{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Pascal-(1,3,1) array."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","-42","13","-5","-13","-1"],["2","-75","-46","73","6","0"],["6","-5","80","7","0","0"],["4","28","0","0","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-44","-22","-119","-217","-29","-1"],["2","-81","-143","-116","-21","-1","0"],["6","-9","24","3","0","0","0"],["4","28","0","0","0","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-2","-35","-114","-204","-28","-1"],["0","-6","-97","-189","-27","-1","0"],["0","-4","-56","-4","0","0","0"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","21","-17","11","1"],["2","14","-5","12","1"],["-2","-3","12","1","0"],["-4","4","0","0","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","21","-17","11","1"],["2","14","-5","12","1"],["-2","-3","12","1","0"],["-4","4","0","0","0"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0","0","0"],["-6","32","-29","10","1"],["10","-15","12","1","0"],["-4","4","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","-139","71","19","1"],["-2","-19","206","-59","-6","0"],["6","89","-80","-7","0","0"],["-4","-28","0","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-6","-71","177","27","1"],["0","-2","-15","181","27","1","0"],["0","4","56","4","0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1","2","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":0,"quotients":[],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["-1","-1"],["2","1"],["-4","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-21","-3"],["1","14","1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-1","-14","-1"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["1"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","21","529","2572","-9284","10382","-5482","812","412","37","1"],["0","0","2","87","1174","3776","-14160","15878","-9768","2072","862","75","2"],["0","0","0","11","293","1768","-2400","1078","-2990","1712","488","39","1"],["0","0","-6","-155","-820","2496","544","-3950","1396","456","38","1","0"],["0","0","-4","-100","-468","1932","-1932","468","100","4","0","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","21","529","2572","-9284","10382","-5482","812","412","37","1"],["0","0","2","87","1174","3776","-14160","15878","-9768","2072","862","75","2"],["0","0","0","11","293","1768","-2400","1078","-2990","1712","488","39","1"],["0","0","-6","-155","-820","2496","544","-3950","1396","456","38","1","0"],["0","0","-4","-100","-468","1932","-1932","468","100","4","0","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","-1","-39","-468","-1180","6162","-6162","1180","468","39","1"],["0","0","0","-3","-117","-1404","-3540","18486","-18486","3540","1404","117","3"],["0","0","0","-3","-117","-1404","-3540","18486","-18486","3540","1404","117","3"],["0","0","0","-1","-39","-468","-1180","6162","-6162","1180","468","39","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","1","37","390","244","-8522","18486","-13504","1892","897","77","2"],["0","0","0","7","267","3042","5452","-50214","80106","-45232","3804","2535","227","6"],["0","0","0","15","579","6786","14892","-99510","129402","-54672","60","2223","219","6"],["0","0","0","13","505","6006","14404","-82466","92430","-27664","-3724","429","65","2"],["0","0","0","4","156","1872","4720","-24648","24648","-4720","-1872","-156","-4","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-1","-32","-193","2174","15358","-46936","31990","4532","-5597","-1208","-85","-2"],["0","0","0","-3","-117","-1401","-3423","19890","-14946","-14946","19890","-3423","-1401","-117","-3"],["0","0","0","-3","-138","-2223","-13368","-6294","110916","-125862","26184","9945","822","21","0"],["0","0","0","-1","-53","-1015","-7771","-10826","78926","-78926","10826","7771","1015","53","1"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","1","53","1015","7771","10826","-78926","78926","-10826","-7771","-1015","-53","-1"],["0","0","0","0","2","106","2030","15542","21652","-157852","157852","-21652","-15542","-2030","-106","-2"],["0","0","0","0","1","53","1015","7771","10826","-78926","78926","-10826","-7771","-1015","-53","-1"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","-1","-31","-140","3212","24058","-24058","-3212","140","31","1"],["0","0","0","-7","-253","-2452","2884","90070","-78926","-11668","260","89","3"],["0","0","0","-15","-573","-6516","-10620","125862","-92430","-15732","-60","81","3"],["0","0","0","-13","-511","-6236","-17044","77746","-44314","-9308","-340","19","1"],["0","0","0","-4","-160","-2032","-6752","17896","-6752","-2032","-160","-4","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","1","32","171","-3072","-27270","0","27270","3072","-171","-32","-1"],["0","0","0","3","118","1411","2696","-34874","-59260","74206","14984","727","-10","-1"],["0","0","0","3","140","2309","14608","12062","-118520","66602","20752","1967","76","1"],["0","0","0","1","54","1069","8840","19666","-59260","19666","8840","1069","54","1"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","-1","-54","-1069","-8840","-19666","59260","-19666","-8840","-1069","-54","-1"],["0","0","0","0","-2","-108","-2138","-17680","-39332","118520","-39332","-17680","-2