{"id":"e-42bdce22a2faaa51","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-8a7a8aaa440e4f07","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"767826d2236da4e1467ecbfd36354745b9288affa6e8b6181a222ab703e39a98","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":16285,"certificate_compressed_size":3124,"conclusion_name":"derivative of Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).","claims":[{"claim":"real_rooted","from_index":1,"region":{}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2ae6a2215aff79d4","role":"sequence","name":"Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"0"},"upper":{"closed":true,"value":"1"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (0, 1]"}],"cited_by":[{"node_id":"n-8a7a8aaa440e4f07","claim":"real_rooted"},{"node_id":"n-8a7a8aaa440e4f07","claim":"interlacing"}]}