{"id":13810,"date":"2020-06-22T18:22:50","date_gmt":"2020-06-22T16:22:50","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=13810"},"modified":"2021-05-31T15:02:27","modified_gmt":"2021-05-31T13:02:27","slug":"back-to-basics-hypergeometric-functions","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2020\/06\/22\/back-to-basics-hypergeometric-functions\/","title":{"rendered":"Back to basics - Hypergeometric functions"},"content":{"rendered":"<figure id=\"attachment_13876\" aria-describedby=\"caption-attachment-13876\" style=\"width: 248px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/en.wikipedia.org\/wiki\/John_Wallis\"><img loading=\"lazy\" class=\"wp-image-13876 size-medium\" src=\"http:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2020\/06\/John_Wallis_by_Sir_Godfrey_Kneller_Bt-248x300.jpg\" alt=\"John Wallis (1616 - 1703) who coined the term hypergeometric series\" width=\"248\" height=\"300\" srcset=\"https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2020\/06\/John_Wallis_by_Sir_Godfrey_Kneller_Bt-248x300.jpg 248w, https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2020\/06\/John_Wallis_by_Sir_Godfrey_Kneller_Bt-852x1030.jpg 852w, https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2020\/06\/John_Wallis_by_Sir_Godfrey_Kneller_Bt-768x928.jpg 768w, https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2020\/06\/John_Wallis_by_Sir_Godfrey_Kneller_Bt-1271x1536.jpg 1271w, https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2020\/06\/John_Wallis_by_Sir_Godfrey_Kneller_Bt-1695x2048.jpg 1695w, https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2020\/06\/John_Wallis_by_Sir_Godfrey_Kneller_Bt-124x150.jpg 124w\" sizes=\"(max-width: 248px) 100vw, 248px\" \/><\/a><figcaption id=\"caption-attachment-13876\" class=\"wp-caption-text\">John Wallis (1616 - 1703) who coined the term <em>hypergeometric series<\/em><\/figcaption><\/figure>\n<p style=\"text-align: justify;\">This tiny post is an invitation to play with hypergeometric functions. These remarkable special functions can be useful to all mathematicians. They are bizarely not known by many, however.<\/p>\n<p style=\"text-align: justify;\"><strong>Newton binomial series<\/strong>. It states that for any $z,\\alpha\\in\\mathbb{C}$ with $|z|&lt;1$, $$\\frac{1}{(1-z)^\\alpha}=\\sum_{n=0}^\\infty(\\alpha)_n\\frac{z^n}{n!}\\quad\\text{where}\\quad (\\alpha)_n:=\\alpha(\\alpha+1)\\cdots(\\alpha+n-1)$$ is the <strong>Pochhammer symbol for rising factorial<\/strong> named after <a href=\"https:\/\/en.wikipedia.org\/wiki\/Leo_August_Pochhammer\">Leo August Pochhammer<\/a> (1841 - 1920), with the convention $(\\alpha)_0=1$ if $\\alpha\\neq0$. Note that $(1)_n=n!$ and $$\\Gamma(\\alpha+n)=(\\alpha)_n\\Gamma(\\alpha)\\quad\\text{where}\\quad\\Gamma(\\alpha):=\\int_0^\\infty t^{\\alpha-1}\\mathrm{e}^{-t}\\mathrm{d}t.$$ When $\\alpha=m$ then this boils down to $(m)_n=(m+n-1)!\/(n-1)!$.<\/p>\n<p style=\"text-align: justify;\"><strong>Hypergeometric functions.