{"id":"e-1670beb9ff565a7b","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-bb74f03367de3ec8","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":"5cc56d7b26d2cf55f54af55c12097e4bf1433776cb51b2ed3985aea6116980f2","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3745,"certificate_compressed_size":1302,"conclusion_name":"Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[],"cited_by":[{"node_id":"n-bb74f03367de3ec8","claim":"real_rooted"}]}