= Shifted polyharmonic Maass form
{title2=$(\Delta-\lambda)^mF=0$}
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 require
$$
(\Delta-\lambda)^mF=0
$$
for an integer $m\ge1$, together with automorphy and the specified growth conditions at each <cusp (automorphic form)>. 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.<Shifted polyharmonic Maass forms for PSL(2,Z) by Nickolas Andersen and others>[Introduction, equations 1.1–1.2 and growth hypothesis]{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 $F=\partial_sE|_{s=2}$ obeys
$$
(\Delta+2)F=-3E(z,2),\qquad(\Delta+2)^2F=0.
$$
The first right-hand side is nonzero, so this gives depth two. Parameter differentiation introduces a generalized eigenfunction while preserving automorphy.<Shifted polyharmonic Maass forms for PSL(2,Z) by Nickolas Andersen and others>[section 6, Eisenstein Taylor coefficients]{ref}
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