138","-108","-2"],["0","0","0","0","-1","-54","-1069","-8840","-19666","59260","-19666","-8840","-1069","-54","-1"]]},"deriv_order":2}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0","0"],["3","6","-1"],["-4","4","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-22","4","2"],["1","13","-13","-1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-1","-13","13","1"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","2","40","-7","-1893","7325","-6621","707","409","37","1"],["0","0","8","208","1195","-1364","9245","-12086","1861","856","75","2"],["0","0","18","508","3663","911","-3021","-4209","1605","485","39","1"],["0","0","20","552","3713","-1658","-4477","1356","455","38","1","0"],["0","0","8","212","1252","-2040","464","100","4","0","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-21","-550","-3122","6162","-4220","1262","450","38","1"],["0","0","-2","-89","-1263","-5039","9121","-6757","3011","939","77","2"],["0","0","0","-11","-304","-2072","328","-750","2240","528","40","1"],["0","0","6","161","981","-1515","-2059","1891","495","39","1","0"],["0","0","4","104","572","-1360","572","104","4","0","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","-1","-39","-468","-1180","6162","-6162","1180","468","39","1"],["0","0","0","-3","-117","-1404","-3540","18486","-18486","3540","1404","117","3"],["0","0","0","-3","-117","-1404","-3540","18486","-18486","3540","1404","117","3"],["0","0","0","-1","-39","-468","-1180","6162","-6162","1180","468","39","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","1","29","79","-3461","-17494","68962","-68962","17494","3461","-79","-29","-1"],["0","0","0","7","239","1953","-7535","-81850","256182","-236254","55330","12099","-85","-83","-3"],["0","0","0","15","543","5385","-1839","-140586","354774","-294990","61026","15531","219","-75","-3"],["0","0","0","13","485","5227","5083","-105598","216850","-157066","26038","8609","377","-17","-1"],["0","0","0","4","152","1716","2848","-29368","49296","-29368","2848","1716","152","4","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-1","-30","-108","3382","20955","-51468","0","51468","-20955","-3382","108","30","1"],["0","0","0","-3","-112","-1178","8","38855","-13184","-157852","192688","-44965","-13520","-746","8","1"],["0","0","0","-3","-134","-2032","-10130","14845","128036","-315704","230972","-27065","-16894","-1816","-74","-1"],["0","0","0","-1","-52","-962","-6756","-3055","89752","-157852","89752","-3055","-6756","-962","-52","-1"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","1","52","962","6756","3055","-89752","157852","-89752","3055","6756","962","52","1"],["0","0","0","0","2","104","1924","13512","6110","-179504","315704","-179504","6110","13512","1924","104","2"],["0","0","0","0","1","52","962","6756","3055","-89752","157852","-89752","3055","6756","962","52","1"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","-1","-23","172","6948","33170","-77746","28188","8596","679","17"],["0","0","0","-7","-225","-1356","16124","124158","-257886","89284","27660","2193","55"],["0","0","0","-15","-537","-5100","6684","173454","-307182","98724","31404","2505","63"],["0","0","0","-13","-491","-5444","-7212","107114","-151690","42348","14212","1147","29"],["0","0","0","-4","-156","-1872","-4720","24648","-24648","4720","1872","156","4"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","1","30","85","-4312","-33038","7604","86530","-43864","-12083","-930","-23","0"],["0","0","0","3","113","1185","-853","-55250","-63718","251986","-98554","-31937","-2875","-99","-1"],["0","0","0","3","136","2115","11230","-11386","-150248","244382","-65516","-27625","-2960","-129","-2"],["0","0","0","1","53","1015","7771","10826","-78926","78926","-10826","-7771","-1015","-53","-1"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","-1","-53","-1015","-7771","-10826","78926","-78926","10826","7771","1015","53","1"],["0","0","0","0","-2","-106","-2030","-15542","-21652","157852","-157852","21652","15542","2030","106","2"],["0","0","0","0","-1","-53","-1015","-7771","-10826","78926","-78926","10826","7771","1015","53","1"]]},"deriv_order":2}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","0","1","26","143","-340","143","26","1"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1","-1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"statement":"w(n) E = sum_nu H_nu S^nu L' (with any remainder removed by row annihilators of the parent), where L' is the parent rule at P'_n = P_(n+0) (valid from base 0) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = phi(n, D) = 1 + (1) D; 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) T_n(P_(n+0)) for every n >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"interlacing":{"from_index":1,"justification":"the parent theorem (LiuWangDerivativeLag) proves P_(m-1) strictly interlacing P_m for every m >= 1; T = phi(n, D) = 1 + (1) D 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.