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The $9j$ symbols of $\mathfrak{su}(1,1)$ are studied within the framework ofthe generic superintegrable system on the 3-sphere. The canonical basescorresponding to the binary coupling schemes of four $\mathfrak{su}(1,1)$representations are constructed explicitly in terms of Jacobi polynomials andare seen to correspond to the separation of variables in different cylindricalcoordinate systems. A triple integral expression for the $9j$ coefficientsexhibiting their symmetries is derived. A double integral formula is obtainedby extending the model to the complex three-sphere and taking the complexradius to zero. The explicit expression for the vacuum coefficients is given.Raising and lowering operators are constructed and are used to recover therelations between contiguous coefficients. It is seen that the $9j$ symbols canbe expressed as the product of the vacuum coefficients and a rational function.The recurrence relations and the difference equations satisfied by the $9j$coefficients are derived.

The Generic Superintegrable System on the 3-Sphere and the $9j$ Symbols of $\mathfrak{su}(1,1)$