{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_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","7"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Pascal-(1,7,1) array."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["1","8","-11","-46"],["1","46","337","-16"],["0","8","120","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-1","-98","-1393","-92"],["0","1","30","-143","-32"],["0","0","8","120","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","-1","-53","-691","-23"],["0","0","0","-8","-240","-8"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["-1","-6","23"],["-1","-14","31"],["0","-8","8"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","-1","-6","23"],["0","-1","-14","31"],["0","0","-8","8"]]},"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"],["-1","2","15"],["0","-8","8"]]},"terms":[{"coeff":{"coeffs":[["0","30","228","30"],["-1","-30","-97","16"],["0","-8","-120","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","45","451","15"],["0","0","8","240","8"]]},"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":"euler_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","7"]]},"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"],["6","1"],["-8","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-45","-3"],["1","30","1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-1","-30","-1"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["1"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","-1","-64","-1108","-1680","27330","-43488","17660","1328","23"],["0","0","-2","-136","-2672","-9160","74396","-106712","41120","3112","54"],["0","0","-1","-80","-2020","-13280","66802","-82960","29260","2240","39"],["0","0","0","-8","-456","-5800","19736","-19736","5800","456","8"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-1","-64","-1108","-1680","27330","-43488","17660","1328","23"],["0","0","0","-2","-136","-2672","-9160","74396","-106712","41120","3112","54"],["0","0","0","-1","-80","-2020","-13280","66802","-82960","29260","2240","39"],["0","0","0","0","-8","-456","-5800","19736","-19736","5800","456","8"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","-1","-87","-2436","-19340","70818","-70818","19340","2436","87","1"],["0","0","0","-3","-261","-7308","-58020","212454","-212454","58020","7308","261","3"],["0","0","0","-3","-261","-7308","-58020","212454","-212454","58020","7308","261","3"],["0","0","0","-1","-87","-2436","-19340","70818","-70818","19340","2436","87","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","1","85","2262","14468","-109498","212454","-160976","36244","4785","173","2"],["0","0","0","11","951","26274","198124","-895038","1203906","-637648","89244","13659","511","6"],["0","0","0","27","2343","65250","507564","-2028126","2336994","-947088","50268","12267","495","6"],["0","0","0","25","2173","60726","478628","-1809130","1912086","-625136","-22220","2697","149","2"],["0","0","0","8","696","19488","154720","-566544","566544","-154720","-19488","-696","-8","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-1","-72","-1129","17374","365790","-1094408","939974","-146028","-75133","-6176","-189","-2"],["0","0","0","-3","-261","-7305","-57759","219762","-154434","-154434","219762","-57759","-7305","-261","-3"],["0","0","0","-3","-306","-11223","-167640","-657846","2974356","-3128790","877608","109881","3918","45","0"],["0","0","0","-1","-117","-5047","-92507","-511818","2034382","-2034382","511818","92507","5047","117","1"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","1","117","5047","92507","511818","-2034382","2034382","-511818","-92507","-5047","-117","-1"],["0","0","0","0","2","234","10094","185014","1023636","-4068764","4068764","-1023636","-185014","-10094","-234","-2"],["0","0","0","0","1","117","5047","92507","511818","-2034382","2034382","-511818","-92507","-5047","-117","-1"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","-1","-71","-1012","22468","462450","-462450","-22468","1012","71","1"],["0","0","0","-11","-917","-23228","-107508","1778982","-1562262","-87596","2332","205","3"],["0","0","0","-27","-2325","-63612","-457332","2562246","-1912086","-127980","924","189","3"],["0","0","0","-25","-2183","-61588","-502268","1637346","-987186","-83044","-1100","47","1"],["0","0","0","-8","-704","-20192","-174912","391632","-174912","-20192","-704","-8","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","1","72","1083","-21456","-484918","0","484918","21456","-1083","-72","-1"],["0","0","0","3","262","7331","54760","-360346","-1424892","1579326","140584","2999","-26","-1"],["0","0","0","3","308","11413","173888","734062","-2849784","1703898","216800","9247","164","1"],["0","0","0","1","118","5165","97672","609490","-1424892","609490","97672","5165","118","1"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","-1","-118","-5165","-97672","-609490","1424892","-609490","-97672","-5165","-118","-1"],["0","0","0","0","-2","-236","-10330","-195344","-1218980","2849784","-1218980","-195344","-10330","-236","-2"],["0","0","0","0","-1","-118","-5165","-97672","-609490","1424892","-609490","-97672","-5165","-118","-1"]]},"deriv_order":2}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0","0"],["7","10","-1"],["-8","8","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-46","12","2"],["1","29","-29","-1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