{"id":"e-c807fb938e5557d1","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-695d6cb541611b1e","premise_count":1,"theorem":"gamma criterion (RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff): if P = sum_j gamma_j x^j (1 + x)^(d - 2j) with deg Gamma <= d/2, then P is nonzero and real-rooted with roots in (-inf, 0] if and only if Gamma(z) = sum_j gamma_j z^j is; Gamma(c z) with c != 0 has the roots of Gamma divided by c","lean_route":null,"lean_declarations":[],"certificate_hash":"0aa9d3e3dc90a5883662cd21290a8b4933eea1e6fdbd7633430cd406a1c3ce15","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"gamma_polynomial","certificate_raw_size":10482,"certificate_compressed_size":2390,"conclusion_name":"gamma_polynomial of Pascal-(1,7,1) array.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-1e2f1b7f566b371c","role":"sequence","name":"Pascal-(1,7,1) array.","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-695d6cb541611b1e","claim":"real_rooted"},{"node_id":"n-695d6cb541611b1e","claim":"one_signed_coefficients"},{"node_id":"n-695d6cb541611b1e","claim":"hadamard_kernel"}]}