Block and Residue Exclusion in Kernel-Root Expansions for Two-Barrier Walks

Abstract

We consider the two-barrier walk on the integers with steps $+y$ and $-b$, stopped on entering either absorbing set. Its finite-state generating functions are rational and admit the standard Schur-function representation in the roots of the normalized kernel $P_z(X)=X^{b+y}-2X^b+z$. Our contribution is a two-stage pruning method for constructing these root expansions without computing terms that cannot affect the final polynomial. First, the Weyl expansion is grouped into complete small-root/large-root blocks, and a $z$-adic valuation bound removes every block beginning above the denominator degree. Second, surviving blocks are split into residue subblocks and projected onto complete cyclic root-label orbits; the resulting congruences eliminate nontrivial characters without any termwise rationality test. In the $(+3,-2)$, $m=12$ example only one mixed-block degree survives, while for $(+5,-2)$, $m=22$, twenty of twenty-one Weyl terms are excluded before expansion.

arXiv comment15 pages