-1","-29","29","1"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","1","66","1255","5052","-5953","-414","-7"],["0","0","2","140","2982","16840","-18642","-1300","-22"],["0","0","1","82","2199","18524","-19425","-1358","-23"],["0","0","0","8","472","6736","-6736","-472","-8"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","1","65","1173","2853","-24477","19011","1351","23"],["0","0","0","2","138","2810","11970","-62426","44286","3166","54"],["0","0","0","1","81","2101","15381","-51421","31539","2279","39"],["0","0","0","0","8","464","6264","-13472","6264","464","8"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","-1","-87","-2436","-19340","70818","-70818","19340","2436","87","1"],["0","0","0","-3","-261","-7308","-58020","212454","-212454","58020","7308","261","3"],["0","0","0","-3","-261","-7308","-58020","212454","-212454","58020","7308","261","3"],["0","0","0","-1","-87","-2436","-19340","70818","-70818","19340","2436","87","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","1","69","871","-24421","-416502","1364882","-1364882","416502","24421","-871","-69","-1"],["0","0","0","11","895","21405","61969","-1970770","5227734","-4815910","1384738","92055","-1925","-199","-3"],["0","0","0","27","2271","58989","332433","-3413298","7493910","-6258438","1655202","129639","-549","-183","-3"],["0","0","0","25","2133","57247","381275","-2580294","4764146","-3528674","822198","80797","1193","-45","-1"],["0","0","0","8","688","18792","135232","-721264","1133088","-721264","135232","18792","688","8","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-1","-70","-940","23550","440923","-948380","0","948380","-440923","-23550","940","70","1"],["0","0","0","-3","-256","-6810","-40360","462535","649440","-4068764","4442960","-1301157","-134560","-3050","24","1"],["0","0","0","-3","-302","-10800","-151370","-397699","4144020","-8137528","6040780","-1279545","-198470","-8920","-162","-1"],["0","0","0","-1","-116","-4930","-87460","-419311","2546200","-4068764","2546200","-419311","-87460","-4930","-116","-1"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","1","116","4930","87460","419311","-2546200","4068764","-2546200","419311","87460","4930","116","1"],["0","0","0","0","2","232","9860","174920","838622","-5092400","8137528","-5092400","838622","174920","9860","232","2"],["0","0","0","0","1","116","4930","87460","419311","-2546200","4068764","-2546200","419311","87460","4930","116","1"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","-1","-55","380","61428","770466","-1637346","718988","83204","2903","33"],["0","0","0","-11","-861","-18348","29564","2877942","-5478582","2311684","269100","9405","107"],["0","0","0","-27","-2253","-57324","-279876","4011030","-6611670","2621124","308076","10797","123"],["0","0","0","-25","-2143","-58084","-402732","2470098","-3336978","1183148","141668","4991","57"],["0","0","0","-8","-696","-19488","-154720","566544","-566544","154720","19488","696","8"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","1","70","893","-27704","-561134","-124572","1909810","-1072952","-120211","-4154","-47","0"],["0","0","0","3","257","6833","37099","-610450","-2283526","5854002","-2657722","-332929","-13355","-211","-1"],["0","0","0","3","304","10987","157310","462502","-4193336","5978574","-2096588","-305225","-14248","-281","-2"],["0","0","0","1","117","5047","92507","511818","-2034382","2034382","-511818","-92507","-5047","-117","-1"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","-1","-117","-5047","-92507","-511818","2034382","-2034382","511818","92507","5047","117","1"],["0","0","0","0","-2","-234","-10094","-185014","-1023636","4068764","-4068764","1023636","185014","10094","234","2"],["0","0","0","0","-1","-117","-5047","-92507","-511818","2034382","-2034382","511818","92507","5047","117","1"]]},"deriv_order":2}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","0","1","58","783","-1684","783","58","1"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","-7"]]},"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 = f(n, theta) = 1 + (1) theta; 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 = f(n, theta) = 1 + (1) theta 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_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"euler_polynomial_roots_nonpositive","statement":"the polynomial in theta is real-rooted with every root <= 0 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 = f(n, theta) = 1 + (1) theta 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":"f(n, theta) = 1 + (1) theta","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_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","7"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (1) theta, 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 multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(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) = 7 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) = 7 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) = (7) 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","7"]]},"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) = (7) 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) = (7) 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 = f(n, theta) = 1 + (1) theta 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"}