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P(1/x), d_n = 0 + 1 (n - 0), for every n >= 0"},"parent_proof":{"route":"hypergeometric_row","statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]","input_root_region":{"upper":{"closed":true,"value":"0"}},"parent_hypothesis":"every parent row used is zero or real-rooted with all roots in (-inf, 0] (coefficients of one sign)","parent_root_region":{"upper":{"closed":true,"value":"0"}},"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically"},"verdict":"proved"},"notes":["the division remainder is removed by 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n >= 2, with A_n(x) = ((-n^2 + 2 n - 1)/(n^2 + 5 n + 6)) + ((8 n^2 + 5 n - 1)/(n^2 + 5 n + 6)) x, V_n(x) = ((9 n + 3)/(n^2 + 5 n + 6)) x + ((-9 n - 3)/(n^2 + 5 n + 6)) x^2, W_n(x) = ((2 n^2 - 6 n + 4)/(n^2 + 5 n + 6)) + ((4 n^2 - 12 n + 8)/(n^2 + 5 n + 6)) x + ((2 n^2 - 6 n + 4)/(n^2 + 5 n + 6)) x^2; open hypotheses: seed_roots_negative: P_1 is zero, not real-rooted, or has a root >= 0 (exact); derivative_degree_step_nonnegative: fails at n = 12: lambda_n = (-n^2 + 11 n + 2)/(n^2 + 5 n + 6) = -1/21 there (exact); multipliers_vanish_at_zero: [x^0] c_2(n, x) = 2 n^2 - 6 n + 4 is not the zero polynomial; constant_term_positive: fails at n = 2: A_n(0) = (-n^2 + 2 n - 1)/(n^2 + 5 n + 6) = -1/20 there (exact); lag_coefficient_nonpositive: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 0 (not a disproof): fails at n = 3: [x^0] W_n = 2/15 there (exact)","route":"liu_wang_derivative_lag","success":false,"variant":"liu_wang_derivative_lag"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + gamma(n) (x - L)(x - U) P'_(n-1)","route":"ma_wang_interval","success":false,"variant":"ma_wang_interval"},{"detail":"Unresolved: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + B_n(x) P_(n-2) for n >= 2, with A_n(x) = ((-n^2 + 2 n - 1)/(n^2 + 5 n + 6)) + ((8 n^2 + 5 n - 1)/(n^2 + 5 n + 6)) x, V_n(x) = ((9 n + 3)/(n^2 + 5 n + 6)) x + ((-9 n - 3)/(n^2 + 5 n + 6)) x^2, B_n(x) = ((2 n^2 - 6 n + 4)/(n^2 + 5 n + 6)) + ((4 n^2 - 12 n + 8)/(n^2 + 5 n + 6)) x + ((2 n^2 - 6 n + 4)/(n^2 + 5 n + 6)) x^2; open hypotheses: degree_law: l_n = ([x^1] A_n + deg P_(n-1) [x^2] V_n) l_(n-1) + ([x^2] B_n) l_(n-2) when deg P_(n-1) - deg P_(n-2) = 1 (then deg P_n - deg P_(n-1) = 1): neither nonnegative step coefficients (fails at n = 12: s_n = -1/21 there (exact); t_n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of t_n at n = 2 + u have one sign); s_n + t_n > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of s_n + t_n at n = 2 + u have one sign)) nor t_n <= 0 for a positive-ratio induction (fails at n = 3: t_n = 2/15 there (exact)) (not a disproof); lag_coefficient_nonpositive: neither everywhere (the quadratic test p_2 <= 0, p_0 <= 0, p_1^2 <= 4 p_2 p_0 does not hold for every n (not a disproof): fails at n = 3: p_2 = [x^2] B_n = 2/15 there (exact); fails at n = 3: p_0 = [x^0] B_n = 2/15 there (exact); p_1^2 - 4 p_2 p_0 <= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of p_1^2 - 4 p_2 p_0 at n = 2 + u have one sign)) nor on x <= 0 (B_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] B_n <= 0 fails for k = 0 (not a disproof): fails at n = 3: [x^0] B_n = 2/15 there (exact)); root_location: seed_roots_negative: a seed of the induction has a root >= 0 (exact); constant_terms_positive: P_n(0) = 0 at n = 1 (exact); nonnegative_coefficients: fails at n = 2: [x^0] A_n = -1/20 there (exact); [x^1] A_n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of [x^1] A_n at n = 2 + u have one sign); [x^0] B_n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of [x^0] B_n at n = 2 + u have one sign); [x^1] B_n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of [x^1] B_n at n = 2 + u have one sign); [x^2] B_n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of [x^2] B_n at n = 2 + u have one sign); [x^1] V_n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of [x^1] V_n at n = 2 + u have one sign); fails at n = 2: [x^2] V_n = -21/20 there (exact)","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"},{"detail":"row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, n >= 0, with p = (-1) (k - 1) (k) (k + 1), q = (-1) (k - n - 3) (k - n - 2) (k - n - 1); valuation v(n) = 1, degree d(n) = n: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0]: for n >= 3 the coefficients U_j (U_j = [x^(j + v(n))] P_n, v(n) = 1) satisfy U_j = U_0(n) prod_(t <= j) R(n, t), R(n, j) = (-1) (j - n - 2) (j - n - 1) (j - n) / ((j) (j + 1) (j + 2)), so sum_j U_j y^j is U_0(n) times the image of (1 + y)^(n - 1) under a nonnegative multiplier sequence, and U_0(n) != 0 for n >= 3 by its scalar recurrence","route":"hypergeometric_row","success":true,"variant":"hypergeometric_row"}],"conditional_on":[],"route":"hypergeometric_row","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 = x^(d_n) P(1/x), d_n = 0 + 1 (n - 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":"preserver_image"}