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Month: June 2020

Back to basics – Hypergeometric functions

John Wallis (1616 - 1703) who coined the term hypergeometric series
John Wallis (1616 – 1703) who coined the term hypergeometric series

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.

Pochhammer symbol for rising factorial. Named after Leo August Pochhammer (1841 – 1920):
$$(z)_k:=z(z+1)\cdots(z+k-1)$$ with the convention $(z)_0:=1$ if $z\neq0$. Note that $(1)_k=k!$. We have $$\Gamma(z+k)=(z)_k\Gamma(z)\quad\text{where}\quad\Gamma(z):=\int_0^\infty t^{z-1}\mathrm{e}^{-t}\mathrm{d}t.$$ When $z=n$ then this boils down to $(n)_k=(n+k-1)!/(n-1)!$.

Hypergeometric function. If $a\in\mathbb{R}^p$, $b\in\mathbb{R}^q$, and $z\in\mathbb{C}$, $|z|<1$, then, when it makes sense,
$${}_pF_q\begin{pmatrix}a_1,\ldots,a_p\\b_1,\ldots,b_q\\z\end{pmatrix}:=\sum_{k=0}^\infty\frac{(a_1)_k\dots(a_p)_k}{(b_1)_k\cdots(b_q)_k}\frac{z^k}{k!},
$$
The formula for ${}_pF_q$ remains valid for more values of $z$ by analytic continuation. Hypergeometric functions where studied by many including notably Leonhard Euler (1707 – 1783) and Carl-Friedrich Gauss (1777 – 1855). This kind of special function contains several others, for instance

  • ${}_2F_1(1,1;2;-z)=\frac{\log(1+z)}{z}$
  • ${}_2F_1(a,b;b;z)=\frac{1}{(1-z)^a}$
  • ${}_2F_1(\frac{1}{2},\frac{1}{2};\frac{3}{2};z^2)=\frac{\arcsin(z)}{z}$

It is also possible to embed Jacobi orthogonal polynomials 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.

Hypergeometric functions admit integral representations, and conversely, certain integrals can be computed using hypergeometric functions. Here is the most basic example.

Euler integral representation formula for ${}_2F_1$ published in 1769. If $a>0$, $b>0$, $|z|\leq 1$, then
$$\int_0^1u^{a-1}(1-u)^{b-1}(1-zu)^{-c}\mathrm{d}u
={}_2F_1\begin{pmatrix}a,c\\a+b\\z\end{pmatrix}
\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$ In other words, for all $a,b,c$ with $c>a>0$ and all $|z|\leq 1$,
$${}_2F_1\begin{pmatrix}a,b\\c\\z\end{pmatrix}=\frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_0^1u^{a-1}(1-u)^{c-a-1}(1-zu)^{-b}\mathrm{d}u.$$

A proof. A binomial series expansion of $(1-zu)^{-c}$ gives
$$
\int_0^1u^{a-1}(1-u)^{b-1}(1-zu)^{-c}\mathrm{d}u
=\sum_{k=0}^\infty\frac{(c)_k}{k!}z^k
\int_0^1u^{a+k-1}(1-u)^{b-1}\mathrm{d}u.
$$
Now 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
$$
\int_0^1u^{a-1}(1-u)^{b-1}(1-zu)^{-c}\mathrm{d}u
=\Gamma(b)\sum_{k=0}^\infty\frac{(c)_k\Gamma(a+k)}{\Gamma(a+b+k)}
\frac{z^k}{k!}.
$$
Finally the formula $\Gamma(z+k)=(z)_k\Gamma(z)$ gives
$$
\int_0^1u^{a-1}(1-u)^{b-1}(1-zu)^{-c}\mathrm{d}u
=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}\sum_{k=0}^\infty\frac{(c)_k(a)_k}{(a+b)_k}
\frac{z^k}{k!}\\
=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}\ {}_2F_1\begin{pmatrix}a,c\\a+b\\z\end{pmatrix}.
$$

