Normalized complex logistic wavelets: Fourier transforms and generating functions

Abstract

We introduce a family of complex logistic mother wavelets generated by higher-order derivatives of an analytically shifted logistic function. Explicit closed-form expressions are obtained for the Fourier transforms of all derivatives, while the higher-order derivatives themselves admit polynomial representations involving Eulerian numbers. Particular attention is devoted to the associated derivative energies, for which we derive an exact differential recurrence, a Rodrigues-type representation, and a generating function. At some values of the shift parameter, these energies reduce to expressions involving Bernoulli and Genocchi numbers, revealing a direct connection between the wavelet construction and classical number-theoretic sequences. These results can be seen as a generalization of the formula by Grosset and Veselov \cite{GV}, which connects a soliton solution of the KdV equation to Bernoulli numbers.The generating function also satisfies a linear wave-type equation, providing a compact description of the entire hierarchy of derivative energies. Using these exact energies as normalization constants, we construct normalized complex logistic mother wavelets of arbitrary order and prove that they satisfy the wavelet admissibility condition. The third-order wavelet corresponding to a purely imaginary logistic shift is presented as an explicit example. The resulting family provides an analytically tractable framework linking complex logistic functions, derivative-energy hierarchies, generating functions, and wavelet analysis.