{"id":"e-71ee257a063243ca","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-c0db4c865e1a9a1a","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":"7202414d59eba79761eec24bc77ba50243cc1cdd5aab343e8f8d24162e8a1bf7","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":9066,"certificate_compressed_size":2374,"conclusion_name":"derivative of Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-01897e65d9bb3867","role":"sequence","name":"Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..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-c0db4c865e1a9a1a","claim":"real_rooted"},{"node_id":"n-c0db4c865e1a9a1a","claim":"one_signed_coefficients"},{"node_id":"n-c0db4c865e1a9a1a","claim":"hadamard_kernel"}]}