Operator is a polynomial in but

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

Let stand for , , , or and stand for , , , or respectively.  If we express as a polynomial in , then we get

where the signs depend on , , , and whether is , , , and . 

These results let us define

The coefficients of these polynomials in , , ,
and are defined when

Operators

are defined for all , , , and .