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

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