{"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) = -1/3 n^2 - 4/3 n != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = -1/3 n^2 - 4/3 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":["-4","-8/3","-1/3"]},"numerator":["0","-4/3","-1/3"],"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 = -4/3 n^2 - 10/3 n - 2 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = -4/3 n^2 - 10/3 n - 2 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":["-14","-26/3","-4/3"]},"numerator":["-2","-10/3","-4/3"],"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 = 1/28 (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 + 3/16 n^3 - 1/16 n^2 - 3/16 n)/(n^4 + 3 n^3 + 11/4 n^2 + 3/4 n) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of beta_n = (1/16 n^4 + 3/16 n^3 - 1/16 n^2 - 3/16 n)/(n^4 + 3 n^3 + 11/4 n^2 + 3/4 n) at n = 3 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["0","3/4","11/4","3","1"],"method":{"kind":"taylor_shift","shifted_denominator":["189","825/4","335/4","15","1"],"shifted_numerator":["9","45/4","5","15/16","1/16"]},"numerator":["0","-3/16","-1/16","3/16","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"}}],"lean_replay":"not_attempted","normal_form":"P_n = (a_n + b_n (x - 2)) P_(n-1) - c_n (x - 2)^2 P_(n-2) for n >= 2, with a_n = (4 n^2 + 10 n + 6)/(n^2 + 4 n), b_n = (2 n^2 + 5 n + 3)/(n^2 + 4 n), c_n = (n^2 + n)/(n^2 + 4 n); Q_n(y) = y^(d_n) P_n(1/y + 2), 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"],["0","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["-4/3"],["-1/3"]]},"terms":[{"coeff":{"coeffs":[["0","-1","0"],["0","-5/3","0"],["0","-2/3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["4/3","-4/3","1/3"],["4/3","-4/3","1/3"]]},"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"}}