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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.

By Nickolas Andersen, Jeffrey C. Lagarias and Robert C. Rhoades.

Verified locator: Page 1, equation (1.1) and footnotes 1–2: classical Maass forms, moderate growth, sign conventions.