]$ we have, in distribution,<\/p>\n<p style=\"text-align:justify;\">\n<ul>\n<li>if $X_n\\sim\\mathrm{Beta}_{[0,1]}(a,n)$ then   $nX_n\\to\\mathrm{Gamma}(a,1)$ <\/li>\n<li>if $X_n\\sim\\mathrm{Beta}_{[0,1]}(n,n)$ then   $\\sqrt{2n}(2X_n-1)\\to\\mathcal{N}(0,1)$ <\/li>\n<\/ul>\n<p style=\"text-align:justify;\">while using Beta distributions on $[-1,1]$ this reads, in distribution,<\/p>\n<p style=\"text-align:justify;\">\n<ul>\n<li>if $X_n\\sim\\mathrm{Beta}_{[-1,1]}(a,n)$ then   $n(\\frac{X_n+1}{2})\\to\\mathrm{Gamma}(a,1)$ <\/li>\n<li>if $X_n\\sim\\mathrm{Beta}_{[-1,1]}(n,n)$ then   $\\sqrt{2n}X_n\\to\\mathcal{N}(0,1)$. <\/li>\n<\/ul>\n<p style=\"text-align:justify;\">This makes the Beta distribution a proxy for the Gamma and the Normal distributions. In other words, Gamma and Normal are translation-dilation deformations of Beta !<\/p>\n<p style=\"text-align:justify;\">This was used for instance by Atle Selberg (1917 - 2007) to get his explicit formula for the normalizing constant of Coulomb gases coming from random matrix theory. This was also used by Dominique Bakry (1954 - ) for the study of univariate Markov diffusion operators<\/p>\n<p style=\"text-align:justify;\">\n<ul>\n<li>Jacobi operator : $(1-x^2)f''+(a-b-(a+b)x)f'$ on $[-1,1]$<br \/>   in other words $\\frac{1}{w}((1-x^2)wf')'$ where $w(x)=(1+x)^{a-1}(1-x)^{b-1}$ <\/li>\n<li>Laguerre operator : $xf''+(a-x)f'$ on $\\mathbb{R}_+$<br \/>   in other words $\\frac{1}{w}(xwf')'$ where $w(x)=x^{a-1}\\mathrm{e}^{-x}$ <\/li>\n<li>Hermite (or Ornstein-Uhlenbeck) operator : $f''-xf'$ on $\\mathbb{R}$<br \/>   in other words $\\frac{1}{w}(wf')'$ where $w(x)=\\mathrm{e}^{-\\frac{x^2}{2}}$ <\/li>\n<\/ul>\n<p style=\"text-align:justify;\">for which these orthogonal polynomials are the eigenfunctions of the operator. The triple of distributions Beta\/Gamma\/Normal corresponds to the triple of operators Jacobi\/Laguerre\/Hermite, via reversible invariant measures and spectral decomposition.<\/p>\n<p style=\"text-align:justify;\">The message is that what matters is the Beta\/Jacobi case, since the Gamma\/Laguerre and Hermite\/Normal cases lie on the boundary, and can be accessed by taking limits.<\/p>\n<p style=\"text-align:justify;\">The Selberg integral is about computing the normalizing constant (or partition function) of Coulomb gases emerging from random matrix theory. Analytic continuation is used to reduce to the case where the inverse temperature is an integer, the interaction term is then a polynomial. This allows to compute the Beta-weight case by reduction to Euler integrals. The Gamma-weight and Normal-weight cases are then obtained by taking limits.<\/p>\n<p style=\"text-align:justify;\">In the same spirit, and depending on the object of interest, one could say that the Beta case with integer parameters is algebra, since the density is a polynomial, analytic continuation then yields the Beta case with real parameters, and the Normal and Gamma cases follow by taking limits. In a sense, this is a reduction to algebra by analysis, for probability.<\/p>\n<p style=\"text-align:justify;\"><strong>Jacobi, Laguerre, Hermite orthogonal polynomials.<\/strong> They are obtained by using the Gram-Schmidt algorithm with the canonical algebraic basis $1,X,X^2,\\ldots$ of $\\mathbb{R}[X]$ and the scalar product of $L^2(\\mu)$ where $\\mu$ is the Beta, Gamma, and Normal distribution respectively. The passage from Beta to Gamma and Normal allows to pass from Jacobi to Laguerre and Hermite polynomials. Jacobi polynomials are proxies for Laguerre and Hermite polynomials.<\/p>\n<p style=\"text-align:justify;\"><strong>Askey scheme.<\/strong> It seems that the observation goes back at least to Ervin Feldheim (1912 - 1944) in the context of orthogonal polynomials, and to Ernst Eduard Kummer (1810 - 1893) in the context of confluent hypergeometric series, something like \\[   \\lim_{b\\to\\infty}{}_2F_1(a,b;c;\\frac{z}{b})={}_1F_1(a;c;z). \\] Nowadays, this is one aspect of the general way of organizing orthogonal polynomials using hypergeometric series, known as the Askey scheme, due to Richard Askey (1933 - 2019). \\begin{align*}   J^{a,b}_n(x)   &=\\frac{(b)_n}{n!