{"id":"e-19b03bdc222aebaf","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-490435c99149594f","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":"c46398a81398a1edfc9c5bafe3c55c7ba68a39af076cc9626a0f6748fab9abb5","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4296,"certificate_compressed_size":1266,"conclusion_name":"Triangle T(n,k) read by rows: coefficient of [x^k] of the polynomial p_n(x)=(5-x)*p_{n-1}(x)-p_{n-2}(x), p_0=1, p_1=5-x.","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"3"},"upper":{"closed":false,"value":"7"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-490435c99149594f","claim":"interlacing"},{"node_id":"n-490435c99149594f","claim":"real_rooted"},{"node_id":"n-490435c99149594f","claim":"one_signed_coefficients"}]}