In mathematics, a bilateral hypergeometric series is a series Σ''a''_''n'' summed over ''all'' integers ''n'', and such that the ratio :''a''_''n''/''a''_''n''+1 of two terms is a

rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...

of ''n''. The definition of the

generalized hypergeometric series In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which ...

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

The bilateral hypergeometric series _''p''H_''p'' is defined by :

_pH_p(a_1,\ldots,a_p;b_1,\ldots,b_p;z)=
_pH_p\left(\begina_1&\ldots&a_p\\b_1&\ldots&b_p\\ \end;z\right)=
\sum_^\infty
\fracz^n

where :

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

is the

rising factorial In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...

Pochhammer symbol In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...

. 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 _''p''H_''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

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 mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b( ...

in ''z'' similar to the hypergeometric differential equation.

Summation formulas

Dougall's bilateral sum

_2H_2(a,b;c,d;1)= \sum_^\infty\frac= \frac

This is sometimes written in the equivalent form :

\sum_^\infty
\frac  = 
\frac 
\frac .

Bailey's formula

gave the following generalization of Dougall's formula: :

_3H_3(a,b, f+1;d,e,f;1)=
 \sum_^\infty\frac= \lambda\frac