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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.

Newton binomial series. It states that for any z,αC with |z|<1, 1(1z)α=n=0(α)nznn!where(α)n:=α(α+1)(α+n1) is the Pochhammer symbol for rising factorial named after Leo August Pochhammer (1841 - 1920), with the convention (α)0=1 if α0. Note that (1)n=n! and Γ(α+n)=(α)nΓ(α)whereΓ(α):=0tα1etdt. When α=m then this boils down to (m)n=(m+n1)!/(n1)!.

Hypergeometric functions. 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 aRp, bRq, and zC, |z|<1, then, when it makes sense,
pFq(a1,,apb1,,bqz):=n=0(a1)n(ap)n(b1)n(bq)nznn!.
We could use from this formula 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

  • 2F1(1,1;2;z)=log(1+z)z
  • 2F1(a,b;b;z)=1(1z)a
  • 2F1(12,12;32;z2)=arcsin(z)z

It is also possible to embed Jacobi orthogonal polynomials into hypergeometric functions and thus several families of orthogonal polynomials, more precisely 2F1(n,a+1+b+n;a+1;x)=n!(a+1)nP(a,b)n(12x). Note that (z)k=0 for large enough k when z is a negative integer, hence pFq(a;b;z) is a polynomial when one of the ai 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 2F1 published in 1769. If a>0, b>0, |z|1, then
10ua1(1u)b1(1zu)cdu=2F1(a,ca+bz)Γ(a)Γ(b)Γ(a+b) In other words, for all a,b,c with c>a>0 and all |z|1,
2F1(a,bcz)=Γ(c)Γ(a)Γ(ca)10ua1(1u)ca1(1zu)bdu.

A proof. A binomial series expansion of (1zu)c gives
10ua1(1u)b1(1zu)cdu=k=0(c)kk!zk10ua+k1(1u)b1du.
Now the beta-gamma formula 10ua+k1(1u)b1du=Γ(a+k)Γ(b)Γ(a+b+k) gives
10ua1(1u)b1(1zu)cdu=Γ(b)k=0(c)kΓ(a+k)Γ(a+b+k)zkk!.
Finally the formula Γ(z+k)=(z)kΓ(z) gives
10ua1(1u)b1(1zu)cdu=Γ(a)Γ(b)Γ(a+b)k=0(c)k(a)k(a+b)kzkk!=Γ(a)Γ(b)Γ(a+b) 2F1(a,ca+bz).

Immediate corollary. By sending z to 1, taking b>c, and using the beta-gamma formula 10ua1(1u)bc1du=Γ(a)Γ(bc)Γ(a+bc), we obtain the following identity discovered by Gauss (1812), for all a,b,c with c>a+b, k=0(a)k(b)k(c)k1k!=2F1(a,bc1)=Γ(cab)Γ(c)Γ(ca)Γ(cb).

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 2F1 hypergeometric function used by Mathematica is 2F1(a,b;c;z)/Γ(c).

Further reading.

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