{"kind":"theorem_check","report":{"conclusion":{"first_index":0,"interlacing":"weak","interlacing_from":1,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":false,"statement":"every P_n, n >= 0, 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 >= 1"},"hypotheses":[{"evidence":"P_0 has degree 0; checked exactly","name":"seed_roots_negative","statement":"P_0 is nonzero and real-rooted with every root < 0","status":{"kind":"verified"}},{"evidence":"lambda_n = 2 > 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of lambda_n = 2 at n = 1 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["2"]},"numerator":["2"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"lambda_n = [x^1] a_n + (deg P_(n-1)) [x^2] b_n > 0 for every n >= 1 (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 >= 1","status":{"kind":"verified"}},{"evidence":"b_n = 0","name":"derivative_coefficient_nonpositive","statement":"b_n(x) <= 0 for every x < 0 and every n >= 1","status":{"kind":"verified"}},{"evidence":"a_n(0) = 3 > 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of a_n(0) = 3 at n = 1 + u have one sign)","name":"constant_term_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["3"]},"numerator":["3"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"a_n(0) > 0 for every n >= 1","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 >= 1, with a_n(x) = (3) + (2) x and b_n(x) = (0) + (0) x + (0) x^2","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","2"]]},"deriv_order":0,"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"}}