4.8: Hypergeometric Functions

Hypergeometric functions are probably the most useful, but least understood, class of functions. They typically do not make it into the undergraduate curriculum and seldom in graduate curriculum. Most functions that you know can be expressed using hypergeometric functions. There are many approaches to these functions and the literature can fill books. See for example Special Functions by G. E. Andrews, R. Askey, and R. Roy, 1999, Cambridge University Press.

In 1812 Gauss published a study of the hypergeometric series

\[ \begin{aligned} y(x)=1 &+\dfrac{\alpha \beta}{\gamma} x+\dfrac{\alpha(1+\alpha) \beta(1+\beta)}{2 ! \gamma(1+\gamma)} x^{2} +\dfrac{\alpha(1+\alpha)(2+\alpha) \beta(1+\beta)(2+\beta)}{3 ! \gamma(1+\gamma)(2+\gamma)} x^{3}+\ldots \end{aligned}\label{4.86} \]

Here \(\alpha, \beta, \gamma\), and \(x\) are real numbers. If one sets \(\alpha=1\) and \(\beta=\gamma\), this series reduces to the familiar geometric series

\[y(x)=1+x+x^{2}+x^{3}+\ldots \nonumber \]

The hypergeometric series is actually a solution of the differential equation

\[x(1-x) y^{\prime \prime}+[\gamma-(\alpha+\beta+1) x] y^{\prime}-\alpha \beta y=0 \nonumber \]

This equation was first introduced by Euler and latter studied extensively by Gauss, Kummer and Riemann. It is sometimes called Gauss’s equation. Note that there is a symmetry in that \(\alpha\) and \(\beta\) may be interchanged without changing the equation. The points \(x=0\) and \(x=1\) are regular singular points. Series solutions may be sought using the Frobenius method. It can be confirmed that the above hypergeometric series results.

A more compact form for the hypergeometric series may be obtained by introducing new notation. One typically introduces the Pochhammer symbol, \((\alpha)_{n}\), satisfying

1. \((\alpha)_{0}=1 \text { if } \alpha \neq 0.\)
2. \((\alpha)_{k}=\alpha(1+\alpha) \ldots(k-1+\alpha), \text { for } k=1,2, \ldots . \)

This symbol was introduced by Leo August Pochhammer ( \(1841-1920)\).

Consider \((1)_{n} . \text { For } n=0,(1)_{0}=1 . \text { For } n>0,\)

\[(1)_{n}=1(1+1)(2+1) \ldots[(n-1)+1] \nonumber \]

This reduces to \((1)_{n}=n !\). In fact, one can show that

\[(k)_{n}=\dfrac{(n+k-1) !}{(k-1) !} \nonumber \]

for \(k\) and \(n\) positive integers. In fact, one can extend this result to noninteger values for \(k\) by introducing the gamma function:

\[(\alpha)_{n}=\dfrac{\Gamma(\alpha+n)}{\Gamma(\alpha)}\nonumber \]

We can now write the hypergeometric series in standard notation as

\[{ }_{2} F_{1}(\alpha, \beta ; \gamma ; x)=\sum_{n=0}^{\infty} \dfrac{(\alpha)_{n}(\beta)_{n}}{n !(\gamma)_{n}} x^{n}\nonumber \]

For \(\gamma>\beta>0\), one can express the hypergeometric function as an integral:

\[{ }_{2} F_{1}(\alpha, \beta ; \gamma ; x)=\dfrac{\Gamma(\gamma)}{\Gamma(\beta) \Gamma(\gamma-\beta)} \int_{0}^{1} t^{\beta-1}(1-t)^{\gamma-\beta-1}(1-t x)^{-\alpha} d t\nonumber \]

Using this notation, one can show that the general solution of Gauss’ equation is

\[y(x)=A_{2} F_{1}(\alpha, \beta ; \gamma ; x)+B x^{1-\gamma}{ }_{2} F_{1}(1-\gamma+\alpha, 1-\gamma+\beta ; 2-\gamma ; x)\nonumber \]

By carefully letting \(\beta\) approach \(\infty\), one obtains what is called the confluent hypergeometric function. This in effect changes the nature of the differential equation. Gauss \(^{\prime}\) equation has three regular singular points at \(x=0,1, \infty\). One can transform Gauss’ equation by letting \(x=u / \beta\). This changes the regular singular points to \(u=0, \beta, \infty\). Letting \(\beta \rightarrow \infty\), two of the singular points merge.

The new confluent hypergeometric function is then given as

\[{ }_{1} F_{1}(\alpha ; \gamma ; u)=\lim _{\beta \rightarrow \infty}{ }_{2} F_{1}\left(\alpha, \beta ; \gamma ; \dfrac{u}{\beta}\right)\nonumber \]

This function satisfies the differential equation

\[x y^{\prime \prime}+(\gamma-x) y^{\prime}-\alpha y=0\nonumber \]

The purpose of this section is only to introduce the hypergeometric function. Many other special functions are related to the hypergeometric function after making some variable transformations. For example, the Legendre polynomials are given by

\[P_{n}(x)=_{2} F_{1}\left(-n, n+1 ; 1 ; \dfrac{1-x}{2}\right)\nonumber \]

In fact, one can also show that

\[\sin ^{-1} x=x_{2} F_{1}\left(\dfrac{1}{2}, \dfrac{1}{2} ; \dfrac{3}{2} ; x^{2}\right)\nonumber \]

The Bessel function \(J_{p}(x)\) can be written in terms of confluent geometric functions as

\[J_{p}(x)=\dfrac{1}{\Gamma(p+1)}\left(\dfrac{z}{2}\right)^{p} e^{-i z}{ }_{1} F_{1}\left(\dfrac{1}{2}+p, 1+2 p ; 2 i z\right)\nonumber \]

These are just a few connections of the powerful hypergeometric functions to some of the elementary functions that you know.