{"id":"e-d98f7e0074b379c4","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-ba8de4e02acfb6ca","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":"d11932a577c4e0f12e701ff5c5e3cfffc316b028d43d004de6620734b655a217","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":2697,"certificate_compressed_size":1057,"conclusion_name":"Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[],"cited_by":[{"node_id":"n-ba8de4e02acfb6ca","claim":"real_rooted"}]}