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interlace weakly (orientation not determined) for every n >= 1"],"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":0,"initial_polynomials":[["8"],["18","12"]],"recurrence":{"denominator":{"coeffs":[["0"],["3"],["10"],["9"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","-12","6"],["7","-14","7"],["-4","8","-4"],["-7","14","-7"],["-2","4","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["12","0"],["22","16"],["16","16"],["4","4"]]},"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":"three_term_recurrence","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":0,"interlacing":"strict","interlacing_from":1,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":true,"statement":"every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)"},"hypotheses":[{"evidence":"f(n) = 2 n^4 + 9 n^3 + 10 n^2 + 3 n != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = 2 n^4 + 9 n^3 + 10 n^2 + 3 n at n = 2 + u have one sign)","name":"left_factor_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["150","215","112","25","2"]},"numerator":["0","3","10","9","2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 2","status":{"kind":"verified"}},{"evidence":"f(n) a_n = 8 n^4 + 32 n^3 + 38 n^2 + 12 n != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = 8 n^4 + 32 n^3 + 38 n^2 + 12 n at n = 2 + u have one sign)","name":"leading_coefficient_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["560","804","422","96","8"]},"numerator":["0","12","38","32","8"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"a_n != 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"Q_0 has degree 0, Q_1 degree 1; simple real roots and strict interlacing checked exactly","name":"seeds_interlace","statement":"Q_0 and Q_1 have simple real roots, degrees e and e + 1, and Q_0 strictly interlaces Q_1","status":{"kind":"verified"}},{"evidence":"beta_2 = 1/25 (exact)","name":"first_coefficient_positive","statement":"beta_2 = c_2 L_0 / (a_2 L_1) > 0","status":{"kind":"verified"}},{"evidence":"beta_n = (1/16 n^8 + 1/4 n^7 + 5/64 n^6 - 41/64 n^5 - 25/64 n^4 + 37/64 n^3 + 1/4 n^2 - 3/16 n)/(n^8 + 4 n^7 + 7/2 n^6 - 7/2 n^5 - 91/16 n^4 - 7/8 n^3 + 19/16 n^2 + 3/8 n) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of beta_n = (1/16 n^8 + 1/4 n^7 + 5/64 n^6 - 41/64 n^5 - 25/64 n^4 + 37/64 n^3 + 1/4 n^2 - 3/16 n)/(n^8 + 4 n^7 + 7/2 n^6 - 7/2 n^5 - 91/16 n^4 - 7/8 n^3 + 19/16 n^2 + 3/8 n) at n = 3 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["0","3/8","19/16","-7/8","-91/16","-7/2","7/2","4","1"],"method":{"kind":"taylor_shift","shifted_denominator":["33075/2","327705/8","701083/16","211631/8","157829/16","4655/2","679/2","28","1"],"shifted_numerator":["3375/4","35175/16","39325/16","98539/64","37835/64","9121/64","1349/64","7/4","1/16"]},"numerator":["0","-3/16","1/4","37/64","-25/64","-41/64","5/64","1/4","1/16"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"beta_n = c_n / (a_n a_(n-1)) > 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"seeds checked exactly; positive-ratio induction at t = 0 with g(n) = (1/4 n + 3/4)/(n + 3/2) (the exact ratio r_n): g(n) > 0 for n >= 2, r_2 >= g(2) exactly, and (t - alpha_n) - beta_n/g(n-1) - g(n) >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of (t - alpha_n) - beta_n/g(n-1) - g(n) at n = 3 + u have one sign)","name":"root_upper_bound","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["0","0","27/16","405/32","1953/64","163/128","-995/8","-29899/128","-9489/64","2185/32","2947/16","1119/8","221/4","23/2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["1406514375/16","22169345625/64","80562804525/128","89489570265/128","4244364843/8","9312801091/32","15235305649/128","4720514741/128","556825065/64","49776817/32","3320041/16","160239/8","5291/4","107/2","1"],"shifted_numerator":[]},"numerator":[],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"every root of Q_n is < 0 for every n >= 0 (region (-inf, 0) for P)","status":{"kind":"verified"}},{"evidence":"on the reflected rows (-1)^(deg) p_n(-y): seeds checked exactly; positive-ratio induction at t = 1 with g(n) = (1/4 n^2 + 5/4 n + 3/2)/(n^2 + 5/2 n + 3/2) (the exact ratio r_n): g(n) > 0 for n >= 2, r_2 >= g(2) exactly, and (t - alpha_n) - beta_n/g(n-1) - g(n) >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of (t - alpha_n) - beta_n/g(n-1) - g(n) at n = 3 + u have one sign)","name":"root_lower_bound","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["0","0","27/16","513/32","3681/64","9595/128","-1461/16","-7697/16","-11837/16","-59115/128","11039/64","18449/32","8307/16","2095/8","317/4","27/2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["1406514375","24982374375/4","206870814675/16","1060337125845/64","1883036766453/128","1228787539993/128","19052636337/4","58665471353/32","70817825657/128","16815432069/128","1565736761/64","113152769/32","6222489/16","251743/8","7067/4","123/2","1"],"shifted_numerator":[]},"numerator":[],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"every root of Q_n is > -1 for every n >= 0 (region (-inf, 0) for P)","status":{"kind":"verified"}}],"lean_replay":"not_attempted","normal_form":"P_n = (a_n + b_n (x - 1)) P_(n-1) - c_n (x - 1)^2 P_(n-2) for n >= 2, with a_n = (4 n^4 + 16 n^3 + 19 n^2 + 6 n)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n), b_n = (2 n^4 + 8 n^3 + 8 n^2)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n), c_n = (n^4 + 7/2 n^3 + 2 n^2 - 7/2 n - 3)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n); Q_n(y) = y^(d_n) P_n(1/y + 1), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2)","recurrence":{"first_index":0,"initial_polynomials":[["8"],["18","12"]],"recurrence":{"denominator":{"coeffs":[["0"],["3"],["10"],["9"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","-12","6"],["7","-14","7"],["-4","8","-4"],["-7","14","-7"],["-2","4","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["12","0"],["22","16"],["16","16"],["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","requested_root_region":{"upper":{"closed":false,"value":"0"}},"schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"},"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":"P_n = (a_n + b_n (x - 1)) P_(n-1) - c_n (x - 1)^2 P_(n-2) for n >= 2, with a_n = (4 n^4 + 16 n^3 + 19 n^2 + 6 n)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n), b_n = (2 n^4 + 8 n^3 + 8 n^2)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n), c_n = (n^4 + 7/2 n^3 + 2 n^2 - 7/2 n - 3)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n); Q_n(y) = y^(d_n) P_n(1/y + 1), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2): every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)","route":"three_term_recurrence","success":true,"variant":"three_term_recurrence"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = (a_n + b_n (x - 1)) P_(n-1) - c_n (x - 1)^2 P_(n-2) for n >= 2, with a_n = (4 n^4 + 16 n^3 + 19 n^2 + 6 n)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n), b_n = (2 n^4 + 8 n^3 + 8 n^2)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n), c_n = (n^4 + 7/2 n^3 + 2 n^2 - 7/2 n - 3)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n); Q_n(y) = y^(d_n) P_n(1/y + 1), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2)","every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)"],"theorem":"three_term_recurrence: three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction"},"kind":"strict","orientation":"previous_interlaces_next"},"route":"three_term_recurrence","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"}