{"id":"e-720cd5df5ac0e3b3","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-30cb3cad216ed189","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":"3e6458774a8c67cb661923272dc6ceaf94c4afb5bad4b844de714e388dbf7695","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3446,"certificate_compressed_size":1179,"conclusion_name":"Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-30cb3cad216ed189","claim":"interlacing"},{"node_id":"n-30cb3cad216ed189","claim":"real_rooted"},{"node_id":"n-30cb3cad216ed189","claim":"one_signed_coefficients"},{"node_id":"n-30cb3cad216ed189","claim":"hadamard_kernel"}]}