{"id":"e-f1f9c50c96bd4da9","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-1eae5c7270ad0739","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":"e9a1474cd328f17f6faca71cb32d45374fb89bdd5f62750b420efbcba6d9a5ba","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4626,"certificate_compressed_size":1366,"conclusion_name":"Triangle read by rows: coefficients of a Bessel polynomial recursion: P(x, n) = 2*(n-1)*P(x, n - 1)/x - n*P(x, n - 2) with substitution x -> 1/y.","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"2"}}},{"claim":"interlacing","from_index":2,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-1eae5c7270ad0739","claim":"interlacing"},{"node_id":"n-1eae5c7270ad0739","claim":"real_rooted"}]}