A sy-stematicanalysisofthe spectral moments method is presenred anddeveloped to compute the response functions of very large harmonic systems. Convergence of the algorirhms is discussed and solutions are proposed to improve lhe results obtained. New developments are proposed. They concern, on the one hand, thedetermination ofthe Green funcrionsorcorrelarion functions ofthe system, and the localizationofeigenvectors and. on theotherhand. thedeterminationofthespectraldensityofvery large homogeneousmatrices by a very simple and powerful technique. These results are illustrated by several examples taken from the main subjects studied by rhe aurhors: conducting polymers. fractals and quasi-crystals. Then comparison with other merhods is discussed.
We provide a systematic method to compute arithmetic sums including some previously computed by Alaca, Besge, Cheng, Glaisher, Huard, Lahiri, Lemire, Melfi, Ou, Ramanujan, Spearman and Williams. Our method is based on quasimodular forms. This extension of modular forms has been constructed by Kaneko and Zagier.
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