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In the weight-zero convention, a Maass form for a discrete subgroup $\Gamma \subseteq {\mathrm{SL}}_2(\mathbb{R})$ is a smooth function on the upper half-plane, invariant under $z\mapsto (az+b)/(cz+d)$, satisfying a hyperbolic Laplacian eigenvalue equation and specified growth conditions at the cusps:Additional conditions specify a cuspidal or square-integrable class. The sign of $\Delta$ is a convention and must be stated.^[ref]

The constant function $f=1$ is an invariant eigenfunction with $\lambda =0$ and moderate growth, but is not cuspidal: its average along a horizontal period is nonzero. A Maass cusp form has zero such constant term at each cusp. The eigenfunction condition allows nonholomorphic automorphic forms, connecting arithmetic symmetries with spectral geometry.

Nickolas Andersen studies analytic number theory, modular forms and L-functions. With Jeffrey C. Lagarias and Robert C. Rhoades, he studied shifted polyharmonic Maass forms, which replace an eigenfunction equation byHere ${\Delta }_k$ is a weight-dependent hyperbolic Laplacian; the functions also obey arithmetic symmetry and controlled growth. The study proves finite dimensionality and describes the role of Eisenstein series and their parameter derivatives. ^[ref][ref]

The mechanism is visible by differentiating a family of eigenfunctions. If $LE(s)=\lambda (s)E(s)$ and $L$ is independent of $s$, thenAt a fixed parameter value, applying $L-\lambda$ again gives zero. A derivative can therefore yield a generalized solution of depth two instead of another ordinary eigenfunction. The arithmetic symmetry and growth requirements still need checking; the operator identity explains the construction's motivation.

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]