Log-concavity and log-convexity in the theory of the Graham--Knuth--Patashnik recurrences

What the math gives to ML

The paper gives a six-parameter family of nonnegative triangular recurrences whose row coefficients are coefficientwise strongly log-concave, while the row-generating polynomials are coefficientwise log-convex and Hankel-TP2. The transferable asset is a differentiable parameterization of positive, unimodal, variation-controlled mixing weights with algebraic guarantees that survive symbolic expansion. A practical neural use is to replace unconstrained learned lag or scale weights in a temporal mixer, state-space layer, or local attention module by recurrence-generated weights. This supplies a cheap structural prior against oscillatory or multi-peaked kernels while retaining learnable content through six parameters per head or channel group.