= Robert C. Rhoades
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<Robert C. Rhoades> coauthored work extending the study of <Maass forms> beyond a single <eigenfunction> equation. With <Jeffrey C. Lagarias>, he studied <polyharmonic Maass forms>: functions with the required arithmetic symmetry and growth conditions that are annihilated by a power of a weight-dependent <hyperbolic Laplacian>, $\Delta_k^m f=0$. Their paper constructs bases using derivatives of <Eisenstein series> with respect to a parameter.<Polyharmonic Maass forms for PSL(2,Z) by Jeffrey C. Lagarias and Robert C. Rhoades>[preprint pages 1–4, Introduction and Theorem 1.1]{ref}
An elementary analogy explains why taking a power changes the problem. For the <derivative> operator $D=d/dt$, the <function> $f(t)=t$ satisfies $D^2f=0$ but $Df=1\ne0$. Thus requiring two applications to give zero allows more functions than requiring one. The arithmetic setting adds essential symmetry and growth restrictions; the analogy alone does not produce a <Maass form>. The paper shows how these restrictions lead to explicit finite-dimensional spaces, rather than the unrestricted solution sets of a <differential equation>.<Polyharmonic Maass forms for PSL(2,Z) by Jeffrey C. Lagarias and Robert C. Rhoades>[preprint page 2, growth restriction and basis construction]{ref}
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