{"certification":{"candidate":{"gauge":{"denominator":["1"],"initial_value":"1","kind":"ratio","numerator":["0","1"]},"index_shift":0,"operator":{"factor":{"terms":[{"kind":"inverse_factorial"}]},"shape":"hadamard_multiplier"},"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"n-71965e79e47a4526"},"identifying_rows":1,"identity":{"certificate":{"certificate":{"factor":{"terms":[{"kind":"inverse_factorial"}]},"hadamard":{"conjugated_input_relation":{"terms":[{"coeff":{"coeffs":[["0","-2","-8"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-1","-10","-16"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","-1","-4","-4"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","2"],["0","2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","0","2"],["0","0","2"]]},"deriv_order":1,"shift":1}]},"conjugation_x_shift":1,"exceptional_bases":[],"factor":{"terms":[{"kind":"inverse_factorial"}]},"gauge":{"denominator":["1"],"initial_value":"1","kind":"ratio","numerator":["0","1"]},"identifying_indices":[0],"input_leader_shift":1,"input_relation":{"terms":[{"coeff":{"coeffs":[["-2","-4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1","-4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-1"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"shift":1}]},"input_valid_from_base":0,"multiplier":["1"],"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","2","8","8"],["0","2","8","8"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","2","8","8"],["0","0","2","8","8"]]},"deriv_order":1}],"shift":0}],"row_annihilator":[{"coeff":{"coeffs":[["0"],["2"]]},"deriv_order":0},{"coeff":{"coeffs":[["-1","-2"]]},"deriv_order":1}],"row_annihilator_quotient":[{"coeff":{"coeffs":[["0","1","8","20","16"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","1","6","12","8"]]},"deriv_order":1}],"row_multiplier":["1","4","4"],"statement":"mu(x) w(n) M'' = sum_nu H_nu S^nu L + Z A (A the row annihilator of the product parent) with M'' the input relation conjugated by lambda(n) 1/k!; so the candidate rows satisfy the input rule at every base n >= 0 (the exceptional bases checked exactly), and agreement on the 1 identifying rows gives Q_n = lambda(n) sum_k u(n,k) [x^k]P_n x^k for every n >= 0"},"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2"]]},"terms":[{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","4"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},"method":"hadamard_conjugation"},"status":"certified"},"input_first_index":0,"interlacing":{"from_index":1,"justification":"the parent theorem (LiuWang) proves P_(m-1) weakly interlacing P_m for every m >= 1; T = x^k -> 1/k! x^k is independent of n and maps every real-rooted polynomial to a real-rooted polynomial or zero, so by the Hermite--Kakeya--Obreschkoff theorem (both directions) T(P_(m-1)) and T(P_m) have weakly interlacing zeros; the nonzero gauge does not move zeros, so Q_(n-1) and Q_n interlace weakly (orientation not determined) for every n >= 1","kind":"weak","orientation":"unordered","status":"claimed"},"lean_declarations":["RealRooted.isMultiplierSequence_inv_factorial"],"lift_summary":{"hypotheses":[{"evidence":"1/k! is defined for every n >= 0","name":"factor_defined","statement":"the factor u(n, k) is defined on the index domain","status":{"kind":"verified"}},{"evidence":"1/k!: sum_k binom(m,k) x^k / k! = L_m(-x) is real-rooted with negative zeros for every m (Branden, Example 41); second condition of Theorem 40","name":"factor_multiplier_sequence","statement":"k -> u(n, k) is a nonnegative multiplier sequence for every n >= n0","status":{"kind":"verified"}},{"evidence":"lambda(n0) = 1 != 0 and lambda(n) / lambda(n - 1) = numerator(n) / denominator(n) != 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of lambda(n) / lambda(n - 1) = numerator(n) / denominator(n) at n = 1 + u have one sign)","name":"gauge_finite_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1","1"]},"numerator":["0","1"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}}],"identity":{"method":"hadamard_conjugation","status":"certified"},"input_first_index":0,"input_interlacing":{"interlacing":{"from_index":1,"justification":{"premises":["the parent theorem (LiuWang) proves P_(m-1) weakly interlacing P_m for every m >= 1; T = x^k -> 1/k! x^k is independent of n and maps every real-rooted polynomial to a real-rooted polynomial or zero, so by the Hermite--Kakeya--Obreschkoff theorem (both directions) T(P_(m-1)) and T(P_m) have weakly interlacing zeros; the nonzero gauge does not move zeros, so Q_(n-1) and Q_n interlace weakly (orientation not determined) for every n >= 1"],"theorem":"Hermite--Kakeya--Obreschkoff theorem (N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, 1963; see S. Fisk, Polynomials, Roots, and Interlacing, arXiv:math/0612833): real polynomials f, g have weakly interlacing zeros iff alpha f + beta g is real-rooted or zero for all real alpha, beta; a linear T independent of n that maps every real-rooted polynomial to a real-rooted polynomial or zero therefore maps interlacing pairs to pairs with weakly interlacing zeros (orientation not determined)"},"kind":"weak","orientation":"unordered"},"status":"proved"},"lean_replay":"not_attempted","operation":{"description":"x^k -> 1/k! x^k","kind":"preserver_image","operator":{"factor":{"terms":[{"kind":"inverse_factorial"}]},"shape":"hadamard_multiplier"}},"parent":{"gauge":{"denominator":["1"],"initial_value":"1","kind":"ratio","numerator":["0","1"]},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = x^k -> 1/k! x^k, for every n >= 0"},"parent_proof":{"route":"periodic_product","statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]","input_root_region":{"upper":{"closed":true,"value":"0"}},"parent_hypothesis":"every parent row used is zero or real-rooted with all roots in (-inf, 0] (coefficients of one sign)","parent_root_region":{"upper":{"closed":true,"value":"0"}},"theorem":"Polya--Schur characterization of multiplier sequences (G. Polya and I. Schur, J. Reine Angew. Math. 144 (1914) 89--113; stated as Theorem 40 of P. Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601): a real sequence (lambda_k) is a multiplier sequence (x^k -> lambda_k x^k maps real-rooted polynomials to real-rooted polynomials or zero) iff sum_k binom(m,k) lambda_k x^k is real-rooted with zeros of one sign for every m, iff sum_k lambda_k x^k / k! (or its reflection) is C x^m e^(a x) prod (1 + alpha_j x) with a, alpha_j >= 0; a product of multiplier sequences is one (composition), and a nonzero factor lambda(n) of the row changes nothing"},"verdict":"proved"},"notes":[],"parent_interlacing_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) = 1 > 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of a_n(0) = 1 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":["1"]},"numerator":["1"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"a_n(0) > 0 for every n >= 1","status":{"kind":"verified"}},{"evidence":"exact","name":"requested_root_region","statement":"the requested region (-inf, 0) contains (-inf, 0), the region this theorem certifies","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) = (1) + (2) x and b_n(x) = (0) + (0) x + (0) x^2","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","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)","requested_root_region":{"upper":{"closed":false,"value":"0"}},"schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"liu_wang"},"parent_proof":{"attempts":[{"detail":"c and the seeds are nonzero, real-rooted and of one sign, so P_(n0 + k m + i) = c^m P_(n0 + i) is too","route":"periodic_product","success":true,"variant":"P_(n+k) = c(x) P_n"}],"conditional_on":[],"route":"periodic_product","rows_checked":10,"statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"proved: Q_n = lambda(n) T_n(P_(n+0)) with T = x^k -> 1/k! x^k for every n >= 0, the side conditions hold for every n, and every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]"},"kind":"preserver_image"}