A set of bi-orthogonal pairs of functions is derived which is especially suited for the calculation of the gravitational field of flat galaxies. The functions are Hankel tranforms of associated Laguerre functions, or ‘Hankel-Laguerre functions’. They can be calculated by means of recurrence relations, which are derived from a generating function. Numerical computations are most efficiently performed using a related set of Chebyshev type functions, which have more economic recurrence relations. The necessary algorithms for efficient calculation of the field are derived using the properties of Hankel-Laguerre functions. This field calculation, can be used for a relatively cheap and simple computer simulation of galaxies.