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"derivative_polynomial_real_rooted","statement":"the polynomial in D is real-rooted for every n >= n0","status":{"kind":"verified"}},{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}},{"evidence":"delta(n) != 0 for every integer n in [0, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 0 + u have one sign)","name":"operator_coefficients_defined","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"0","last":null}},"decision":"holds"},"statement":"the operator coefficients are finite at every n >= n0 (their denominators do not vanish)","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":0,"input_interlacing":{"interlacing":{"from_index":1,"justification":{"premises":["the parent theorem (LiuWangDerivativeLag) proves P_(m-1) strictly interlacing P_m for every m >= 1; T = phi(n, D) = 1 + (1) D 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":"phi(n, D) = 1 + (1) D","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = phi(n, D) = 1 + (1) D, for every n >= 0"},"parent_proof":{"route":"liu_wang_derivative_lag","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":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n"},"verdict":"proved"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"parent_interlacing_report":{"conclusion":{"first_index":0,"interlacing":"strict","interlacing_from":1,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":true,"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, P_n has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 2"},"hypotheses":[{"evidence":"checked exactly","name":"seed_roots_negative","statement":"P_0 and P_1 are nonzero and real-rooted with every root < 0","status":{"kind":"verified"}},{"evidence":"degrees 0 and 1, leading coefficients of one sign","name":"seed_degree_step","statement":"deg P_1 = deg P_0 + 1, with leading coefficients of one sign","status":{"kind":"verified"}},{"evidence":"Bezout test, exact","name":"seed_strict_interlacing","statement":"P_0 strictly interlaces P_1 (simple real roots, no common root)","status":{"kind":"verified"}},{"evidence":"lambda_n = 1 > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of lambda_n = 1 at n = 2 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"lambda_n = [x^1] A_n + (deg P_(n-1)) [x^2] V_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 and [x^0] c_2 are zero polynomials","name":"multipliers_vanish_at_zero","statement":"V_n(0) = 0 and W_n(0) = 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"A_n(0) = 1 > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of A_n(0) = 1 at n = 2 + 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":"2","last":null}},"decision":"holds"},"statement":"A_n(0) > 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"V_n = 0","name":"derivative_coefficient_nonpositive","statement":"V_n(x) <= 0 for every x < 0 and every n >= 2","status":{"kind":"verified"}},{"evidence":"W_n(x) = gamma(n) w(x) / f(n) with w(x) = 3 x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 2 + u have one sign)","name":"lag_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"W_n(x) <= 0 for every x < 0 and every n >= 2","status":{"kind":"verified"}},{"evidence":"W_n(x) = gamma(n) w(x) / f(n) with w(x) = 3 x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 2 + u have one sign)","name":"derivative_or_lag_coefficient_negative","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"for every n >= 2 and x < 0, V_n(x) < 0 or W_n(x) < 0 (V_n < 0 on (-inf, 0) for every n, or W_n < 0 there for every n)","status":{"kind":"verified"}}],"lean_declarations":["RealRooted.threeTerm_interlaces_of_nonnegCoeffs","RealRooted.LiuWang.strictInterl_nonpos_lag_sequence"],"lean_replay":"not_attempted","lean_route":"sign","normal_form":"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) = (1) + (1) x, V_n(x) = 0, W_n(x) = (3) x","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":2,"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 the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), 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_derivative_lag"},"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":"no theorem matched (weighted_renewal_window: support must satisfy offset = 1 + ell * t_degree; fixed_delay_predecessor_window: more than one weighted lag)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: c_2(n, x) = gamma(n) w(x) with w neither constant nor a constant times a square (x + v_0)^2","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":"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) = (1) + (1) x, V_n(x) = 0, W_n(x) = (3) x: 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, P_n has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 2","route":"liu_wang_derivative_lag","success":true,"variant":"liu_wang_derivative_lag"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: 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) = (1) + (1) x, V_n(x) = 0, W_n(x) = (3) x","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, P_n has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 2"],"theorem":"liu_wang_derivative_lag: 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 the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)"},"kind":"strict","orientation":"previous_interlaces_next"},"route":"liu_wang_derivative_lag","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 = phi(n, D) = 1 + (1) D 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"}