{"chains":[{"edge":{"id":"e-8d805b4c99a48239","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-ad08ac6743aea7a3","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"89a7dd7cf01352965bf6f32beee34782f946961938b397c468d6f119676d9c56","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ff69dadb0811f5e6","role":"sequence","name":"Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-ff69dadb0811f5e6","role":"sequence","name":"Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"},"chains":[{"edge":{"id":"e-a526bc158f95a9e3","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-ff69dadb0811f5e6","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":"9b55553c1ca89452e512ff125a94bdbe438a3d2e9f523b6e9f6102352d2ddaf4","verification_polytool":"replayed","verification_lean":"not_attempted","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":[]},"premises":[]}]}]}],"truncated":false,"expanded":2}