Fibonacci-Weighted Bicomplex Hardy Spaces: Reproducing Kernels, Shift Bounds, and Germ Sheaves

Motivated by the fact that the Fibonacci sequence is the simplest nontrivial second-order recurrence with a rational generating function, we develop a Fibonacci-weighted Hardy theory for bicomplex holomorphic functions. Starting from the coefficient norm ∑ n ≥ 0 | a n | 2 / F n + 1 , we obtain a bicomplex Hilbert module whose reproducing kernel is governed by ( 1 − t − t 2 ) − 1 and whose maximal disk of holomorphy is determined sharply by the nearest kernel singularity, giving the radius ρ F = φ − 1 / 2 (the square-root inverse of the golden ratio φ ). The arithmetic recurrence makes several objects fully explicit: we derive closed formulas for the kernels through the idempotent decomposition of BC , compute exact norms of the shift powers and a golden-ratio spectral radius, and package the local theory into a sheaf of Fibonacci-holomorphic germs that are compatible with the bicomplex idempotent splitting. We also treat ( p , q ) -Fibonacci weights, obtaining a one-parameter family of rational kernels ( 1 − p t − q t 2 ) − 1 and corresponding operator bounds. In addition to providing a concrete bicomplex model within weighted Hardy theory, the resulting explicit kernels furnish benchmark examples for kernel-based interpolation and for the operator theory of unilateral weighted shifts.