New expansions of confluent hypergeometric function

Last week Bujanda et al published a paper [1] that gives new expansions for the confluent hypergeometric function. I’ll back up explain what that means before saying more about the new paper.

Hypergeometric functions

Hypergeometric functions are something of a “grand unified theory” of special functions. Many functions that come up in application are special cases of hypergeometric function, as shown in the diagram below. (Larger version here.)

special function relationships

I give a brief introduction to hypergeometric functions in these notes (4-page pdf).

Confluent hypergeometric functions

The confluent hypergeometric function corresponds to Hypergeometric 1F1 in the chart above [2]. In Mathematica it goes by Hypergeometric1F1 and in Python goes by scipy.special.hyp1f1.

This function is important in probability because you can use it to compute the CDFs for the normal and gamma distributions.

New expansions

Bujanda et al give series expansions of the confluent hypergeometric functions in terms of incomplete gamma functions. That may not sound like progress because the incomplete gamma functions are still special functions, but the confluent hypergeometric function is a function of three variables M(ab; z) whereas the incomplete gamma function γ(sz) is a function of two variables.

They also give expansions in terms of elementary functions which may be of more interest. The authors give the series expansion below.

M_n(a, b; z) = \frac{\Gamma(b)}{\Gamma(a)\,\Gamma(b-a)} \sum_{k=0}^{n-1} A_k(a, b)\, F_k(z)

The A functions are given recursively by

and

A_n(a, b) = \frac{2}{n} \left( (2a - b)A_{n-1}(a, b) + 2(n-b)A_{n-2}(a,b) \right) 

for n > 1.

The F functions are given by

F_n(z) = \frac{n!}{(-z)^{n+1}} \Big( e_n(z/2) - \exp(z) e_n(-z/2) \Big)

where

e_n(z) = \sum_{k=0}^n \frac{z^k}{k!}

The authors give approximation error bounds in [1]. In the plot below we show that n = 3 makes a good approximation for M(3, 4.1, z). The error converges to zero uniformly as n increases.

Related posts

[1] Blanca Bujanda, José L. López, and Pedro J. Pagola. Convergence expansions of the confluent hypergeometric functions in terms of elementary functions. Mathematics of computation. DOI https://doi.org/10.1090/mcom/3389

[2] “Confluent” is a slightly archaic name, predating a more systematic naming for hypergeometric functions. The name mFn means the hypergeometric function has m upper parameters and n lower parameters. The confluent hypergeometric function has one of each, so it corresponds to 1F1. For more details, see these notes.

2 thoughts on “New expansions of confluent hypergeometric function

  1. Nice to know! Although I would have liked to see the ratio/error rather than absolute comparison. Just a predilection of mine. Does the paper include a minimal/dominant discussion of the recursion?

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