Application To The Hypergeometric Equation
See also: Legendre equation and Hypergeometric equationThe Mehler–Fock transform concerns the eigenfunction expansion associated with the Legendre differential operator D
on (1,∞). The eigenfunctions are the Legendre functions
with eigenvalue λ ≥ 0. The two Mehler–Fock transformations are
and
(Often this is written in terms of the variable τ = √λ.)
Mehler and Fock studied this differential operator because it arose as the radial component of the Laplacian on 2-dimensional hyperbolic space. More generally, consider the group G = SU(1,1) consisting of complex matrices of the form
with determinant |α|2 − |β|2 = 1.
Read more about this topic: Spectral Theory Of Ordinary Differential Equations
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