<\/strong> Inspired by the Newton binomial series expansion, they try to catch, via a unique parametrized series, a large variety of special functions. More precisely, if $a\\in\\mathbb{R}^p$, $b\\in\\mathbb{R}^q$, and $z\\in\\mathbb{C}$, $|z|&lt;1$, then, when it makes sense,<br \/>\n$${}_pF_q\\begin{pmatrix}a_1,\\ldots,a_p\\\\b_1,\\ldots,b_q\\\\z\\end{pmatrix}:=\\sum_{n=0}^\\infty\\frac{(a_1)_n\\dots(a_p)_n}{(b_1)_n\\cdots(b_q)_n}\\frac{z^n}{n!}.<br \/>\n$$<br \/>\nWe could use from this formula analytic continuation. <a href=\"https:\/\/en.wikipedia.org\/wiki\/Hypergeometric_function\">Hypergeometric functions<\/a> where studied by many including notably <a href=\"https:\/\/en.wikipedia.org\/wiki\/Leonhard_Euler\">Leonhard Euler<\/a> (1707 - 1783) and <a href=\"https:\/\/en.wikipedia.org\/wiki\/Carl_Friedrich_Gauss\">Carl-Friedrich Gauss<\/a> (1777 - 1855). This kind of special function contains several others, for instance<\/p>\n<ul>\n<li>${}_2F_1(1,1;2;-z)=\\frac{\\log(1+z)}{z}$<\/li>\n<li>${}_2F_1(a,b;b;z)=\\frac{1}{(1-z)^a}$<\/li>\n<li>${}_2F_1(\\frac{1}{2},\\frac{1}{2};\\frac{3}{2};z^2)=\\frac{\\arcsin(z)}{z}$<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">It is also possible to embed <a href=\"https:\/\/en.wikipedia.org\/wiki\/Jacobi_polynomials\">Jacobi orthogonal polynomials<\/a> into hypergeometric functions and thus several families of orthogonal polynomials, more precisely $${}_{2}F_{1}(-n,a+1+b+n;a+1;x)={\\frac {n!}{(a+1)_{n}}}P_{n}^{(a,b)}(1-2x).$$ Note that $(z)_k=0$ for large enough $k$ when $z$ is a negative integer, hence ${}_pF_q(a;b;z)$ is a polynomial when one of the $a_i$ is a negative integer.<\/p>\n<p>Hypergeometric functions admit integral representations, and conversely, certain integrals can be computed using hypergeometric functions. Here is the most basic example.<\/p>\n<p><strong>Euler integral representation formula for ${}_2F_1$ published in 1769.<\/strong> If $a&gt;0$, $b&gt;0$, $|z|\\leq 1$, then<br \/>\n$$\\int_0^1u^{a-1}(1-u)^{b-1}(1-zu)^{-c}\\mathrm{d}u<br \/>\n={}_2F_1\\begin{pmatrix}a,c\\\\a+b\\\\z\\end{pmatrix}<br \/>\n\\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)}$$ In other words, for all $a,b,c$ with $c&gt;a&gt;0$ and all $|z|\\leq 1$,<br \/>\n$${}_2F_1\\begin{pmatrix}a,b\\\\c\\\\z\\end{pmatrix}=\\frac{\\Gamma(c)}{\\Gamma(a)\\Gamma(c-a)}\\int_0^1u^{a-1}(1-u)^{c-a-1}(1-zu)^{-b}\\mathrm{d}u.$$<\/p>\n<p><strong>A proof.<\/strong> A binomial series expansion of $(1-zu)^{-c}$ gives<br \/>\n$$<br \/>\n\\int_0^1u^{a-1}(1-u)^{b-1}(1-zu)^{-c}\\mathrm{d}u<br \/>\n=\\sum_{k=0}^\\infty\\frac{(c)_k}{k!}z^k<br \/>\n\\int_0^1u^{a+k-1}(1-u)^{b-1}\\mathrm{d}u.<br \/>\n$$<br \/>\nNow the beta-gamma formula $\\displaystyle \\int_0^1u^{a+k-1}(1-u)^{b-1}\\mathrm{d}u=\\frac{\\Gamma(a+k)\\Gamma(b)}{\\Gamma(a+b+k)}$ gives<br \/>\n$$<br \/>\n\\int_0^1u^{a-1}(1-u)^{b-1}(1-zu)^{-c}\\mathrm{d}u<br \/>\n=\\Gamma(b)\\sum_{k=0}^\\infty\\frac{(c)_k\\Gamma(a+k)}{\\Gamma(a+b+k)}<br \/>\n\\frac{z^k}{k!}.<br \/>\n$$<br \/>\nFinally the formula $\\Gamma(z+k)=(z)_k\\Gamma(z)$ gives<br \/>\n$$<br \/>\n\\int_0^1u^{a-1}(1-u)^{b-1}(1-zu)^{-c}\\mathrm{d}u<br \/>\n=\\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)}\\sum_{k=0}^\\infty\\frac{(c)_k(a)_k}{(a+b)_k}<br \/>\n\\frac{z^k}{k!