On a remarkable functional equation in the theory of generalized Dunkl operators and transformations of elliptic genera (Q1358303)

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.