Let

The differential equation for becomes

Now is a polynomial in but

so can also be expressed as a polynomial in terms of shift operators and converting the differential equation into a difference equation among contiguous instances of which we call a contiguity relation.  Operators

are defined if and respectively. 

If we express as a polynomial in , then we get

which has degree . 

If we express as a polynomial in , we get

which has degree at most . 

These results let us define

The coefficients of these polynomials in and are defined when

Operators and for are defined if and respectively.  Hence, is defined if and is distinct from all .  is defined if and is distinct from all .  Recall that .