Generalized Hypergeometric Function - Contiguous Function and Related Identities

Contiguous Function and Related Identities

Let be the operator . From the differentiation formulas given above, the linear space spanned by and contains each of

,

.

Since the space has dimension 2, any three of these functions are linearly dependent. These dependencies can be written out to generate a large number of identities involving .

For example, in the simplest non-trivial case,

,

,

,

So

.

This, and other important examples,

,

,

,

,

,

can be used to generate continued fraction expressions known as Gauss's continued fraction.

Similarly, by applying the differentiation formulas twice, there are such functions contained in, which has dimension three so any four are linearly dependent. This generates more identities and the process can be continued. The identities thus generated can be combined with each other to produce new ones in a different way.

A function obtained by adding to exactly one of the parameters in is called contiguous to . Using the technique outlined above, an identity relating and its two contiguous functions can be given, six identities relating and any two of its four contiguous functions, and fifteen identities relating and any two of its six contiguous functions have been found. (The first one was derived in the previous paragraph. The last fifteen were given by Gauss in his 1812 paper.)