  The calculation of pair correlations and density profiles of quasiholes are
routine steps in the study of proposed fractional quantum Hall states.
Nevertheless, the field has not adopted a standard way to present the results
of such calculations in an easily reproducible form. We develop a polynomial
expansion that allows for easy quantitative comparison between different
candidate wavefunctions, as well as reliable scaling of correlation and
quasihole profiles to the thermodynamic limit. We start from the well-known
expansion introduced by Girvin [PRB, 30 (1984)] (see also [Girvin, MacDonald
and Platzman, PRB, 33 (1986)]), which is physically appealing but, as we
demonstrate, numerically unstable.0 We orthogonalize their basis set to obtain
a new basis of modified Jacobi polynomials, whose coefficients can be stably
calculated. We then apply our expansion to extract pair correlation expansion
coefficients and quasihole profiles in the thermodynamic limit for a wide range
of fractional quantum Hall wavefunctions. These include the Laughlin series,
composite fermion states with both reverse and direct flux attachment, the
Moore-Read Pfaffian state, and BS hierarchy states. The expansion procedure
works for both abelian and non-abelian quasiholes, even when the density at the
core is not zero. We find that the expansion coefficients for all quantum Hall
states considered can be fit remarkably well using a cosine oscillation with
exponentially decaying amplitude. The frequency and the decay length are
related in an intuitive, but not elementary way to the filling fraction.
Different states at the same filling fraction can have distinct values for
these parameters. Finally, we also use our scaled correlation functions to
calculate estimates for the magneto-roton gaps of the various states.
