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interlacing conclusion","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"phi(n, D) = 1 + (1) D","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"],["2","4"]],"recurrence":{"denominator":{"coeffs":[["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["1","2","1"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["-4","-4"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = phi(n, D) = 1 + (1) D, for every n >= 0"},"parent_proof":{"route":"liu_wang_three_term","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":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n"},"verdict":"proved"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"parent_proof":{"attempts":[{"detail":"the leader is not a nonzero 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 + 1)) P_(n-1) - c_n (x + 1)^2 P_(n-2) for n >= 3, with a_n = (0)/(n^2 - 3 n + 2), b_n = (2 n^2 - 4 n)/(n^2 - 3 n + 2), c_n = (n^2 - n)/(n^2 - 3 n + 2); Q_n(y) = y^(d_n) P_n(1/y - 1), d_n = deg P_1 + n - 1, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2); open hypotheses: leading_coefficient_nonzero: fails at n = 3: f(n) a_n = 0 = 0 there (exact); first_coefficient_positive: a_3 or c_3 is undefined or a_3 = 0; recurrence_coefficient_positive: fails at n = 4: the denominator of beta_n = n^4 - 6 n^3 + 11 n^2 - 6 n vanishes there; earlier_seeds: P_0 is 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; 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 = 1 (not a disproof): fails at n = 3: [x^1] W_n = -6 there (exact); derivative_or_lag_coefficient_negative: 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 = 1 (not a disproof): fails at n = 3: [x^1] W_n = -6 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":"P_n = A_n(x) P_(n-1) + B_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, B_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: P_0 is zero; every other P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); for n >= 2, P_(n-1) interlaces P_n (weakly, in the sense of Liu and Wang), and deg P_n - deg P_(n-1) is 0 or 1 for n >= 3","route":"liu_wang_three_term","success":true,"variant":"liu_wang_three_term"}],"conditional_on":[],"interlacing":{"from_index":2,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = A_n(x) P_(n-1) + B_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, B_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","P_0 is zero; every other P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); for n >= 2, P_(n-1) interlaces P_n (weakly, in the sense of Liu and Wang), and deg P_n - deg P_(n-1) is 0 or 1 for n >= 3"],"theorem":"liu_wang_three_term: L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction; Proposition 3.3 (the case B_n(x) = -(c_n x + d_n)^2)"},"kind":"weak","orientation":"previous_interlaces_next"},"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 = phi(n, D) = 1 + (1) D 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"}