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zero or not real-rooted with every root in (-inf, 0) (exact); root_region: the region contains the pole x = -1 of the coordinate change y = 1/(x + 1)","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"HypothesisFailed: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 3, with A_n(x) = ((2 n^2 - 4 n)/(n^2 - 3 n + 2)) + ((2 n^2 - 4 n)/(n^2 - 3 n + 2)) x, V_n(x) = 0, W_n(x) = ((-n^2 + n)/(n^2 - 3 n + 2)) + ((-2 n^2 + 2 n)/(n^2 - 3 n + 2)) x + ((-n^2 + n)/(n^2 - 3 n + 2)) x^2; open hypotheses: lag_degree_step_nonnegative: fails at n = 3: mu_n = (-n^2 + n)/(n^2 - 3 n + 2) = -3 there (exact); degree_step_positive: fails at n = 3: lambda_n + mu_n = (n^2 - 3 n)/(n^2 - 3 n + 2) = 0 there (exact); multipliers_vanish_at_zero: [x^0] c_2(n, x) = -n^2 + n is not the zero polynomial; 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