Shifted polyharmonic Maass form ($(\Delta -\lambda {)}^{m}F=0$)

A shifted polyharmonic Maass form is a smooth automorphic function that satisfies a generalized eigenfunction equation for a hyperbolic Laplacian. In weight zero, choose the convention $\Delta =-y^2({\partial }_x^2+{\partial }_y^2)$ and requirefor an integer $m\ge 1$, together with automorphy and the specified growth conditions at each cusp. Moderate growth means at most polynomial growth in cusp coordinates. The weight-$k$ version replaces $\Delta$ by the corresponding Weight-k hyperbolic Laplacian and uses the weight-$k$ transformation law.^[ref]

The least possible $m$ is the exact depth. For $m=1$ this is an ordinary eigenfunction condition; higher depth allows a chain of generalized eigenfunctions. It is not merely an equation with a different eigenvalue. The cited paper uses the opposite Laplacian sign, so its eigenvalue parameter is the negative of the one used here.

For an illuminating construction, let a nonholomorphic Eisenstein series $E(z,s)$ satisfy $\Delta E=s(1-s)E$. At a regular value $s=2$, differentiate in $s$. The function $F={\partial }_{s}E{|}_{s=2}$ obeysThe first right-hand side is nonzero, so this gives depth two. Parameter differentiation introduces a generalized eigenfunction while preserving automorphy.^[ref]