}\\ {}_2F_1(-n;n+a+b-1;b;\\tfrac{1-x}{2})\\\\   &=\\frac{\\Gamma(b+n)}{n!\\Gamma(a+b+n-1)}\\sum_{m=0}^n\\binom{n}{m}\\frac{\\Gamma(a+b+n+m-1)}{\\Gamma(b+m)}\\Bigr(\\frac{x-1}{2}\\Bigr)^m\\\\   L_n^a(x)&=(-1)^n\\lim_{b\\to\\infty}J^{a,b}_n(\\tfrac{2x}{b}-1)\\\\   H_n(x)&=n!2^n\\lim_{a\\to\\infty}a^{-\\frac{n}{2}}J^{a,a}_n(\\tfrac{x}{\\sqrt{2a}}) \\end{align*}<\/p>\n<p style=\"text-align:justify;\"><strong>Random matrices.<\/strong> Let $N$ be an $n\\times m$ random matrix with iid entries of law $\\mathcal{N}(0,1)$, a matrix analogue of the Normal distribution. The law of \\[   A=NN^\\top \\] is known as the Wishart ensemble, a matrix analogue of $\\chi^2$ hence Gamma distribution. If $B$ is an independent copy of $A$, then the law of \\[   C=(A+B)^{-1\/2}A(A+B)^{-1\/2} \\] is a matrix analogue of the Beta distribution. The eigenvalues of $A$ and $C$ turn out to be Coulomb gases with weight Gamma and Beta, known as Laguerre and Jacobi ensembles. These are Coulomb gases studied notably by Freeman Dyson (1923 - 2020) and for which the normalizing constant was computed by Atle Selberg (1917 - 2007). We refer to Edelman and Rao for more.<\/p>\n<p style=\"text-align:justify;\"><strong>Ervin Feldheim (1912 - 1944).<\/strong> A Hungarian mathematician. He studied in Paris from 1931 to 1934 and obtained a doctorate in 1937 from Universit\u00e9 de Paris with a committee formed by \u00c9mile Borel (1871 - 1956), Georges Darmois (1888 - 1960), and Maurice Fr\u00e9chet (1878 - 1973). He knew Wolfgang D\u00f6blin or Vincent Doblin (1915 - 1940), and they helped Paul L\u00e9vy (1886 - 1971) proofreading his famous manuscript <em>Th\u00e9orie de l'addition de variables   al\u00e9atoires<\/em>. After returning to Hungary in 1934, Ervin Feldheim worked in insurance while continuing research in mathematics. This was common at that time in Europe. He wrote notably on characteristic functions as well as on special functions. According to L\u00e1szl\u00f3 Lov\u00e1sz and Vera T. S\u00f3s (2015), he was, together with Paul Erd\u0151s (1913 - 1996), Tibor Gallai (1912 - 1992), P\u00e1l Tur\u00e1n (1910 - 1976), and some few others, a member of the legendary <em>Anonymus   group<\/em>, meeting regularly during their university years at the Statue of Anonymus in City Park in Budapest. Lifelong friendships were formed between them, with deep impact on their professional lives. According to Bernard Bru (1992), Ervin Feldheim was arrested in summer 1942, deported until late 1943, returned to Hungary in poor health, then deported again in 1944, and probably killed during the evacuation massacres.<\/p>\n<p style=\"text-align:justify;\"><strong>Further reading.<\/strong><\/p>\n<p style=\"text-align:justify;\">\n<ul>\n<li>Richard Askey and James Wilson<br \/>   <strong>Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials<\/strong><br \/>   Memoirs of the American Mathematical Society 54 (1985) <\/li>\n<li>Dominique Bakry<br \/>   <strong>Remarques sur les semigroupes de Jacobi<\/strong><br \/>   Ast\u00e9risque 236 23-39 (1996) <\/li>\n<li>Bernard Bru<br \/>   <strong>La vie et l\u2019\u0153uvre de W. Doeblin (1915-1940) d\u2019apr\u00e8s les archives parisiennes<\/strong><br \/>   Math\u00e9matiques et sciences humaines 119 5-51 (1992) <\/li>\n<li>Alan Edelman and N. Raj Rao<br \/>   <strong>Random matrix theory<\/strong><br \/>   Acta Numerica 14 233-297 (2005) <\/li>\n<li>Ervin Feldheim<br \/>   <strong>Relations entre les polyn\u00f4mes de Jacobi, Laguerre et Hermite<\/strong><br \/>   Acta Mathematica 75 117-138 (1942) <\/li>\n<li>Ervin Feldheim<br \/>   <strong>On the positivity of certain sums of ultraspherical polynomials<\/strong><br \/>   Journal d\u2019Analyse Math\u00e9matique 