Weighted Binomial Transforms of Signatures of Linear Recurrent Sequences

This preprint introduces and develops the weighted binomial transform of signatures associated with linear recurrent sequences. Extending the ordinary binomial transform through a scalar weight parameter, the transformation is studied from generating-function, characteristic-root, and signature-level perspectives. The paper shows that the weighted transform preserves C-finite sequences and induces an affine transformation of characteristic roots. A general coefficient formula is derived for signatures of arbitrary order, together with explicit formulas in orders 2, 3, and 4 and an order-8 visualization of the resulting alternating Pascal structure. Examples include Fibonacci, Tribonacci, Pell-type cubic, and repeated-root signatures. The interaction between weighted binomial transforms and geometric scaling is also investigated, leading to irregular Hadamard products and a generic symmetric reduction in the number of distinct characteristic roots. The resulting affine family is related to generalized binomial interpolated operators for linear recurrent sequences. Particular attention is given to the negative-weight case, including the structural connection with Euler’s transformation of alternating series. The work extends the authors’ previous study of ordinary binomial root transforms, geometric scalings, and irregular Hadamard products by introducing a weighted affine framework and identifying further directions involving weighted binomial convolution and transforms of ordinary convolutions.