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E = sum_nu H_nu S^nu L' (with any remainder removed by row annihilators of the parent), where L' is the parent rule at P'_n = P_(n+0) (valid from base 1) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = f(n, theta) = 1 + (2) theta + (1) theta^2; so the candidate rows satisfy the input rule at every base n >= 1 (the exceptional bases checked exactly), and agreement on the 2 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"interlacing":{"reason":"the parent theorem does not cover every parent row used","status":"none"},"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 2, 1] is fixed; exact check: real-rooted = true, coefficients 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vanish)","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":0,"input_interlacing":{"reason":"the parent theorem does not cover every parent row used","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"f(n, theta) = 1 + (2) theta + (1) theta^2","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-4"],["3"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","-2"],["6","4"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"],["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (2) theta + (1) theta^2, for every n >= 0"},"parent_proof":{"route":"liu_wang_derivative_lag","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 multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n"},"verdict":"proved"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"parent_interlacing_report":{"conclusion":{"first_index":1,"interlacing":"strict","interlacing_from":2,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":true,"statement":"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 has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 3; the rule is applied from n = 3 on in the induction (start moved 1 row(s) past the supplied seeds; P_0, ..., P_2 enter as seed rows, checked exactly)"},"hypotheses":[{"evidence":"f(n) = n^2 + 3 n - 4 != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) = n^2 + 3 n - 4 at n = 3 + u have one sign)","name":"left_factor_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["14","9","1"]},"numerator":["-4","3","1"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 3","status":{"kind":"verified"}},{"evidence":"checked exactly","name":"seed_roots_negative","statement":"P_1 and P_2 are nonzero and real-rooted with every root < 0","status":{"kind":"verified"}},{"evidence":"degrees 0 and 1, leading coefficients of one sign","name":"seed_degree_step","statement":"deg P_2 = deg P_1 + 1, with leading coefficients of one sign","status":{"kind":"verified"}},{"evidence":"Bezout test, exact","name":"seed_strict_interlacing","statement":"P_1 strictly interlaces P_2 (simple real roots, no common root)","status":{"kind":"verified"}},{"evidence":"checked exactly for P_0, ..., P_0 (1 zero)","name":"earlier_seeds","statement":"the seed rows P_0, ..., P_0 are each zero or real-rooted with every root in (-inf, 0)","status":{"kind":"verified"}},{"evidence":"lambda_n = (n^2 + 5 n + 4)/(n^2 + 3 n - 4) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of lambda_n = (n^2 + 5 n + 4)/(n^2 + 3 n - 4) at n = 3 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["-4","3","1"],"method":{"kind":"taylor_shift","shifted_denominator":["14","9","1"],"shifted_numerator":["28","11","1"]},"numerator":["4","5","1"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"lambda_n = [x^1] A_n + (deg P_(n-1)) [x^2] V_n > 0 for every n >= 3 (deg P_n = deg P_(n-1) + 1 with leading coefficients of one sign)","status":{"kind":"verified"}},{"evidence":"[x^0] c_1 and [x^0] c_2 are zero polynomials","name":"multipliers_vanish_at_zero","statement":"V_n(0) = 0 and W_n(0) = 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"A_n(0) = (n^2 + 6 n + 8)/(n^2 + 3 n - 4) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of A_n(0) = (n^2 + 6 n + 8)/(n^2 + 3 n - 4) at n = 3 + u have one sign)","name":"constant_term_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["-4","3","1"],"method":{"kind":"taylor_shift","shifted_denominator":["14","9","1"],"shifted_numerator":["35","12","1"]},"numerator":["8","6","1"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"A_n(0) > 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"V_n(x) = gamma(n) w(x) / f(n) with w(x) = -3 x^2 + 