{"id":"e-67c88de355bdef8b","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-f3e0631eb8740f76","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":"9bfed8380c7c4ba4e3927a6f7ef1caa23594be8b2be88b903b4fa5009f10235c","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4469,"certificate_compressed_size":1354,"conclusion_name":"Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-f3e0631eb8740f76","claim":"real_rooted"},{"node_id":"n-f3e0631eb8740f76","claim":"interlacing"},{"node_id":"n-f3e0631eb8740f76","claim":"one_signed_coefficients"},{"node_id":"n-f3e0631eb8740f76","claim":"hadamard_kernel"}]}