The series:

with is geometric if the ratio of two consecutive terms is a constant (with respect to ):

then we get:

It is hypergeometric if the ratio is a rational function (with respect to ):

where and are polynomials in , which we can completely factor into the form

(1)

where is a constant and the factor is just a convention (if the polynomial does not contain the factor we can just add it to both numerator and denominator and absorb the “1” into ). The hypergeometric series is then given by:

where

is the rising factorial function (also called the Pochhammer symbol).

To write a function as a hypergeometric series, we simply expand it in series and then write the ratio in the form (1) and immediately identify the proper function. If the ratio cannot be put into the form (1) then the function is not hypergeometric.

If any , then the series is a polynomial of degree .

If any then the denominators eventually become 0 (unless the series is terminated as a polynomial before that, due to the previous point) and the series is undefined.

Except the previous two cases, the radius of convergence of the hypergeometric series is:

The hypergeometric functions for low and have special names:

confluent hypergeometric limit function | |

Kummer’s confluent hypergeometric function of the first kind | |

Gauss’ hypergeometric function |

Most common functions can be expressed using as follows:

Elementary functions:

Bessel function:

Spherical Bessel function of the first kind:

Modified Bessel functions:

Elementary functions:

Lower incomplete gamma function:

Error function:

Hermite polynomials:

Laguerre polynomials:

(2)

Solution of the radial Schrödinger equation in the Coulomb potential (we use (2) in the second equation below):

Elementary functions:

Legendre polynomials (and associated Legendre polynomials):

Chebyshev polynomials:

Gegenbauer polynomials:

Jacobi polynomials:

Complete elliptic integrals:

By writing out the series expansion for the ratio we can prove that:

The left hand side is equal to:

We simplify the term:

We calculate the ratio as well as to get the normalization:

From which we read the arguments of the hypergeometric function on the right hand side and we need to multiply it by the normalization factor .

By writing out the series expansion for the ratio we can prove that:

We can also use the substitution :

Which is a special case of

for .

One way to express is:

using the previous example, this is equal to:

So the lowest hypergeometric function that can express is .