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
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