Immediate corollary. By sending $z$ to $1$, taking $b>c$, 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 $c>a+b$, $$\sum_{k=0}^\infty\frac{(a)_k(b)_k}{(c)_k}\frac{1}{k!}={}_2F_1\begin{pmatrix}a,b\\c\\1\end{pmatrix}=\frac{\Gamma(c-a-b)\Gamma(c)}{\Gamma(c-a)\Gamma(c-b)}.$$

Maple, Mathematica, and Maxima. All implement hypergeometric functions. Here is an example with the Euler integral formula with Mathematica:

In[1]:= Integrate[u^{a-1}*(1 - u)^{b-1}*(1 - z*u)^{-c}, {u, 0, 1}]

Out[1]= {ConditionalExpression[
Gamma[a] Gamma[b] Hypergeometric2F1Regularized[a, c, a + b, z],
Re[a] > 0 && Re[b] > 0 && (Re[z] <= 1 || z ∉ ℝ)]}

The regularized ${}_2F_1$ hypergeometric function used by Mathematica is ${}_2F_1(a,b;c;z)/\Gamma(c)$.

Further reading.

Last Updated on 2020-08-21

2 Comments

Online conferences

Online RMTA 2020
Online RMTA 2020

It is a great time for scientific conferences over the Internet, a discovery for many colleagues and communities. Ideally, such events should be organized using software platforms implementing virtual reality, just like for certain video games, with virtual buildings, virtual rooms, virtual characters, virtual discussions, virtual restaurants, etc. Unfortunately, such platforms do not seem to be available yet with the expected level of quality and flexibility, even if I have heard that some colleagues from PSL University managed to organize a virtual poster session using a software initially designed for virtual museums!

Online scientific conferences are very easy to organize, cost almost nothing, and have an excellent carbon footprint in comparison with traditional face-to-face on-site conferences. The carbon footprint of Internet is not zero, but the fraction that is used for an online scientific event has nothing to do with what is used for an on-site conference with plenty of participants coming from far away by airplanes. The main drawback of online scientific events is of course the limited interactions between participants, and the fact that they do not extract the participants from their daily life and duties. Online conferences are an excellent way to maintain the social link inside scientific communities. The term webinar/webconference is sometimes used but emphasizes the web which is not the heart of the concept.

It would be unfortunate to make all scientific conferences online. The best would be to reduce the number of traditional conferences, and try to increase their quality, for instance by asking participants to stay longer. Also I would not be surprised to see the development of blended or hybrid conferences mixing on-site participants and remote online participants.

I had recently the opportunity to co-organize with a few colleagues an online scientific conference on Random Matrices and Their Applications, in replacement of a conventional on-site face-to-face conference in New York canceled due to COVID-19. The initial conference was supposed to last a whole week, with about thirty talks and a poster session. For the online replacement, we have decided to keep the same week. About twelve of the initial speakers accepted to give an online talk. For simplicity, we have then decided to put three 45 minutes talks per day on Monday, Tuesday, Thursday, and Friday, and to give up the poster session. We have used the same schedule for each day, with a first talk at 10 am New York local time. We had between 80 and 150 participants per talk, from all over the world. In short:

  • Schedule. Few talks per day, compatible with as many local times as possible
  • Website. Speakers, titles, abstracts, slides, registration
  • Talks. To improve speakers experience, turn-on few cameras along the talks, typically the chairperson/organizers/coauthors. The integrated text chat can be used for questions
  • Workspace. An online collaborative workspace in parallel is useful. A private channel can host the discussions between organizers, replacing emails, a general channel can host the interaction with the speakers and the participants, and between them, etc. 

On the technical side, we have decided to use current social standards instead of best quality solutions, namely Dokuwiki for the website, Zoom for the talks, and Slack for the workspace.

Last Updated on 2020-08-21

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