Hypergeometric Differential Equation - Gauss' Contiguous Relations

Gauss' Contiguous Relations

The six functions

are called contiguous to 2F1(a,b;c;z). Gauss showed that 2F1(a,b;c;z) 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 Differential Equation

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)

    “Consciousness, we shall find, is reducible to relations between objects, and objects we shall find to be reducible to relations between different states of consciousness; and neither point of view is more nearly ultimate than the other.”
    —T.S. (Thomas Stearns)