Finite and infinite product transformations
Martin Nicholson
TL;DR
This paper investigates infinite products with symmetry properties, their relation to modular forms, and finite versions motivated by boundary problems, providing elementary proofs of modular transformations.
Contribution
It introduces finite product formulas related to infinite products satisfying $f(eta)=f(1/eta)$, connecting them to modular forms and boundary problem solutions.
Findings
Infinite products satisfying $f(eta)=f(1/eta)$ are studied.
Finite product formulas are derived and linked to modular transformations.
Elementary proofs of modular transformations are provided.
Abstract
Several infinite products are studied that satisfy the transformation relation of the type . For certain values of the parameters these infinite products reduce to modular forms. Finite counterparts of these infinite products are motivated by solution of Dirichlet boundary problem on a rectangular grid. These finite product formulas give an elementary proof of several modular transformations.
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Taxonomy
TopicsMathematics and Applications · semigroups and automata theory · Analytic Number Theory Research