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

where

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[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

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]

(Dougall 1907)

This is sometimes written in the equivalent form

Bailey's formula[edit]

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

where

See also[edit]

References[edit]