{"id":"e-be0b062a2a4f328c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-8dfceeb7e33f2ee0","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"547f39802875d3e9084bf80f9fc372139f0bdbb18306d8cdd34a68ab100f9bf3","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":10284,"certificate_compressed_size":2874,"conclusion_name":"theta_plus_one of shift_x_plus_one of Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d6d713aa669b74c1","role":"sequence","name":"shift_x_plus_one of Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.","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-8dfceeb7e33f2ee0","claim":"real_rooted"},{"node_id":"n-8dfceeb7e33f2ee0","claim":"one_signed_coefficients"},{"node_id":"n-8dfceeb7e33f2ee0","claim":"hadamard_kernel"}]}