{"chains":[{"edge":{"id":"e-0732c8d06b4db3af","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-2d155baa50423f8c","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":"3c051f7d230ef13833407f271ed9c87bd88a81a36c75a432834a8239c8277daa","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2821c21537bbbcdd","role":"sequence","name":"Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (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]"}]},"premises":[{"premise":{"position":0,"node_id":"n-2821c21537bbbcdd","role":"sequence","name":"Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (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]"},"chains":[{"edge":{"id":"e-3cb5f3d64c4bb317","operation":{"map":"A056242","operation":"basis_map"},"operation_detail":"A056242","conclusion_id":"n-2821c21537bbbcdd","premise_count":1,"theorem":"affine current multiplier with an affine initial row (module documentation, \"The affine-current theorem\"): with beta = b/d, K = beta^2 + a beta - c > 0 and eta = m/d >= 0, every real polynomial with all roots in E = [-beta - sqrt(K), -beta + sqrt(K)] has a real-rooted image, and no interval of positive length outside E works (Theorem 1); for eta > 0 the images lie in I = [(-a - 2 beta - 2 sqrt(K))/d, (-a - 2 beta + 2 sqrt(K))/d] if and only if eta >= 1/2 and (s - (2 eta - 1) beta)^2 <= (2 eta - 1)^2 K, s = l - eta a (Theorem 2); with a >= 0, K >= beta^2, a + 2 beta >= 0, a^2 + 4c >= 0, l >= 0 and the scalar criterion, PF inputs with roots in [-R, 0], R = beta + sqrt(K), have PF images, sharply (the common-interlacer proof; the Chebyshev case is Iserles and Saff, Math. Proc. Cambridge Philos. Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"02cb55ae901e2ce091cc2f0a0b356e3fb98e174fbdf710f8981844185c7f7a6b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-1bf99ad40f056d33","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-1bf99ad40f056d33","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]"},"chains":[]}]}]}]}],"truncated":false,"expanded":2}