Bilateral hypergeometric series

Definition

The bilateral hypergeometric series pHp is defined by :{}_pH_p(a_1,\ldots,a_p;b_1,\ldots,b_p;z)= {}pH_p\left(\begin{matrix}a_1&\ldots&a_p\b_1&\ldots&b_p\ \end{matrix};z\right)= \sum{n=-\infty}^\infty \frac{(a_1)_n(a_2)_n\ldots(a_p)_n}{(b_1)_n(b_2)_n\ldots(b_p)_n}z^n

where

:(a)_n=a(a+1)(a+2)\cdots(a+n-1),

is the rising factorial or Pochhammer symbol.

Usually the variable z is taken to be 1, in which case it is omitted from the notation. It is possible to define the series pHq with different p and q in a similar way, but this either fails to converge or can be reduced to the usual hypergeometric series by changes of variables.

Convergence and analytic continuation

Suppose that none of the variables a or b are integers, so that all the terms of the series are finite and non-zero. Then the terms with n0 diverge if |z| 1, so the series cannot converge unless |z|=1. When |z|=1, the series converges if :\Re(b_1+\cdots b_n -a_1-\cdots - a_n) 1.

The bilateral hypergeometric series can be analytically continued to a multivalued meromorphic function of several variables whose singularities are branch points at z = 0 and z=1 and simple poles at a**i = −1, −2,... and b**i = 0, 1, 2, ... This can be done as follows. Suppose that none of the a or b variables are integers. The terms with n positive converge for |z| 1 to a function satisfying an inhomogeneous linear equation with singularities at z = 0 and z=1, so can also be continued to a multivalued function with these points as branch points. The sum of these functions gives the analytic continuation of the bilateral hypergeometric series to all values of z other than 0 and 1, and satisfies a linear differential equation in z similar to the hypergeometric differential equation.

Summation formulas

Dougall's bilateral sum

: {}2H_2(a,b;c,d;1)= \sum{n=-\infty}^\infty\frac{(a)_n(b)_n}{(c)_n(d)_n}= \frac{\Gamma(d)\Gamma(c)\Gamma(1-a)\Gamma(1-b)\Gamma(c+d-a-b-1)}{\Gamma(c-a)\Gamma(c-b)\Gamma(d-a)\Gamma(d-b)}

This is sometimes written in the equivalent form :\sum_{n=-\infty}^\infty \frac {\Gamma(a+n) \Gamma(b+n)}{\Gamma(c+n)\Gamma(d+n)} = \frac {\pi^2}{\sin (\pi a) \sin (\pi b)} \frac {\Gamma (c+d-a-b-1)}{\Gamma(c-a) \Gamma(d-a) \Gamma(c-b) \Gamma(d-b)}.

Bailey's formula

gave the following generalization of Dougall's formula:

: {}3H_3(a,b, f+1;d,e,f;1)= \sum{n=-\infty}^\infty\frac{(a)_n(b)_n(f+1)_n}{(d)_n(e)_n(f)_n}= \lambda\frac{\Gamma(d)\Gamma(e)\Gamma(1-a)\Gamma(1-b)\Gamma(d+e-a-b-2)}{\Gamma(d-a)\Gamma(d-b)\Gamma(e-a)\Gamma(e-b)} where : \lambda=f^{-1}\left[(f-a)(f-b)-(1+f-d)(1+f-e)\right].

References

• (there is a 2008 paperback with )