{"id":"e-68ac132d04cd33c2","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-5af4ea1ce3f85b64","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":"9e5459a32ee68fe44e27dbee7c39c6694b487b73fff3b34bbcbf7b5ad02fcadd","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":9599,"certificate_compressed_size":2406,"conclusion_name":"derivative of T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n)/k!, triangle read by rows, n >= 0 and 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-01512c1ccffbc04e","role":"sequence","name":"T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n)/k!, triangle read by rows, n >= 0 and 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-5af4ea1ce3f85b64","claim":"real_rooted"},{"node_id":"n-5af4ea1ce3f85b64","claim":"one_signed_coefficients"},{"node_id":"n-5af4ea1ce3f85b64","claim":"hadamard_kernel"}]}