{"id":"e-e3b364f94f508324","operation":{"factor":"1/k!","operation":"hadamard_factor"},"operation_detail":"1/k!","conclusion_id":"n-f25b32370a8d627e","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":"be95ee08374957235d078ec3104ccf9c2cdaffb31dee941ec1cc87a4554b5647","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"hadamard_factor","certificate_raw_size":6471,"certificate_compressed_size":1832,"conclusion_name":"Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d938c1937a93c4df","role":"sequence","name":"Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 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]"}],"cited_by":[{"node_id":"n-f25b32370a8d627e","claim":"real_rooted"},{"node_id":"n-f25b32370a8d627e","claim":"one_signed_coefficients"},{"node_id":"n-f25b32370a8d627e","claim":"hadamard_kernel"}]}