It’s occasionally necessary to evaluate a hypergeometric function at a large negative argument. I was working on a project today that involved evaluating *F*(*a*, *b*; *c*; z) where *z* is a large negative number.

The hypergeometric function *F*(*a*, *b*; *c*; *z*) is defined by a power series in *z* whose coefficients are functions of *a*, *b*, and *c*. However, this power series has radius of convergence 1. This means you can’t use the series to evaluate *F*(*a*, *b*; *c*; *z*) for *z < −*1.

It’s important to keep in mind the difference between a function and its power series representation. The former may exist where the latter does not. A simple example is the function *f*(*z*) = 1/(1 − *z*). The power series for this function has radius 1, but the function is defined everywhere except at *z* = 1.

Although the series defining *F*(*a*, *b*; *c*; *z*) is confined to the unit disk, the function itself is not. It can be extended analytically beyond the unit disk, usually with a branch cut along the real axis for *z* ≥ 1.

It’s good to know that our function *can* be evaluated for large negative *x*, but *how* do we evaluate it?

## Linear transformation formulas

Hypergeometric functions satisfy a huge number of identities, the simplest of which are known as the linear transformation formulas even though they are not linear transformations of *z*. They involve *bilinear* transformations *z*, a.k.a. fractional linear transformations, a.k.a. Möbius transformations. [1]

One such transformation is the following, found in A&S 15.3.4 [2].

If *z* < 1, then 0 < *z*/(*z* − 1) < 1, which is inside the radius of convergence. However, as *z* goes off to −∞, *z*/(*z* − 1) approaches 1, and the convergence of the power series will be slow.

A more complicated, but more efficient, formula is A&S 15.3.7, a linear transformation formula relates *F* at *z* to two other hypergeometric functions evaluated at 1/*z*. Now when *z* is large, 1/*z* is small, and these series will converge quickly.

## Related posts

[1] It turns out these transformations *are* linear, but not as functions of a complex argument. They’re linear as transformations on a projective space. More on that here.

[2] A&S refers to the venerable Handbook of Mathematical Functions by Abramowitz and Stegun.