Hypergeometric Function - 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 Function

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)

    “The land is the appointed remedy for whatever is false and fantastic in our culture. The continent we inhabit is to be physic and food for our mind, as well as our body. The land, with its tranquilizing, sanative influences, is to repair the errors of a scholastic and traditional education, and bring us to just relations with men and things.”
    —Ralph Waldo Emerson (1803–1882)