{"kind":"theorem_check","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) = n^2 + 6 n != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = n^2 + 6 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":["16","10","1"]},"numerator":["0","6","1"],"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 = 4 n^2 + 18 n + 20 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = 4 n^2 + 18 n + 20 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":["72","34","4"]},"numerator":["20","18","4"],"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 = 1/36 (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^4 + 7/16 n^3 + 9/16 n^2 - 7/16 n - 5/8)/(n^4 + 7 n^3 + 71/4 n^2 + 77/4 n + 15/2) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of beta_n = (1/16 n^4 + 7/16 n^3 + 9/16 n^2 - 7/16 n - 5/8)/(n^4 + 7 n^3 + 71/4 n^2 + 77/4 n + 15/2) at n = 3 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["15/2","77/4","71/4","7","1"],"method":{"kind":"taylor_shift","shifted_denominator":["495","1691/4","539/4","19","1"],"shifted_numerator":["20","43/2","63/8","19/16","1/16"]},"numerator":["-5/8","-7/16","9/16","7/16","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 + 3/2)/(n + 5/2) (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":["1875/4","7625/4","52775/16","25425/8","29871/16","2739/4","153","19","1"],"method":{"kind":"taylor_shift","shifted_denominator":["598950","3695945/4","9933803/16","1898415/8","902691/16","34167/4","804","43","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 + 3/2)/(n + 5/2) (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":["1875/4","7625/4","52775/16","25425/8","29871/16","2739/4","153","19","1"],"method":{"kind":"taylor_shift","shifted_denominator":["598950","3695945/4","9933803/16","1898415/8","902691/16","34167/4","804","43","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^2 + 18 n + 20)/(n^2 + 6 n), b_n = (2 n^2 + 9 n + 10)/(n^2 + 6 n), c_n = (n^2 + 3 n + 2)/(n^2 + 6 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":[["1"],["3","3"]],"recurrence":{"denominator":{"coeffs":[["0"],["6"],["1"]]},"terms":[{"coeff":{"coeffs":[["10","10"],["9","9"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","4","-2"],["-3","6","-3"],["-1","2","-1"]]},"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","status":"proved","theorem":"three_term_recurrence"}}