{"id":"e-dee2eaa32f0764ef","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"hadamard_multiplier\",\"factor\":{\"terms\":[{\"kind\":\"inverse_factorial\"}]}}"},"operation_detail":"hadamard_multiplier, shift 0","conclusion_id":"n-8b66c9f49e6d2f0a","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":["RealRooted.isMultiplierSequence_inv_factorial"],"certificate_hash":"975d63d962b2ecfb2f6ecd281bb351919562b81890c7df46f83e525337eb0a6d","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":11940,"certificate_compressed_size":3064,"conclusion_name":"Interpolating the factorial and the powers of 2. Triangle read by rows, T(n, k) for 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-71965e79e47a4526","role":"sequence","name":"Triangle of coefficients in expansion of (1+2*x)^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-8b66c9f49e6d2f0a","claim":"real_rooted"},{"node_id":"n-8b66c9f49e6d2f0a","claim":"one_signed_coefficients"},{"node_id":"n-8b66c9f49e6d2f0a","claim":"hadamard_kernel"},{"node_id":"n-8b66c9f49e6d2f0a","claim":"interlacing"}]}