Bilateral hypergeometric series

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

an/an+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 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 hypergeomtric 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 ai = −1, −2,... and bi = 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]

 {}_2H_2(a,b;c,d;1)= \sum_{-\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:

 {}_3H_3(a,b, f+1;d,e,f;1)=
 \sum_{-\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].

See also[edit]

References[edit]