On a remarkable functional equation in the theory of generalized Dunkl operators and transformations of elliptic genera
type: Item
instance of: scholarly article
title: On a remarkable functional equation in the theory of generalized Dunkl operators and transformations of elliptic genera
zbMATH Open document ID: 0870.39009
DOI: 10.1007/PL00004572
published in: Mathematische Zeitschrift
publication date: 1997-07-03T00:00:00Z
full work available at URL: https://eudml.org/doc/174940
review text: The functional equation \[ f(x)(g(x+y)+g(x-y))+f(y)(g(x+y)-g(x-y))=0, \] arising from the theory of the generalised Dunkl operators [cf. \textit{C. F. Dunkl}, Trans. Am. Math. Soc. 311, No. 1, 167-183 (1989; Zbl 0652.33004)] is considered. It turns out that \(f\) and \(g\) in general are the certain elliptic functions related by the composition of the Landen's and imaginary Jacobi's transformations (LJ-transformation). This leads to some new relations in the theory of elliptic genera in topology, which imply certain divisibility relations of a new type for the signatures of the stably almost complex manifold \(M^{4n}\) and its canonical virtual submanifolds \(\overline P_q[M^{4n}]\), corresponding to Pontryagin classes of the tangent bundle.
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 58J26
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 1028272
zbMATH Keywords: Landen transformation
zbMATH Keywords: Jacobi transformation
zbMATH Keywords: functional equation
zbMATH Keywords: Dunkl operators
zbMATH Keywords: elliptic functions
zbMATH Keywords: elliptic genera
zbMATH Keywords: divisibility relations
zbMATH Keywords: Pontryagin classes
zbMATH Keywords: tangent bundle
author: V. M. Buchstaber
author: A. P. Veselov
MaRDI profile type: Publication
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OpenAlex ID: W1991087146
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description: scientific article; zbMATH DE number 1028272