{"id":"e-b0148bc76d6ae768","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-4172472abe953cea","premise_count":0,"theorem":"index-dependent scalar term: P_(k+1) = (L + k s) P_k with L f = (a + b t) f + kappa (t - h) f' gives T = U_g o R_s (RealRooted basisMap_factorization_of_scalarIndex_recurrence), where R_s(u^k) = u (u + s) ... (u + (k-1) s) keeps every root <= 0 for s >= 0 (Su, Yang and Zhang, Australas. J. Combin. 56 (2013), Theorem 4; Brenti, Mem. AMS 413 (1989), Theorem 2.4.3; RealRooted NewtonParametersAffine.preservesNonposRooted) and U_g(u^k) = L^k g sends (-inf, a + b h + kappa m] into (-inf, h] (RealRooted rayBasisMap_maps_realRootedIn_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"5219a7754ac70c79604351eb41bbd647aac1194098e00630282c1af52f8a3f92","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":2630,"certificate_compressed_size":1080,"conclusion_name":"Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, 0]","statement":"(-inf, 0] -> (-inf, 0] through R_s, s = 4, and U_g"}],"premises":[],"cited_by":[{"node_id":"n-4172472abe953cea","claim":"basis_map_preserver"}]}