{"id":"e-b87132cd573fb9e2","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-b31aa1a3fd93d174","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":"83e7ad4e4ce3b9b5b1ca32b94dd205c6f0771a0bc7433064eb03fa37d37ce223","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3485,"certificate_compressed_size":1201,"conclusion_name":"Delannoy paths counted by number of weak peaks.","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[],"cited_by":[{"node_id":"n-b31aa1a3fd93d174","claim":"real_rooted"}]}