Mehler–Fock transform

Summary

The Mehler–Fock transform is an integral transform on the half-line $[1, \infty)$ whose kernel is a conical Legendre function $P_{-1/2 + i\tau}(x)$. It is an index transform: the transform variable $\tau$ sits in the degree of the Legendre function, not in its argument. It plays for the Legendre operator the role that the Fourier transform plays for $d^2/dx^2$. It is used to solve boundary value problems for cones and in harmonic analysis on the hyperbolic plane.