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constant","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"not applicable: expected one nonzero lag term","route":"product_recurrence","success":false,"variant":"product_recurrence"},{"detail":"no theorem matched (weighted_renewal_window: LHS must be a nonzero constant; fixed_delay_predecessor_window: LHS must be a nonzero constant)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"HypothesisFailed: P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 4, with a_n = (0)/(n^2 - 5 n + 6), b_n = (2 n^2 - 6 n)/(n^2 - 5 n + 6), c_n = (n^2 - n)/(n^2 - 5 n + 6); Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_2 + n - 2, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2); open hypotheses: leading_coefficient_nonzero: fails at n = 4: f(n) a_n = 0 = 0 there (exact); first_coefficient_positive: a_4 or c_4 is undefined or a_4 = 0; recurrence_coefficient_positive: fails at n = 5: the denominator of beta_n = n^4 - 8 n^3 + 19 n^2 - 12 n vanishes there; earlier_seeds: P_0 is zero or not real-rooted with every root in (-inf, 0) (exact); root_upper_bound: not attempted: the real-rootedness hypotheses do not all hold","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 >= 4, with A_n(x) = ((2 n^2 - 6 n)/(n^2 - 5 n + 6)) x, V_n(x) = 0, W_n(x) = ((-n^2 + n)/(n^2 - 5 n + 6)) x^2; open hypotheses: lag_degree_step_nonnegative: fails at n = 4: mu_n = (-n^2 + n)/(n^2 - 5 n + 6) = -6 there (exact); degree_step_positive: fails at n = 4: lambda_n + mu_n = (n^2 - 5 n)/(n^2 - 5 n + 6) = -2 there (exact); constant_term_positive: fails at n = 4: A_n(0) = (0)/(n^2 - 5 n + 6) = 0 there 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rows, checked exactly)","route":"liu_wang_three_term","success":true,"variant":"liu_wang_three_term"}],"conditional_on":[],"route":"liu_wang_three_term","rows_checked":10,"statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"proved: Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (0)) for every n >= 0, the side conditions hold for every n, and every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]"},"kind":"definition_equality"}