TeachingSpecial Functions – Hypergeometric functions
The Pochhammer symbol or the shifted factorial \((a)_n\) is defined by
\[(a)_n:=a(a+1)(a+2)\cdots(a+n-1),\quad n=1,2,3,\ldots\quad\text{and}\quad(a)_0:=1.\]A hypergeometric function is the sum of a hypergeometric series, which is defined as follows.
Definition: A series \(\sum c_n\) is called hypergeometric if the ratio \(\displaystyle\frac{c_{n+1}}{c_n}\) is a rational function of \(n\).
By factorization this means that
\[\frac{c_{n+1}}{c_n}=\frac{(n+a_1)(n+a_2)\cdots(n+a_p)z}{(n+b_1)(n+b_2)\cdots(n+b_q)(n+1)},\quad n=0,1,2,\ldots.\]The factor \(z\) appears because the polynomials involved need not to be monic. The factor \((n+1)\) in the denominator is convenient in the sequel. This factor may result from the factorization, or it may not. If not, this extra factor can be compensated by one of the factors \((n+a_i)\) in the numerator (choose \(a_i=1\) for some \(i\)).
Iteration leads to
\[c_n=\frac{(a_1)_n(a_2)_n\cdots(a_p)_nz^n}{(b_1)_n(b_2)_n\cdots(b_q)_n\,n!}\,c_0,\quad n=0,1,2,\ldots.\]Hence
\[\sum_{n=0}^{\infty}c_n=c_0\sum_{n=0}^{\infty}\frac{(a_1)_n(a_2)_n\cdots(a_p)_n}{(b_1)_n(b_2)_n\cdots(b_q)_n}\cdot\frac{z^n}{n!}.\]This leads to:
Definition: The hypergeometric function \({}_pF_q(a_1,a_2,\ldots,a_p;\,b_1,b_2,\ldots,b_q;\,z)\) is defined by means of a hypergeometric series as
\[{}_pF_q\left(\genfrac{}{}{0pt}{}{a_1,\,a_2,\,\ldots,\,a_p}{b_1,\,b_2,\,\ldots,\,b_q}\,;\,z\right) =\sum_{n=0}^{\infty}\frac{(a_1)_n(a_2)_n\cdots(a_p)_n}{(b_1)_n(b_2)_n\cdots(b_q)_n}\cdot\frac{z^n}{n!}.\]Of course, the parameters must be such that the denominator factors in the terms of the series are never zero. When one of the numerator parameters \(a_i\) equals \(-N\), where \(N\) is a nonnegative integer, the hypergeometric function is a polynomial in \(z\). Otherwise, the radius of convergence \(\rho\) of the hypergeometric series is given by
\[\rho=\left\{\begin{array}{ll}\infty&\text{if}\quad p < q+1\\[2.5mm]1&\text{if}\quad p=q+1\\[2.5mm]0&\text{if}\quad p > q+1.\end{array}\right.\]This follows directly from the ratio test. In fact, we have
\[\lim\limits_{n\to\infty}\left|\frac{c_{n+1}}{c_n}\right|=\left\{\begin{array}{ll} 0&\text{if}\quad p < q+1\\[2.5mm]|z|&\text{if}\quad p=q+1\\[2.5mm]\infty&\text{if}\quad p > q+1.\end{array}\right.\]In the case that \(p=q+1\) the situation that \(|z|=1\) is of special interest.
The hypergeometric series \({}_{q+1}F_q(a_1,a_2,\ldots,a_{q+1};\,b_1,b_2,\ldots,b_q;\,z)\) with \(|z|=1\) converges absolutely if \(\text{Re}\left(\sum b_i-\sum a_j\right)>0\).
The series converges conditionally if \(|z|=1\) with \(z\ne 1\) and \(-1<\text{Re}\left(\sum b_i-\sum a_j\right)\leq0\) and the series diverges if \(\text{Re}\left(\sum b_i-\sum a_j\right)\leq-1\).
Last modified on May 15, 2021
