{"kind":"theorem_check","report":{"conclusion":{"first_index":1,"interlacing":"weak","interlacing_from":2,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":false,"statement":"every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Liu--Wang) for n >= 2"},"hypotheses":[{"evidence":"f(n) = 2 n^2 + n != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = 2 n^2 + 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":["10","9","2"]},"numerator":["0","1","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":"P_1 has degree 0; checked exactly","name":"seed_roots_negative","statement":"P_1 is nonzero and real-rooted with every root < 0","status":{"kind":"verified"}},{"evidence":"lambda_n = (n^2 + 1/2 n)/(n^2 + 1/2 n) > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of lambda_n = (n^2 + 1/2 n)/(n^2 + 1/2 n) at n = 2 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["0","1/2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["5","9/2","1"],"shifted_numerator":["5","9/2","1"]},"numerator":["0","1/2","1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"lambda_n = [x^1] a_n + (deg P_(n-1)) [x^2] b_n > 0 for every n >= 2 (deg P_n = deg P_(n-1) + 1 with leading coefficients of one sign)","status":{"kind":"verified"}},{"evidence":"[x^0] c_1(n, x) is the zero polynomial","name":"derivative_coefficient_vanishes_at_zero","statement":"b_n(0) = 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"b_n(x) = gamma(n) w(x) / f(n) with w(x) = x^2 - 4 x, which is positive on (-inf, 0) (no root there, exact): gamma(n)/f(n) = (-2 n + 1/2)/(n^2 + 1/2 n) <= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = (-2 n + 1/2)/(n^2 + 1/2 n) at n = 2 + u have one sign)","name":"derivative_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonpositive","denominator":["0","1/2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["5","9/2","1"],"shifted_numerator":["-7/2","-2"]},"numerator":["1/2","-2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"b_n(x) <= 0 for every x < 0 and every n >= 2","status":{"kind":"verified"}},{"evidence":"a_n(0) = (4 n^2 - 1)/(n^2 + 1/2 n) > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of a_n(0) = (4 n^2 - 1)/(n^2 + 1/2 n) at n = 2 + u have one sign)","name":"constant_term_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["0","1/2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["5","9/2","1"],"shifted_numerator":["15","16","4"]},"numerator":["-1","0","4"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"a_n(0) > 0 for every n >= 2","status":{"kind":"verified"}}],"lean_replay":"not_attempted","normal_form":"P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 2, with a_n(x) = ((4 n^2 - 1)/(n^2 + 1/2 n)) + ((3 n^2 - 4 n + 1)/(n^2 + 1/2 n)) x and b_n(x) = ((0)/(n^2 + 1/2 n)) + ((8 n - 2)/(n^2 + 1/2 n)) x + ((-2 n + 1/2)/(n^2 + 1/2 n)) x^2","recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"],["2"]]},"terms":[{"coeff":{"coeffs":[["-2","2"],["0","-8"],["8","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","1"],["0","16","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"liu_wang"}}