}\\\\<br \/>\n=\\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)}\\ {}_2F_1\\begin{pmatrix}a,c\\\\a+b\\\\z\\end{pmatrix}.<br \/>\n$$<\/p>\n<p><strong>Immediate corollary.<\/strong> By sending $z$ to $1$, taking $b&gt;c$, and using the beta-gamma formula $$\\int_0^1u^{a-1}(1-u)^{b-c-1}\\mathrm{d}u =\\frac{\\Gamma(a)\\Gamma(b-c)}{\\Gamma(a+b-c)},$$ we obtain the following identity discovered by Gauss (1812), for all $a,b,c$ with $c&gt;a+b$, $$\\sum_{k=0}^\\infty\\frac{(a)_k(b)_k}{(c)_k}\\frac{1}{k!}={}_2F_1\\begin{pmatrix}a,b\\\\c\\\\1\\end{pmatrix}=\\frac{\\Gamma(c-a-b)\\Gamma(c)}{\\Gamma(c-a)\\Gamma(c-b)}.$$<\/p>\n<p><strong>Maple, Mathematica, and Maxima.<\/strong> All implement hypergeometric functions. Here is an example with the Euler integral formula with Mathematica:<\/p>\n<pre class=\"prettyprint\">In[1]:= Integrate[u^{a-1}*(1 - u)^{b-1}*(1 - z*u)^{-c}, {u, 0, 1}]\r\n\r\nOut[1]= {ConditionalExpression[\r\nGamma[a] Gamma[b] Hypergeometric2F1Regularized[a, c, a + b, z],\r\nRe[a] &gt; 0 &amp;&amp; Re[b] &gt; 0 &amp;&amp; (Re[z] &lt;= 1 || z \u2209 \u211d)]}<\/pre>\n<p>The regularized ${}_2F_1$ hypergeometric function used by Mathematica is ${}_2F_1(a,b;c;z)\/\\Gamma(c)$.<\/p>\n<p><strong>Further reading.<\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/zbmath.org\/?q=an%3A0920.33001\">George Andrews, Richard\u00a0Askey, Ranjan Roy<br \/>\n<em>Special functions<\/em> (Theorem 2.2.1 pages 65)<br \/>\nEncyclopedia of Mathematics and Its Applications 71<br \/>\nCambridge University Press. xvi, 664 p. (1999)<\/a><\/li>\n<li><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=107725\">Rainville, Earl D.<br \/>\n<em>Special functions.<\/em><br \/>\nThe Macmillan Company, New York\u00a01960\u00a0xii+365 pp<\/a><\/li>\n<li><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=2280366\">Kathy A. Driver, Sarah Jane Johnston<br \/>\n<em>An integral representation of some hypergeometric functions<\/em><br \/>\nElectron. Trans. Numer. Anal. 25 (2006), 115-120<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>This tiny post is an invitation to play with hypergeometric functions. These remarkable special functions can be useful to all mathematicians. They are bizarely not&#8230;<\/p>\n<div class=\"more-link-wrapper\"><a class=\"more-link\" href=\"https:\/\/djalil.chafai.net\/blog\/2020\/06\/22\/back-to-basics-hypergeometric-functions\/\">Continue reading<span class=\"screen-reader-text\">Back to basics - Hypergeometric functions<\/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":182},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/13810"}],"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=13810"}],"version-history":[{"count":129,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/13810\/revisions"}],"predecessor-version":[{"id":15087,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/13810\/revisions\/15087"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=13810"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=13810"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=13810"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}