{"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) = 2 n^2 + 7 n + 3 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = 2 n^2 + 7 n + 3 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":["25","15","2"]},"numerator":["3","7","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^2 + 16 n + 6 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = 8 n^2 + 16 n + 6 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":["70","48","8"]},"numerator":["6","16","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^4 + 1/8 n^3 - 7/64 n^2 - 11/64 n + 3/32)/(n^4 + 2 n^3 + 1/2 n^2 - 1/2 n - 3/16) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of beta_n = (1/16 n^4 + 1/8 n^3 - 7/64 n^2 - 11/64 n + 3/32)/(n^4 + 2 n^3 + 1/2 n^2 - 1/2 n - 3/16) at n = 3 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["-3/16","-1/2","1/2","2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["2205/16","329/2","145/2","14","1"],"shifted_numerator":["225/32","595/64","281/64","7/8","1/16"]},"numerator":["3/32","-11/64","-7/64","1/8","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":["-27/64","-351/128","-333/64","77/32","351/16","259/8","89/4","15/2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["6251175/128","13276305/128","6122277/64","1602077/32","260301/16","26899/8","1727/4","63/2","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":["-27/64","-459/128","-711/64","-1375/128","1379/64","2517/32","1743/16","675/8","153/4","19/2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["6251175/8","32803785/16","308374479/128","213765665/128","48414773/64","7488429/32","801213/16","58563/8","2799/4","79/2","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^2 + 8 n + 3)/(n^2 + 7/2 n + 3/2), b_n = (2 n^2 + 4 n)/(n^2 + 7/2 n + 3/2), c_n = (n^2 + 1/2 n - 3/2)/(n^2 + 7/2 n + 3/2); 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":[["2"],["3","2"]],"recurrence":{"denominator":{"coeffs":[["3"],["7"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","0"],["8","8"],["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["3","-6","3"],["-1","2","-1"],["-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","status":"proved","theorem":"three_term_recurrence"}}