Bilateral hypergeometric series

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 n<0 diverge if |z| <1, and the terms with n>0 diverge if |z| >1, so the series cannot converge unless |z|=1. When |z|=1, the series converges if

${\displaystyle \mathrm{\Re }(b_1+\cdots b_n-a_1-\cdots -a_n)>1.}$

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 be continued to a multivalued function with these points as branch points. Similarly the terms with n negative 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

${\displaystyle {}_2{H}_2(a,b;c,d;1)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n(d{)}_n}=\frac{\mathrm{\Gamma }(d)\mathrm{\Gamma }(c)\mathrm{\Gamma }(1-a)\mathrm{\Gamma }(1-b)\mathrm{\Gamma }(c+d-a-b-1)}{\mathrm{\Gamma }(c-a)\mathrm{\Gamma }(c-b)\mathrm{\Gamma }(d-a)\mathrm{\Gamma }(d-b)}}$

(Dougall 1907)

This is sometimes written in the equivalent form

${\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }(a+n)\mathrm{\Gamma }(b+n)}{\mathrm{\Gamma }(c+n)\mathrm{\Gamma }(d+n)}=\frac{{\pi }^2}{\sin (\pi a)\sin (\pi b)}\frac{\mathrm{\Gamma }(c+d-a-b-1)}{\mathrm{\Gamma }(c-a)\mathrm{\Gamma }(d-a)\mathrm{\Gamma }(c-b)\mathrm{\Gamma }(d-b)}.}$

Bailey's formula

(Bailey 1959) gave the following generalization of Dougall's formula:

${\displaystyle {}_3{H}_3(a,b,f+1;d,e,f;1)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n(f+1{)}_n}{(d{)}_n(e{)}_n(f{)}_n}=\lambda \frac{\mathrm{\Gamma }(d)\mathrm{\Gamma }(e)\mathrm{\Gamma }(1-a)\mathrm{\Gamma }(1-b)\mathrm{\Gamma }(d+e-a-b-2)}{\mathrm{\Gamma }(d-a)\mathrm{\Gamma }(d-b)\mathrm{\Gamma }(e-a)\mathrm{\Gamma }(e-b)}}$

where

${\displaystyle \lambda =f^{-1}\left[(f-a)(f-b)-(1+f-d)(1+f-e)\right].}$