# Hypergeometric functions are key

From Orthogonal Polynomials and Special Functions by Richard Askey:

At first the results we needed were in the literature but after a while we ran out of known results and had to learn something about special functions. This was a very unsettling experience for there were very few places to go to really learn about special functions. At least that is what we thought. Actually there were many, but the typical American graduate education which we had did not include anything about hypergeometric functions. And hypergeometric functions are the key to this subject, as I have found out after many years of fighting them.

Askey’s book was written in 1975, and he was describing his experience from ten years before that. Special functions, and in particular hypergeometric functions, went from being common knowledge among mathematicians at the beginning of the 20th century to being arcane by mid century.

I learned little about special functions and nothing about hypergeometric functions as a graduate student. I first ran into hypergeometric functions reading in Concrete Mathematics how they are used in combinatorics and in calculating sums in closed form. Then when I started working in statistics I found that they are everywhere.

Hypergeometric functions are very useful, but not often taught anymore. Like a lot of useful mathematics, they fall between two stools. They’re considered too advanced or arcane for the undergraduate curriculum, and not a hot enough area of research to be part of the graduate curriculum.

## 2 thoughts on “Hypergeometric functions are key”

1. Hi, John.

“They’re considered too advanced or arcane for the undergraduate curriculum, . . .”

So true. In fact, the same can probably be said of fractional derivatives. One of your re-tweets (Oct 20, Dan Piponi) was my introduction to the concept of fractional derivatives. Mind blown! I had no idea such concepts existed. And now that I know about it, I am still not aware of what level of school this material is taught. Is this graduate level material? Masters? PhD? Or does a student have to specifically pursue this field to learn about it? I am certain there is a lot more behind this and we are only just scratching the surface.

2. If you read physics literature from before (say) 1930, a lot of the effort went into simplifying the model in such a way that it was solvable in terms of special functions. Now physicists just find a numerical analyst to give them a graph. It’s not super surprising that special functions waned in importance, but you allude to a good point: Math education is *super* homogeneous in the US. Everyone uses the same texts and takes the same set of classes. I think it would be awesome if a department said “mathletes from our school will learn a disjoint set of tools from everywhere else-continued fractions, special functions, elliptic curves-and they won’t take all the standard classes”. It’d be like going to football summer camp just for kickers-all teams need them but it’s not popular enough to get a full camp running.