{"kind":"theorem_check","report":{"conclusion":{"first_index":0,"simple_roots":true,"statement":"every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; Q_(n-1) strictly interlaces Q_n in y for n >= 1"},"hypotheses":[{"evidence":"f(n) = n^2 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = n^2 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":["4","4","1"]},"numerator":["0","0","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 = 16 n^2 - 16 n + 4 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = 16 n^2 - 16 n + 4 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":["36","48","16"]},"numerator":["4","-16","16"],"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 = 1/12 (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 - 1/4 n^3 + 23/64 n^2 - 7/32 n + 3/64)/(n^4 - 4 n^3 + 11/2 n^2 - 3 n + 9/16) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of beta_n = (1/16 n^4 - 1/4 n^3 + 23/64 n^2 - 7/32 n + 3/64)/(n^4 - 4 n^3 + 11/2 n^2 - 3 n + 9/16) at n = 3 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["9/16","-3","11/2","-4","1"],"method":{"kind":"taylor_shift","shifted_denominator":["225/16","30","47/2","8","1"],"shifted_numerator":["15/16","31/16","95/64","1/2","1/16"]},"numerator":["3/64","-7/32","23/64","-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"}}],"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 = (16 n^2 - 16 n + 4)/(n^2), b_n = (8 n^2 - 8 n + 2)/(n^2), c_n = (16 n^2 - 32 n + 12)/(n^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":[["1"],["2","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","2"],["-8","-8"],["8","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-12","24","-12"],["32","-64","32"],["-16","32","-16"]]},"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","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"}}