Bilateral hypergeometric series

In mathematics, a bilateral hypergeometric series is a series Σa_n summed over all integers n, and such that the ratio

a_n/a_n+1

of two terms is a rational function of n. The definition of the generalized hypergeometric series is similar, except that the terms with negative n must vanish; the bilateral series will in general have infinite numbers of non-zero terms for both positive and negative n.

The bilateral hypergeometric series fails to converge for most rational functions, though it can be analytically continued to a function defined for most rational functions. There are several summation formulas giving its values for special values where it does converge.

Definition[edit]

The bilateral hypergeometric series _pH_p is defined by

{}_{p}H_{p}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{p};z)={}_{p}H_{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 _pH_q 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[edit]

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

\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[edit]

Dougall's bilateral sum[edit]

{}_{2}H_{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)}}

(Dougall 1907)

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[edit]

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

{}_{3}H_{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)}}