{"kind":"theorem_check","report":{"conclusion":{"first_index":0,"interlacing":"strict","interlacing_from":2,"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 = 1 on; P_(n-1) strictly interlaces P_n for n >= 2; every root lies in (-inf, 0); 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) = 2 n^2 - 3 n != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) = 2 n^2 - 3 n 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":["9","9","2"]},"numerator":["0","-3","2"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 3","status":{"kind":"verified"}},{"evidence":"f(n) a_n = 8 n^2 - 16 n + 6 != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = 8 n^2 - 16 n + 6 at n = 3 + u have one sign)","name":"leading_coefficient_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["30","32","8"]},"numerator":["6","-16","8"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"a_n != 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"Q_1 has degree 0, Q_2 degree 1; simple real roots and strict interlacing checked exactly","name":"seeds_interlace","statement":"Q_1 and Q_2 have simple real roots, degrees e and e + 1, and Q_1 strictly interlaces Q_2","status":{"kind":"verified"}},{"evidence":"beta_3 = 1/18 (exact)","name":"first_coefficient_positive","statement":"beta_3 = c_3 L_1 / (a_3 L_2) > 0","status":{"kind":"verified"}},{"evidence":"beta_n = (1/16 n^4 - 3/8 n^3 + 49/64 n^2 - 39/64 n + 5/32)/(n^4 - 6 n^3 + 25/2 n^2 - 21/2 n + 45/16) > 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of beta_n = (1/16 n^4 - 3/8 n^3 + 49/64 n^2 - 39/64 n + 5/32)/(n^4 - 6 n^3 + 25/2 n^2 - 21/2 n + 45/16) at n = 4 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["45/16","-21/2","25/2","-6","1"],"method":{"kind":"taylor_shift","shifted_denominator":["525/16","115/2","73/2","10","1"],"shifted_numerator":["63/32","225/64","145/64","5/8","1/16"]},"numerator":["5/32","-39/64","49/64","-3/8","1/16"],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"beta_n = c_n / (a_n a_(n-1)) > 0 for every n >= 4","status":{"kind":"verified"}},{"evidence":"checked exactly for P_0, ..., P_0","name":"earlier_seeds","statement":"the seed rows P_0, ..., P_0 are nonzero and real-rooted with every root in (-inf, 0)","status":{"kind":"verified"}},{"evidence":"seeds checked exactly; positive-ratio induction at t = 0 with g(n) = (1/4 n^2)/(n^2 - 3/2 n + 1/2) (the exact ratio r_n): g(n) > 0 for n >= 3, r_3 >= g(3) exactly, and (t - alpha_n) - beta_n/g(n-1) - g(n) >= 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of (t - alpha_n) - beta_n/g(n-1) - g(n) at n = 4 + u have one sign)","name":"root_upper_bound","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["135/128","-1539/128","7659/128","-21901/128","19925/64","-12057/32","4917/16","-1335/8","231/4","-23/2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["3472875/128","12932325/128","21469455/128","20946579/128","6655661/64","1440279/32","215085/16","21897/8","1455/4","57/2","1"],"shifted_numerator":[]},"numerator":[],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"every root of Q_n is < 0 for every n >= 1 (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)/(n - 1/2) (the exact ratio r_n): g(n) > 0 for n >= 3, r_3 >= g(3) exactly, and (t - alpha_n) - beta_n/g(n-1) - g(n) >= 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of (t - alpha_n) - beta_n/g(n-1) - g(n) at n = 4 + u have one sign)","name":"root_lower_bound","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["135/128","-1269/128","2493/64","-2665/32","1693/16","-655/8","151/4","-19/2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["385875/128","1179675/128","778085/64","289719/32","66693/16","9729/8","879/4","45/2","1"],"shifted_numerator":[]},"numerator":[],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"every root of Q_n is > -1 for every n >= 1 (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 >= 3, with a_n = (4 n^2 - 8 n + 3)/(n^2 - 3/2 n), b_n = (2 n^2 - 4 n + 2)/(n^2 - 3/2 n), c_n = (n^2 - 5/2 n + 1)/(n^2 - 3/2 n); 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)","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["2","4"],["-8","-8"],["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","4","-2"],["5","-10","5"],["-2","4","-2"]]},"deriv_order":0,"offset":2,"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","start_delay":1,"status":"proved","theorem":"three_term_recurrence"}}