{"id":"e-835093125a0f64b6","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"hadamard_multiplier\",\"factor\":{\"terms\":[{\"kind\":\"binomial\",\"index_offset\":0,\"coefficient_offset\":0}]}}"},"operation_detail":"hadamard_multiplier, shift 0","conclusion_id":"n-fabb0a68e001a863","premise_count":1,"theorem":"Polya--Schur characterization of multiplier sequences (G. Polya and I. Schur, J. Reine Angew. Math. 144 (1914) 89--113; stated as Theorem 40 of P. Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601): a real sequence (lambda_k) is a multiplier sequence (x^k -> lambda_k x^k maps real-rooted polynomials to real-rooted polynomials or zero) iff sum_k binom(m,k) lambda_k x^k is real-rooted with zeros of one sign for every m, iff sum_k lambda_k x^k / k! (or its reflection) is C x^m e^(a x) prod (1 + alpha_j x) with a, alpha_j >= 0; a product of multiplier sequences is one (composition), and a nonzero factor lambda(n) of the row changes nothing","lean_route":null,"lean_declarations":[],"certificate_hash":"45519ff56043690e7de04bf293b9b8ba042021c1604d7be50d905e469b604c4b","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":21379,"certificate_compressed_size":4275,"conclusion_name":"Triangle of cubed binomial coefficients, T(n,k) = C(n,k)^3, read by rows.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b13531ad79e7c629","role":"sequence","name":"Square the entries of Pascal's triangle.","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-fabb0a68e001a863","claim":"real_rooted"},{"node_id":"n-fabb0a68e001a863","claim":"one_signed_coefficients"},{"node_id":"n-fabb0a68e001a863","claim":"hadamard_kernel"}]}