Lommel polynomials and explicitly solvable prediction problems on the unit circle

Abstract

To each finite symmetric measure $σ$ on the real line, with support a compact subset of $(-2,2)$, we associate a measure $μ$ on the unit circle by transporting mass at $\pm x$ to $e^{\pm iθ(x)}$, $θ(x)=2\arcsin(x/2)$. The linear prediction errors, Verblunsky coefficients, and Toeplitz determinants of $μ$ are then expressed through the orthogonal polynomial data of $σ$ at the single edge point $x=2$. In particular $E_m(μ)=\tfrac12 t_m\|P_m\|_σ^2$ with $t_m=P_{m+1}(2)/P_m(2)$. Under a condition on the first coefficients, decay of the recurrence coefficients of $σ$ forces the $t_m$ to increase from $t_1$ onward, a Turán-type monotonicity in the degree placing every prediction margin of $μ$ past the first above the corresponding coefficient of $σ$. Taking $σ$ to be the Lommel-polynomial measures, with atoms at rescaled reciprocals of the zeros of the Bessel function $J_ν$, yields a two-parameter family of purely atomic circle measures with a normalized determinant limit equal to a value of $J_ν$.

Disclosure

“Disclosure of AI Tool Use This research was human-directed and carried out with assistance from AI tools. The systems used were Claude Opus 4.8 and Claude Fable 5 (Anthropic), and GPT 5.5 and GPT 5.6 Sol (OpenAI). Prompted by the author, Claude orchestrated high-precision numerical experiments in Python using mpmath, and the identification central to Theorem 3 emerged serendipitously in the course of a question the author had posed for other reasons. Claude was used interactively”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T05:11:58.203986Z

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