{"kind":"theorem_check","report":{"conclusion":{"first_index":1,"interlacing":"strict","interlacing_from":2,"root_region":{"lower":{"closed":false,"value":"0"},"upper":{"closed":false,"value":"1"}},"simple_roots":true,"statement":"every P_n, n >= 1, 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 (0, 1)"},"hypotheses":[{"evidence":"f(n) a_n = -1 != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = -1 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":["-1"]},"numerator":["-1"],"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 (exact)","name":"first_coefficient_positive","statement":"beta_3 = c_3 L_1 / (a_3 L_2) > 0","status":{"kind":"verified"}},{"evidence":"beta_n = n^2 - 4 n + 4 > 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of beta_n = n^2 - 4 n + 4 at n = 4 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["4","4","1"]},"numerator":["4","-4","1"],"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":"seeds checked exactly; positive-ratio induction at t = -1 with g(n) = n - 1 (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":["-2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["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 (0, 1) 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 = -1, b_n = -2 n + 2, c_n = n^2 - 4 n + 4; 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":1,"initial_polynomials":[["1"],["1","-2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","2"],["2","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-4","8","-4"],["4","-8","4"],["-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":{"lower":{"closed":false,"value":"0"},"upper":{"closed":false,"value":"1"}},"schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"}}