{"id":"e-0c1e3d0b7f1bd773","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-c30cb06ad75c5e21","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"05cde71e32ebeaafad3ebe80852a052db7da3c2a443190cca30cf8eb427382b9","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":2697,"certificate_compressed_size":1052,"conclusion_name":"Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[],"cited_by":[{"node_id":"n-c30cb06ad75c5e21","claim":"real_rooted"}]}