3 x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = (2/3 n + 1)/(n^2 + 3 n - 4) >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = (2/3 n + 1)/(n^2 + 3 n - 4) at n = 3 + u have one sign)","name":"derivative_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["-4","3","1"],"method":{"kind":"taylor_shift","shifted_denominator":["14","9","1"],"shifted_numerator":["3","2/3"]},"numerator":["1","2/3"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"V_n(x) <= 0 for every x < 0 and every n >= 3","status":{"kind":"verified"}},{"evidence":"W_n = 0","name":"lag_coefficient_nonpositive","statement":"W_n(x) <= 0 for every x < 0 and every n >= 3","status":{"kind":"verified"}},{"evidence":"V_n(x) = gamma(n) w(x) / f(n) with w(x) = -3 x^2 + 3 x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = (2/3 n + 1)/(n^2 + 3 n - 4) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = (2/3 n + 1)/(n^2 + 3 n - 4) at n = 3 + u have one sign)","name":"derivative_or_lag_coefficient_negative","sign_decision":{"certificate":{"condition":"positive","denominator":["-4","3","1"],"method":{"kind":"taylor_shift","shifted_denominator":["14","9","1"],"shifted_numerator":["3","2/3"]},"numerator":["1","2/3"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"for every n >= 3 and x < 0, V_n(x) < 0 or W_n(x) < 0 (V_n < 0 on (-inf, 0) for every n, or W_n < 0 there for every n)","status":{"kind":"verified"}}],"lean_declarations":["RealRooted.derivLag_strictInterl_of_nonnegCoeffs","RealRooted.derivLag_strictInterl_and_noCommonRoot","RealRooted.derivLag_eval_zero_pos"],"lean_replay":"not_attempted","lean_route":"halfGrowth","normal_form":"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) = ((n^2 + 6 n + 8)/(n^2 + 3 n - 4)) + ((3 n^2 + 4 n - 2)/(n^2 + 3 n - 4)) x, V_n(x) = ((2 n + 3)/(n^2 + 3 n - 4)) x + ((-2 n - 3)/(n^2 + 3 n - 4)) x^2, W_n(x) = 0","recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-4"],["3"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","-2"],["6","4"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"],["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","schema":"polytool.recurrence-real-rootedness-theorem.v1","start_delay":1,"status":"proved","theorem":"liu_wang_derivative_lag"},"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":"not applicable: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_2(n, x) P_(n-2) with both terms nonzero","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"HypothesisFailed: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 2, with a_n(x) = ((n^2 + 6 n + 8)/(n^2 + 3 n - 4)) + ((3 n^2 + 4 n - 2)/(n^2 + 3 n - 4)) x and b_n(x) = ((0)/(n^2 + 3 n - 4)) + ((2 n + 3)/(n^2 + 3 n - 4)) x + ((-2 n - 3)/(n^2 + 3 n - 4)) x^2; open hypotheses: earlier_seeds: P_0 is zero or not real-rooted with every root in (-inf, 0) (exact)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"induction start moved 1 row(s) past the seeds: 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) = ((n^2 + 6 n + 8)/(n^2 + 3 n - 4)) + ((3 n^2 + 4 n - 2)/(n^2 + 3 n - 4)) x, V_n(x) = ((2 n + 3)/(n^2 + 3 n - 4)) x + ((-2 n - 3)/(n^2 + 3 n - 4)) x^2, W_n(x) = 0: 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 has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 3; the rule is applied from n = 3 on in the induction (start moved 1 row(s) past the supplied seeds; P_0, ..., P_2 enter as seed rows, checked exactly)","route":"liu_wang_derivative_lag","success":true,"variant":"liu_wang_derivative_lag"}],"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) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 3, with A_n(x) = ((n^2 + 6 n + 8)/(n^2 + 3 n - 4)) + ((3 n^2 + 4 n - 2)/(n^2 + 3 n - 4)) x, V_n(x) = ((2 n + 3)/(n^2 + 3 n - 4)) x + ((-2 n - 3)/(n^2 + 3 n - 4)) x^2, W_n(x) = 0","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 has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 3; the rule is applied from n = 3 on in the induction (start moved 1 row(s) past the supplied seeds; P_0, ..., P_2 enter as seed rows, checked exactly)"],"theorem":"liu_wang_derivative_lag: L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)"},"kind":"strict","orientation":"previous_interlaces_next"},"route":"liu_wang_derivative_lag","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 = f(n, theta) = 1 + (2) theta + (1) theta^2 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"}