Complex logistic wavelets: Exact normalization via Bernoulli, Genocchi and Eulerian numbers

We introduce a new family of normalized complex logistic mother wavelets generated by higher-order derivatives of a shifted complex logistic function. Explicit formulas are derived for their Fourier transforms and derivative polynomials, the latter being expressed in terms of Eulerian numbers through the Riccati equation. Closed-form normalization constants are obtained using Bernoulli and Genocchi numbers, yielding normalized wavelets of arbitrary order. We prove that the proposed functions satisfy the wavelet admissibility condition and therefore constitute a new analytically tractable class of complex mother wavelets. In contrast to most commonly used complex wavelets, the entire construction is given in explicit closed form, including the Fourier transform, normalization constants, and higher-order derivatives. The third-order complex logistic wavelet is presented as an illustrative example. The proposed family provides a mathematically transparent framework for continuous wavelet analysis and offers promising perspectives for applications in signal processing, time-frequency analysis, and wavelet coherence.