{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-2"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","-4","0"],["1","0","0"],["1","0","7"],["0","4","2"],["0","0","-9"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","4","-32","0"],["0","3","0","0"],["0","3","-16","73"],["0","0","-4","62"],["0","0","0","-43"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","5","-38","0"],["0","0","1","0","0"],["0","0","1","-20","99"],["0","0","0","-16","108"],["0","0","0","0","-29"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","0","1","-12","0"],["0","0","0","0","0","0"],["0","0","0","0","-4","34"],["0","0","0","0","-4","42"],["0","0","0","0","0","-4"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","0","-1","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","3"],["0","0","0","0","0","0","4"]]},"deriv_order":4,"shift":0},{"coeff":{"coeffs":[["0","0"],["-1","0"],["-2","0"],["0","-4"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","-4","0"],["0","-3","0"],["0","-6","16"],["0","0","4"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","-5","0"],["0","0","-1","0"],["0","0","-2","20"],["0","0","0","16"]]},"deriv_order":2,"shift":1},{"coeff":{"coeffs":[["0","0","0","-1","0"],["0","0","0","0","0"],["0","0","0","0","4"],["0","0","0","0","4"]]},"deriv_order":3,"shift":1},{"coeff":{"coeffs":[["0"],["0"],["1"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["0","0"],["0","0"],["0","3"]]},"deriv_order":1,"shift":2},{"coeff":{"coeffs":[["0","0","0"],["0","0","0"],["0","0","1"]]},"deriv_order":2,"shift":2}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1,2],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["4"],["-4"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","36","132"],["3","-48","-284"],["-1","24","221"],["0","-4","-74"],["0","0","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-11","124"],["0","0","32","-244"],["0","0","-20","167"],["0","0","4","-46"],["0","0","0","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","1","20"],["0","0","0","0","-36"],["0","0","0","0","21"],["0","0","0","0","-4"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","-32"],["-7","48"],["2","-24"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","16"],["0","0","-32"],["0","0","20"],["0","0","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-2"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0"],["-1"],["-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","-4","0"],["0","-3","0"],["0","-3","7"],["0","0","9"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","-5","0"],["0","0","-1","0"],["0","0","-1","13"],["0","0","0","17"]]},"deriv_order":2},{"coeff":{"coeffs":[["0","0","0","-1","0"],["0","0","0","0","0"],["0","0","0","0","3"],["0","0","0","0","4"]]},"deriv_order":3}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["0"],["0"],["1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0"],["0","0"],["0","3"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0"],["0","0","0"],["0","0","1"]]},"deriv_order":2}],"shift":1}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","1"],["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"statement":"w(n) 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 >= 1"},"method":"operator_division"},"status":"certified"},"input_first_index":1,"interlacing":{"from_index":2,"justification":"the parent theorem (LiuWang) proves P_(m-1) weakly interlacing P_m for every m >= 2; T = f(n, theta) = 1 + (2) theta + (1) theta^2 is independent of n and maps every real-rooted polynomial to a real-rooted polynomial or zero, so by the Hermite--Kakeya--Obreschkoff theorem (both directions) T(P_(m-1)) and T(P_m) have weakly interlacing zeros; the nonzero gauge does not move zeros, so Q_(n-1) and Q_n interlace weakly (orientation not determined) for every n >= 2","kind":"weak","orientation":"unordered","status":"claimed"},"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 of one sign = true","name":"euler_polynomial_roots_nonpositive","statement":"the polynomial in theta is real-rooted with every root <= 0 for every n >= n0","status":{"kind":"verified"}},{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}},{"evidence":"delta(n) != 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 1 + u have one sign)","name":"operator_coefficients_defined","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"the operator coefficients are finite at every n >= n0 (their denominators do not vanish)","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":1,"input_interlacing":{"interlacing":{"from_index":2,"justification":{"premises":["the parent theorem (LiuWang) proves P_(m-1) weakly interlacing P_m for every m >= 2; T = f(n, theta) = 1 + (2) theta + (1) theta^2 is independent of n and maps every real-rooted polynomial to a real-rooted polynomial or zero, so by the Hermite--Kakeya--Obreschkoff theorem (both directions) T(P_(m-1)) and T(P_m) have weakly interlacing zeros; the nonzero gauge does not move zeros, so Q_(n-1) and Q_n interlace weakly (orientation not determined) for every n >= 2"],"theorem":"Hermite--Kakeya--Obreschkoff theorem (N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, 