{"kind":"theorem_check","report":{"conclusion":{"first_index":0,"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); for n >= 1, Q_(n-1) interlaces Q_n (weakly, in the sense of Liu and Wang; no interlacing of the P_n is claimed)"},"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","1"]},"numerator":["2","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":"degrees 0 and 1; real-rooted and Q_0 << Q_1 checked exactly","name":"seed_rows","statement":"Q_0 and Q_1 are nonzero and real-rooted, of degrees e and e or e + 1, and Q_0 << Q_1 (weak interlacing)","status":{"kind":"verified"}},{"evidence":"l_n = ([x^1] A_n) l_(n-1) + (0) l_(n-2) when deg P_(n-1) - deg P_(n-2) = 1 (then deg P_n - deg P_(n-1) = 1): positive seeds and nonnegative step coefficients: s_n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of s_n at n = 2 + u have one sign); t_n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of t_n at n = 2 + u have one sign); s_n + t_n > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of s_n + t_n at n = 2 + u have one sign)","name":"degree_law","statement":"deg P_n - deg P_(n-1) is 0 or 1 and P_n, P_(n-2) have positive leading coefficients (after the sign normalization) for every n >= 2: full growth (deg P_n = deg P_(n-1) + 1); l_n = ([x^1] A_n) l_(n-1) + (0) l_(n-2) when deg P_(n-1) - deg P_(n-2) = 1 (then deg P_n - deg P_(n-1) = 1)","status":{"kind":"verified"}},{"evidence":"B_n = (-n + 1)/(n + 2) <= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of B_n = (-n + 1)/(n + 2) at n = 2 + u have one sign)","name":"lag_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonpositive","denominator":["2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["4","1"],"shifted_numerator":["-1","-1"]},"numerator":["1","-1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"B_n(x) <= 0 for every real x and every n >= 2","status":{"kind":"verified"}},{"evidence":"checked exactly","name":"seed_roots_negative","statement":"Q_0 and Q_1 have every root < 0","status":{"kind":"verified"}},{"evidence":"positive-ratio induction with g(n) = 1 (the exact ratio): g(n) > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of g(n) at n = 2 + u have one sign); u_2/u_1 = 1 >= g(2) = 1 exactly; t_n <= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of t_n at n = 3 + u have one sign); and s_n + t_n / g(n-1) - g(n) >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of s_n + t_n / g(n-1) - g(n) at n = 3 + u have one sign)","name":"constant_terms_positive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["5","1"],"shifted_numerator":[]},"numerator":[],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"P_n(0) > 0 for every n >= 2 (normalized rows), with P_n(0) = A_n(0) P_(n-1)(0) + B_n(0) P_(n-2)(0)","status":{"kind":"verified"}}],"lean_declarations":["RealRooted.MaWang.strictInterl_of_interlaces_evalCoeff_nonpos"],"lean_replay":"not_attempted","normal_form":"Q_n = A_n(x) Q_(n-1) + B_n(x) Q_(n-2) for n >= 2, with A_n(x) = ((2 n + 1)/(n + 2)) + ((4 n + 2)/(n + 2)) x, B_n(x) = ((-n + 1)/(n + 2)); Q_n = P_n / (x + 1)^(n - 0) for n >= 0, which satisfies the rule below","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","3","2"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","3","2"],["2","6","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","2","1"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"liu_wang_three_term"}}