Here we derive contiguity relations for the Gauss hypergeometric function. This function is known to Maple as:
It is for
When considering the summand, we introduce the following algebra:
The bivariate sequence vanishes at both of the following operators:
(General algorithms exist to find such operators.)
From the previous first-order recurrences, we derive relations on in the mixed differential difference algebra.
The equations are obtained by multiplication of the recurrences by , followed by summation over all non-negative . Formally, this corresponds to changing into and into .
Therefore, we set
The linear differential operator is called a step-up operator. It relates the forward shift of to derivatives of by the following equation.
The elimination of between this step-up operator and the differential equation yields a contiguity relation for --a purely recurrence equation. It is obtained by the extended skew gcd algorithm:
In other words, the Gauss hypergeometric function satisfies the following equation:
More interestingly, the extended Euclidean algorithm yields a step-down operator for - a relation between an inverse shift of and its derivatives. This is obtained by computing an inverse of modulo .
From this result, we have and . In particular,
is the step-down operator: