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
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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