1963; see S. Fisk, Polynomials, Roots, and Interlacing, arXiv:math/0612833): real polynomials f, g have weakly interlacing zeros iff alpha f + beta g is real-rooted or zero for all real alpha, beta; a linear T independent of n that maps every real-rooted polynomial to a real-rooted polynomial or zero therefore maps interlacing pairs to pairs with weakly interlacing zeros (orientation not determined)"},"kind":"weak","orientation":"unordered"},"status":"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":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-2"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"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 >= 1"},"parent_proof":{"route":"liu_wang","statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 1, 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":[],"parent_interlacing_report":{"conclusion":{"first_index":1,"interlacing":"weak","interlacing_from":2,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":false,"statement":"every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Liu--Wang) for n >= 3; the rule is applied from n = 3 on in the induction (start moved 1 row(s) past the supplied seeds; P_1, ..., P_2 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 2, ..., 2"},"hypotheses":[{"evidence":"P_2 has degree 0; checked exactly","name":"seed_roots_negative","statement":"P_2 is nonzero and real-rooted with every root < 0","status":{"kind":"verified"}},{"evidence":"checked exactly for P_1, ..., P_1","name":"earlier_seeds","statement":"the seed rows P_1, ..., P_1 are nonzero and real-rooted with every root in (-inf, 0)","status":{"kind":"verified"}},{"evidence":"lambda_n = 1 > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of lambda_n = 1 at n = 3 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"lambda_n = [x^1] a_n + (deg P_(n-1)) [x^2] b_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(n, x) is the zero polynomial","name":"derivative_coefficient_vanishes_at_zero","statement":"b_n(0) = 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"b_n(x) = gamma(n) w(x) / f(n) with w(x) = -x^2, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 3 + u have one sign)","name":"derivative_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"b_n(x) <= 0 for every x < 0 and every n >= 3","status":{"kind":"verified"}},{"evidence":"a_n(0) = 1 > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of a_n(0) = 1 at n = 3 + u have one sign)","name":"constant_term_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"a_n(0) > 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"exact","name":"requested_root_region","statement":"the requested region (-inf, 0) contains (-inf, 0), the region this theorem certifies","status":{"kind":"verified"}}],"lean_replay":"not_attempted","normal_form":"P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 3, with a_n(x) = (1) + (n - 2) x and b_n(x) = (0) + (0) x + (-1) x^2","recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-2"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"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 g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","requested_root_region":{"upper":{"closed":false,"value":"0"}},"schema":"polytool.recurrence-real-rootedness-theorem.v1","start_delay":1,"status":"proved","theorem":"liu_wang"},"parent_proof":{"attempts":[{"detail":"the rule is not P_(n+k) = c(x) P_n with c free of n","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"no theorem matched (weighted_renewal_window: coefficients must be index-independent; fixed_delay_predecessor_window: derivatives and alternating signs are unsupported)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: the theorem needs at least two seed rows","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"induction start moved 1 row(s) past the seeds: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 3, with a_n(x) = (1) + (n - 2) x and b_n(x) = (0) + (0) x + (-1) x^2: every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Liu--Wang) for n >= 3; the rule is applied from n = 3 on in the induction (start moved 1 row(s) past the supplied seeds; P_1, ..., P_2 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 2, ..., 2","route":"liu_wang","success":true,"variant":"liu_wang"}],"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-1) for n >= 3, with a_n(x) = (1) + (n - 2) x and b_n(x) = (0) + (0) x + (-1) x^2","every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Liu--Wang) for n >= 3; the rule is applied from n = 3 on in the induction (start moved 1 row(s) past the supplied seeds; P_1, ..., P_2 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 2, ..., 2"],"theorem":"liu_wang: 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 g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)"},"kind":"weak","orientation":"previous_interlaces_next"},"route":"liu_wang","rows_checked":10,"statement":"every parent row P_n, n >= 1, 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 >= 1, the side conditions hold for every n, and every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, 0]"},"kind":"preserver_image"}