{"id":"e-84f080937a1c1ea0","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-fea97fcbe0a965c8","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":"234e9b681ce7fa7a2c6b804276519f802fb1e11351cdc3e0916491a373ac3642","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14647,"certificate_compressed_size":2815,"conclusion_name":"basis_map_A008277 of Exponential Riordan array [x*exp(x),x].","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7b5887f058255aa1","role":"sequence","name":"Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-fea97fcbe0a965c8","claim":"real_rooted"},{"node_id":"n-fea97fcbe0a965c8","claim":"one_signed_coefficients"},{"node_id":"n-fea97fcbe0a965c8","claim":"hadamard_kernel"}]}