Hypergeometric Functions: From Euler to Appell and Beyond

This is an excerpt of a post written by Tigran Ishkhanyan and originally published on the Wolfram Blog. The full post can be viewed here.

Hypergeometric series appeared in the mid-seventeenth century; since then, they have played an important role in the development of mathematical and physical theories. Most of the elementary and special functions are members of the large hypergeometric class.

Hypergeometric functions have been a part of Wolfram Language since Version 1.0. The following plot shows the implementation timeline of different hypergeometric functions during the evolution of our system:

The Gauss hypergeometric 2F1, Kummer hypergeometric 1F1 and confluent hypergeometric 0F1 functions were implemented in Wolfram Language Version 1.0, and in Versions 3.0, 4.0 and 7.0, powerful updates were made that implemented four very general functions: the generalized hypergeometric pFq function, the “monster” superfunction MeijerG, the AppellF1 function and the so-called q-hypergeometric function, implemented as QHypergeometricPFQ. All these general functions significantly increased the integration, summation and other symbolic manipulation capabilities of Wolfram Language.

During the last three years, we have made a strong effort to implement the remaining computable hypergeometric functions. Three Appell functions (AppellF2, AppellF3 and AppellF4) were implemented in Version 13.3; further generalization of MeijerG — the FoxH function — was implemented a little earlier, in Version 12.3; and, finally, for Version 14.0, we’re presenting the doubly infinite hypergeometric function of one variable — the so-called bilateral hypergeometric function — as BilateralHypergeometricPFQ.

A Bit of History

The term “hypergeometric series” appears to have first been used by John Wallis in his 1655 book Arithmetica Infinitorum, and then these hypergeometric series were treated by Leonhard Euler.

Starting from the works of Carl Gauss and continuing with Ernst Kummer, Bernhard Riemann, Paul Appell and other great scholars, these functions were systematically studied, along with the differential equations they satisfy and their vast applications in different engineering, physical and other applications.

Hypergeometric Series

A hypergeometric series is a power series, seen here…

…where the ratio of successive coefficients is a rational function of n:

Let’s take a look at the Taylor series of the exponential function:

Calculate the ratio of successive coefficients (this can be done via DiscreteRatio):

This ratio is obviously a rational function of n, and for this case A(n) = 1, B(n) = n + 1, hence the Taylor series of Exp is hypergeometric.

In fact, various well-known series are hypergeometric, so having a comprehensive theory of such series-based functions is interesting as well as very useful in different areas of science. So let’s switch to the class of hypergeometric functions and start with the leading one — the generalized hypergeometric function pFq — and then move on to the well-known Kummer 1F1 and Gauss 2F1 hypergeometric functions that frequently arise in different physical and mathematical applications.

The Generalized Hypergeometric Function

The main function of the hypergeometric class is the generalized hypergeometric function pFq, which is defined by the following series:

…where (ai)n is the Pochhammer symbol or the rising factorial.

The ratio of successive terms of pFq is obviously rational:

The generalized hypergeometric function pFq is implemented in Wolfram Language as HypergeometricPFQ[a;b;z]. Here, the number of parameters in the a and b lists is not fixed; they might even be empty lists.

The q-analog of pFq is the basic hypergeometric function rΦs, which has the series expansion…

…where (a;q)n is the q-Pochhammer symbol. The basic hypergeometric function rΦs, implemented in Wolfram Language as QHypergeometricPFQ, becomes the generalized hypergeometric function pFq in the limit q → 1.

pFq plays an important role in the theory of differential equations. A large set of ordinary differential equations (ODEs) can be solved in terms of pFq functions (we refer to such equations as hypergeometric ODEs). Following, we present such an ODE that is solved in terms of pFq functions:

pFq has a well-developed theory and various fundamental applications in science (one might take a look at the Applications section of the HypergeometricPFQ reference page).

Another remarkable application example is the trinomial equation xn — x + t = 0 that, in the general form, is solved in terms of pFq functions:

The trinomial equation has n roots. Let’s generate one of them for, say, n = 5 and t = 2:

Now we generate a table of five solutions and check that they really solve the trinomial equation:

pFq is extensively used for integration and summation as well as for symbolic expression simplification. For example, here is a seemingly simple integration example:

And here is an example of an infinite sum:

Other hypergeometric functions can be written in terms of HypergeometricPFQ:

The following table shows some special cases of pFq:

Although pFq is a very general and important function, its special cases are even more popular. They significantly affected mathematical and physical theories of the nineteenth and twentieth centuries. Two of the most famous special cases are the Gauss hypergeometric function 2F1 and the Kummer confluent hypergeometric function 1F1.

Want to dive deeper in hypergeometric functions? Check out the original post for so much more!