 Polyharmonic Maass form

In the integer-depth convention, a polyharmonic Maass form is a smooth automorphic function annihilated by a positive integer power of its weight-appropriate hyperbolic Laplacian, with specified growth at the cusps. In weight zero, using $\Delta =-y^2({\partial }_x^2+{\partial }_y^2)$, the equation is ${\Delta }^{m}F=0$. It is the zero-shift case of a shifted polyharmonic Maass form. Exact depth is the least such integer $m$. Some refined conventions also use half-integer depths, so the convention must be stated.^[ref]

A global depth-two example comes from the finite part of a completed nonholomorphic Eisenstein series. Near $s=1$, writewhere $c$ is a nonzero constant and $\Delta E^{\ast }=s(1-s)E^{\ast }$. Comparing constant Laurent coefficients gives $\Delta F=-c$, hence ${\Delta }^{2}F=0$ but $\Delta F\ne 0$. The finite part is automorphic and has moderate growth.^[ref]

This demonstrates why higher depth is substantive: the space contains functions that are not harmonic but become harmonic after one application of the Laplacian. Merely solving a local differential equation is insufficient; automorphy and cusp growth remain part of the definition.