{"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":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"},"parent_name":"theta_plus_one of Exponential Riordan array [x*exp(x),x]."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-1","-4","-6","-4","-1"],["-1","-11","-21","-15","-4"],["0","-7","-21","-20","-6"],["0","0","-6","-10","-4"],["0","0","0","-1","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-3","-12","-18","-12","-3"],["0","-3","-33","-63","-45","-12"],["0","0","-21","-63","-60","-18"],["0","0","0","-18","-30","-12"],["0","0","0","0","-3","-3"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","-1","-4","-6","-4","-1"],["0","0","-1","-11","-21","-15","-4"],["0","0","0","-7","-21","-20","-6"],["0","0","0","0","-6","-10","-4"],["0","0","0","0","0","-1","-1"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","0","0"],["1","-4","1","0"],["0","7","-4","0"],["0","0","6","0"],["0","0","0","1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","0","0","0","0"],["0","3","-12","3","0"],["0","0","21","-12","0"],["0","0","0","18","0"],["0","0","0","0","3"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","1","-4","1","0"],["0","0","0","7","-4","0"],["0","0","0","0","6","0"],["0","0","0","0","0","1"]]},"deriv_order":2,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1","11","-11","1"],["1","-18","27","-4"],["0","7","-22","6"],["0","0","6","-4"],["0","0","0","1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0","0"],["1","-3","-3","1","0"],["0","7","3","-4","0"],["0","0","6","6","0"],["0","0","0","1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["0","0","0","0","0","0","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"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"},"parent_leader_shift":2,"parent_valid_from_base":1,"quotients":[],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["2","-2"],["-3","-1"],["-1","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1","0","-1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0"],["0"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["1","-1","-8","-8","-1","1","0","0","0","0"],["0","10","14","-15","-29","-7","3","0","0","0"],["0","0","30","66","21","-33","-15","3","0","0"],["0","0","0","41","107","76","-4","-13","1","0"],["0","0","0","0","28","80","72","16","-4","0"],["0","0","0","0","0","9","27","27","9","0"],["0","0","0","0","0","0","1","3","3","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","3","-3","-24","-24","-3","3","0","0","0","0"],["0","0","30","42","-45","-87","-21","9","0","0","0"],["0","0","0","90","198","63","-99","-45","9","0","0"],["0","0","0","0","123","321","228","-12","-39","3","0"],["0","0","0","0","0","84","240","216","48","-12","0"],["0","0","0","0","0","0","27","81","81","27","0"],["0","0","0","0","0","0","0","3","9","9","3"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","1","-1","-8","-8","-1","1","0","0","0","0"],["0","0","0","10","14","-15","-29","-7","3","0","0","0"],["0","0","0","0","30","66","21","-33","-15","3","0","0"],["0","0","0","0","0","41","107","76","-4","-13","1","0"],["0","0","0","0","0","0","28","80","72","16","-4","0"],["0","0","0","0","0","0","0","9","27","27","9","0"],["0","0","0","0","0","0","0","0","1","3","3","1"]]},"deriv_order":2}],"step_multiplier":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","0"],["1","2","1","0","0"],["0","2","4","2","0"],["0","0","1","2","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","0","0","0","0"],["2","4","2","0","0","0"],["3","13","17","7","0","0"],["2","14","30","26","8","0"],["1","7","22","30","17","3"],["0","2","8","14","12","4"],["0","0","1","3","3","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["1","2","0","-2","-1","0","0"],["1","4","4","-2","-5","-2","0"],["0","2","5","2","-4","-4","-1"],["0","0","1","2","0","-2","-1"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","0","0","0","0"],["-1","0","2","0","-1","0"],["0","-1","0","2","0","-1"],["1","0","-2","0","1","0"],["0","1","0","-2","0","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["-1","-1","2","2","-1","-1","0"],["-1","-2","1","4","1","-2","-1"],["0","-1","-1","2","2","-1","-1"]]},"deriv_order":1}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["2","0"],["-2","1"],["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1","1","0"],["0","1","1"]]},"deriv_order":1,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["-1","5","4","-24","-31","-5","4","0","0","0"],["0","-6","-1","-10","-77","-91","-22","7","0","0"],["0","0","-13","-41","-82","-127","-104","-28","3","0"],["0","0","0","-23","-92","-150","-128","-59","-12","0"],["0","0","0","0","-26","-91","-118","-68","-16","-1"],["0","0","0","0","0","-12","-37","-39","-15","-1"],["0","0","0","0","0","0","-1","-3","-3","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","-3","8","19","-16","-37","-8","5","0","0","0"],["0","0","-25","-10","58","-8","-97","-38","8","0","0"],["0","0","0","-53","-74","36","24","-89","-54","2","0"],["0","0","0","0","-52","-114","-55","-2","-36","-28","-1"],["0","0","0","0","0","-31","-90","-88","-34","-9","-4"],["0","0","0","0","0","0","-12","-38","-42","-18","-2"],["0","0","0","0","0","0","0","-2","-6","-6","-2"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","-1","2","7","0","-7","-2","1","0","0","0"],["0","0","0","-9","-6","22","14","-15","-8","2","0","0"],["0","0","0","0","-21","-30","23","40","-3","-10","1","0"],["0","0","0","0","0","-20","-36","8","40","12","-4","0"],["0","0","0","0","0","0","-8","-16","0","16","8","0"],["0","0","0","0","0","0","0","-1","-2","0","2","1"]]},"deriv_order":2}],"step_multiplier":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","0"],["1","2","1","0","0"],["0","2","4","2","0"],["0","0","1","2","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","0","0","0","0"],["-2","-5","-4","-1","0","0"],["-2","-10","-16","-10","-2","0"],["0","-5","-16","-18","-8","-1"],["0","0","-4","-10","-8","-2"],["0","0","0","-1","-2","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["1","4","6","4","1","0","0"],["1","7","17","19","10","2","0"],["0","3","14","25","21","8","1"],["0","0","3","11","15","9","2"],["0","0","0","1","3","3","1"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","0","0","0","0"],["1","4","6","4","1","0"],["5","17","24","18","7","1"],["5","24","43","37","16","3"],["1","13","35","39","20","4"],["0","2","11","19","13","3"],["0","0","1","3","3","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["-1","-3","-2","2","3","1","0"],["-1","-5","-7","0","7","5","1"],["0","-2","-6","-4","4","6","2"],["0","0","-1","-2","0","2","1"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["1","3","3","1","0","0","0"],["0","3","9","9","3","0","0"],["0","0","3","9","9","3","0"],["0","0","0","1","3","3","1"]]},"start_base":1}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["2","4","2"],["3","6","3"],["1","2","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0"],["-4","-4"],["-2","-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0"],["1"],["1"]]},"deriv_order":0,"shift":2}]},"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 >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"interlacing":{"reason":"no parent theorem report with an interlacing conclusion","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 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 [0, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 0 + 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":"0","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":0,"input_interlacing":{"reason":"no parent theorem report with an interlacing conclusion","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"],["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 = f(n, theta) = 1 + (2) theta + (1) theta^2, 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 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_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 = 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"}