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)

    “When one walks, one is brought into touch first of all with the essential relations between one’s physical powers and the character of the country; one is compelled to see it as its natives do. Then every man one meets is an individual. One is no longer regarded by the whole population as an unapproachable and uninteresting animal to be cheated and robbed.”
    —Aleister Crowley (1875–1947)