{"input":{"first_index":0,"initial_polynomials":[["1"],["0","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","8"],["0","-4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"],["0","0","4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"kind":"hadamard_factor","lift":{"factor":{"terms":[{"coefficient_offset":0,"index_offset":1,"kind":"binomial"}]},"factor_text":"binom(n+1, k)","gauge":{"kind":"none"},"hidden_coefficients":"zero","identity":{"certificate":{"conjugated_input_relation":{"terms":[{"coeff":{"coeffs":[["0","0","-18"],["0","0","-8"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","0","-36"],["0","0","0","-16"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","-9"],["0","0","0","0","-4"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0"],["0","8"],["0","4"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","0","0"],["0","0","8"],["0","0","4"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["6"],["5"],["1"]]},"deriv_order":0,"shift":2}]},"conjugation_x_shift":0,"exceptional_bases":[],"factor":{"terms":[{"coefficient_offset":0,"index_offset":1,"kind":"binomial"}]},"gauge":{"kind":"none"},"identifying_indices":[0,1],"input_leader_shift":2,"input_relation":{"terms":[{"coeff":{"coeffs":[["0","0","-9"],["0","0","-4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0"],["0","4"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"input_valid_from_base":0,"multiplier":["1","2","1"],"parent":{"first_index":0,"initial_polynomials":[["1"],["0","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","0","6"],["0","0","17"],["0","0","17"],["0","0","7"],["0","0","1"]]},"deriv_order":0}],"shift":0}],"row_annihilator_system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","-2"],["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","12"],["0","0","0","30"],["0","0","0","24"],["0","0","0","6"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","8"],["0","0","0","0","12"],["0","0","0","0","4"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["1"],["1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","1"],["0","1"]]},"deriv_order":0}],[]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","-27"],["0","0","0","0","-59"],["0","0","0","0","-41"],["0","0","0","0","-9"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","-9"],["0","0","0","0","0","-22"],["0","0","0","0","0","-17"],["0","0","0","0","0","-4"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","1"],["0","2"],["0","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","-1"],["0","-2"],["0","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","1"],["0","0","1"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","2"],["0","0","3"],["0","0","1"]]},"deriv_order":0}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","0","1"]]},"start_base":0},"row_multiplier":["1"],"statement":"mu(x) w(n) M'' = sum_nu H_nu S^nu L + sum_l Z_l A_l (an invariant row-annihilator system of the parent) with M'' the input relation conjugated by lambda(n) binom(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 2 identifying rows gives Q_n = lambda(n) sum_k u(n,k) [x^k]P_n x^k for every n >= 0"},"status":"certified"},"lift_summary":{"hypotheses":[{"evidence":"binom(n+1, k): sum_k u(n,k) x^k / k! = L_(A-0)^(0)(-x) with A = n + a (zero when A < 0), a Laguerre polynomial with parameter 0 > -1, whose zeros are real and positive (Szego, Orthogonal Polynomials, Theorem 3.3.1); third condition of Theorem 40","name":"factor_multiplier_sequence","statement":"k -> binom(n+1, k) is a nonnegative multiplier sequence for every n >= 0","status":{"kind":"verified"}}],"identity":{"method":"conjugation_division","status":"certified"},"input_first_index":0,"input_interlacing":{"reason":"the factor binom(n+1, k) depends on n, so consecutive rows are mapped by different operators","status":"not_proved"},"lean_replay":"not_attempted","operation":{"factor":"binom(n+1, k)","kind":"hadamard_factor"},"parent":{"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) sum_k binom(n+1, k) [x^k]P_n x^k for every n >= 0"},"parent_proof":{"route":"liu_wang_three_term","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]: a nonnegative multiplier sequence keeps real-rootedness and one coefficient sign","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"},"multiplier_sequence":"binom(n+1, k): sum_k u(n,k) x^k / k! = L_(A-0)^(0)(-x) with A = n + a (zero when A < 0), a Laguerre polynomial with parameter 0 > -1, whose zeros are real and positive (Szego, Orthogonal Polynomials, Theorem 3.3.1); third condition of Theorem 40","parent":{"first_index":0,"initial_polynomials":[["1"],["0","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_proof":{"attempts":[{"detail":"the rule is not P_(n+k) = c(x) P_n with c free of n","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"not applicable: expected one nonzero lag term","route":"product_recurrence","success":false,"variant":"product_recurrence"},{"detail":"no theorem matched (weighted_renewal_window: coefficient of P_(n-1) must be 1 after normalization; fixed_delay_predecessor_window: coefficient of P_(n-1) must be 1 (+ w(n) t) after normalization)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0), induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal, induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0), induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal, induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0), induction start moved 3 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 3 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal, induction start moved 3 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 3 row(s)"},{"detail":"HypothesisFailed: P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 2, with a_n = 0, b_n = 2, c_n = 1; Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2); open hypotheses: leading_coefficient_nonzero: fails at n = 2: f(n) a_n = 0 = 0 there (exact); seeds_interlace: deg Q_1 != deg Q_0 + 1 = 1; first_coefficient_positive: not checked: the seeds do not satisfy the seed hypothesis; recurrence_coefficient_positive: fails at n = 3: the denominator of beta_n = 1 vanishes there; root_upper_bound: not attempted: the real-rootedness hypotheses do not all hold","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"HypothesisFailed: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 2, with A_n(x) = (2) x, V_n(x) = 0, W_n(x) = (-1) x^2; open hypotheses: seed_roots_negative: P_1 is zero, not real-rooted, or has a root >= 0 (exact); lag_degree_step_nonnegative: fails at n = 2: mu_n = -1 = -1 there (exact); constant_term_positive: fails at n = 2: A_n(0) = 0 = 0 there (exact)","route":"liu_wang_derivative_lag","success":false,"variant":"liu_wang_derivative_lag"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + gamma(n) (x - L)(x - U) P'_(n-1)","route":"ma_wang_interval","success":false,"variant":"ma_wang_interval"},{"detail":"Q_n = A_n(x) Q_(n-1) + B_n(x) Q_(n-2) for n >= 2, with A_n(x) = (2), B_n(x) = (-1); Q_n = P_n / x^(n - 0) for n >= 0, which satisfies the rule below: 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)","route":"liu_wang_three_term","success":true,"variant":"liu_wang_three_term"}],"conditional_on":[],"route":"liu_wang_three_term","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}}