Hypergeometric Functions - Gauss' Contiguous Relations

Gauss' Contiguous Relations

The six functions, and are called contiguous to . Gauss showed that can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of a,b,c, and z. This gives (6
2)=15 relations, given by identifying any two lines on the right hand side of

$\begin{align} z\frac{dF}{dz} = z\frac{ab}{c}F(a+,b+,c+) &=a(F(a+)-F)\\ &=b(F(b+)-F)\\ &=(c-1)(F(c-)-F)\\ &=\frac{(c-a)F(a-)+(a-c+bz)F}{1-z}\\ &=\frac{(c-b)F(b-)+(b-c+az)F}{1-z}\\ &=z\frac{(c-a)(c-b)F(c+)+c(a+b-c)F}{c(1-z)}\\ \end{align}$

In the notation above, and so on.

Repeatedly applying these relations gives a linear relation over C(z) between any three functions of the form, where m, n, and l are integers.

Read more about this topic: Hypergeometric Functions

Famous quotes containing the words contiguous and/or relations:

“A CAUSE is an object precedent and contiguous to another, and so united with it that the idea of the one determines the mind to form the idea of the other, and the impression of the one to form a more lively idea of the other.”
—David Hume (1711–1776)

“Happy will that house be in which the relations are formed from character; after the highest, and not after the lowest order; the house in which character marries, and not confusion and a miscellany of unavowable motives.”
—Ralph Waldo Emerson (1803–1882)