11 275-284 (1963) (postumous publication)<br \/>   Includes an editorial note by G\u00e1bor Szeg\u0151 (1895 - 1985) explaining that it   is essentially a letter dated March 12, 1944, found among Lip\u00f3t Fej\u00e9r (1880   - 1959) papers by P\u00e1l Tur\u00e1n (1910 - 1976). <\/li>\n<li>Peter J. Forrester and S. Ole Warnaar<br \/>   <strong>The importance of the Selberg integral<\/strong><br \/>   Bulletin of the American Mathematical Society 45 489-534 (2008) <\/li>\n<li>Tom H. Koornwinder<br \/>   <strong>Group theoretic interpretations of Askey's scheme of hypergeometric     orthogonal polynomials<\/strong><br \/>   Orthogonal polynomials and their applications (Segovia, 1986) 46-72<br \/>   Lecture Notes in Mathematics 1329 Springer (1988) <\/li>\n<li>L\u00e1szl\u00f3 Lov\u00e1sz and Vera T. S\u00f3s<br \/>   <strong>Erd\u0151s Centennial<\/strong><br \/>   Notices of the American Mathematical Society 62(2) (February 2015) <\/li>\n<\/ul>\n<p style=\"text-align:justify;\"><strong>Further reading on this blog.<\/strong><\/p>\n<p style=\"text-align:justify;\">\n<ul>\n<li><a href=\"https:\/\/djalil.chafai.net\/blog\/2024\/10\/20\/back-to-basics-student-and-barenblatt\/\">Back to basics : Student and Barenblatt<\/a> (2024) <\/li>\n<li><a href=\"https:\/\/djalil.chafai.net\/blog\/2024\/03\/18\/archimedes-theorem-sphere-cylinder\/\">Archimedes theorem on sphere and cylinder<\/a> (2024) <\/li>\n<li><a href=\"https:\/\/djalil.chafai.net\/blog\/2021\/05\/22\/the-funk-hecke-formula\/\">The Funk-Hecke formula<\/a> (2021) <\/li>\n<li><a href=\"https:\/\/djalil.chafai.net\/blog\/2020\/06\/22\/back-to-basics-hypergeometric-functions\/\">Back to basics - Hypergeometric functions<\/a> (2020)   <\/li>\n<li><a href=\"https:\/\/djalil.chafai.net\/blog\/2011\/03\/17\/beta-laws-arcsine-uniform-semicircle\/\">Beta laws: arcsine, uniform, semicircle<\/a> (2011) <\/li>\n<\/ul>\n<figure id=\"attachment_22336\" aria-describedby=\"caption-attachment-22336\" style=\"width: 200px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/en.wikipedia.org\/wiki\/Ervin_Feldheim\"><img loading=\"lazy\" class=\"wp-image-22336\" src=\"http:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2025\/12\/Feldheim.jpg\" alt=\"Photo of Ervin Feldheim (1912 - 1944)\" width=\"200\" height=\"246\" srcset=\"https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2025\/12\/Feldheim.jpg 261w, https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2025\/12\/Feldheim-244x300.jpg 244w\" sizes=\"(max-width: 200px) 100vw, 200px\" \/><\/a><figcaption id=\"caption-attachment-22336\" class=\"wp-caption-text\">Ervin Feldheim (1912 - 1944). Early explorer of the Jacobi \/ Laguerre \/ Hermite trilogy.<\/figcaption><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>This post is about the Beta\/Gamma\/Normal or Jacobi\/Laguerre\/Hermite trilogy. Beta. The Beta distribution $\\mathrm{Beta}_{[0,1]}(a,b)$ on $[0,1]$ has density proportional to \\[ x\\mapsto x^{a-1}(1-x)^{b-1}\\mathbf{1}_{x\\in [0,1]}. \\]&#8230;<\/p>\n<div class=\"more-link-wrapper\"><a class=\"more-link\" href=\"https:\/\/djalil.chafai.net\/blog\/2025\/12\/21\/beta-gamma-normal-and-jacobi-laguerre-hermite\/\">Continue reading<span class=\"screen-reader-text\">Beta\/Gamma\/Normal and Jacobi\/Laguerre\/Hermite<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":95},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/22329"}],"collection":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/comments?post=22329"}],"version-history":[{"count":71,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/22329\/revisions"}],"predecessor-version":[{"id":22455,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/22329\/revisions\/22455"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=22329"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=22329"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=22329"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}