{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"1","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["8"],["18","12"]],"recurrence":{"denominator":{"coeffs":[["0"],["3"],["10"],["9"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","-12","6"],["7","-14","7"],["-4","8","-4"],["-7","14","-7"],["-2","4","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["12","0"],["22","16"],["16","16"],["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"reversal of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n)."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-36","-24"],["-33","-41"],["-10","-20"],["-1","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-15","15"],["0","-11","11"],["0","-2","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["16"],["20"],["8"],["1"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["8"]],"recurrence":{"denominator":{"coeffs":[["3"],["7"],["5"],["1"]]},"terms":[{"coeff":{"coeffs":[["12","0"],["26","10"],["18","11"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-6","-6"],["0","-7","-7"],["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["16","20","8","1"],"operator":{"intercept":"1","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["8"],["18","12"]],"recurrence":{"denominator":{"coeffs":[["0"],["3"],["10"],["9"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","-12","6"],["7","-14","7"],["-4","8","-4"],["-7","14","-7"],["-2","4","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["12","0"],["22","16"],["16","16"],["4","4"]]},"deriv_order":0,"offset":1,"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":[["-36","-24"],["-33","-41"],["-10","-20"],["-1","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-15","15"],["0","-11","11"],["0","-2","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["16"],["20"],["8"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["16"],["20"],["8"],["1"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["0","-256","256"],["0","-640","640"],["0","-656","656"],["0","-352","352"],["0","-104","104"],["0","-16","16"],["0","-1","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","-26880","53760","-26880"],["0","-116096","232192","-116096"],["0","-219024","438048","-219024"],["0","-238416","476832","-238416"],["0","-166168","332336","-166168"],["0","-77592","155184","-77592"],["0","-24609","49218","-24609"],["0","-5239","10478","-5239"],["0","-717","1434","-717"],["0","-57","114","-57"],["0","-2","4","-2"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","67200","-134400","67200"],["0","0","216320","-432640","216320"],["0","0","305576","-611152","305576"],["0","0","248540","-497080","248540"],["0","0","128238","-256476","128238"],["0","0","43521","-87042","43521"],["0","0","9714","-19428","9714"],["0","0","1375","-2750","1375"],["0","0","112","-224","112"],["0","0","4","-8","4"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","-1008000","2016000","-1008000"],["0","0","-3984000","7968000","-3984000"],["0","0","-7097560","14195120","-7097560"],["0","0","-7522076","15044152","-7522076"],["0","0","-5268662","10537324","-5268662"],["0","0","-2560513","5121026","-2560513"],["0","0","-880917","1761834","-880917"],["0","0","-214521","429042","-214521"],["0","0","-36233","72466","-36233"],["0","0","-4042","8084","-4042"],["0","0","-268","536","-268"],["0","0","-8","16","-8"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["1"],["1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-3","0"],["0","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-1","1"]]},"deriv_order":2,"shift":0}]},"quotient":[],"step_multiplier":{"coeffs":[["0","0","4096","-8192","4096"],["0","0","15360","-30720","15360"],["0","0","25344","-50688","25344"],["0","0","24128","-48256","24128"],["0","0","14592","-29184","14592"],["0","0","5808","-11616","5808"],["0","0","1520","-3040","1520"],["0","0","252","-504","252"],["0","0","24","-48","24"],["0","0","1","-2","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","76800","-153600","76800"],["0","0","417280","-834560","417280"],["0","0","1020864","-2041728","1020864"],["0","0","1488896","-2977792","1488896"],["0","0","1442960","-2885920","1442960"],["0","0","979440","-1958880","979440"],["0","0","477556","-955112","477556"],["0","0","168536","-337072","168536"],["0","0","42723","-85446","42723"],["0","0","7585","-15170","7585"],["0","0","895","-1790","895"],["0","0","63","-126","63"],["0","0","2","-4","2"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","-115200","153600","38400","-76800"],["0","0","-453120","527360","304640","-378880"],["0","0","-794016","756608","868832","-831424"],["0","0","-815760","558336","1330608","-1073184"],["0","0","-543792","181216","1268944","-906368"],["0","0","-245592","-35072","806920","-526256"],["0","0","-76050","-62328","352806","-214428"],["0","0","-15933","-29456","106711","-61322"],["0","0","-2160","-7742","21964","-12062"],["0","0","-171","-1212","2937","-1554"],["0","0","-6","-106","230","-118"],["0","0","0","-4","8","-4"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","-38400","115200","-115200","38400"],["0","0","0","-151040","453120","-453120","151040"],["0","0","0","-264672","794016","-794016","264672"],["0","0","0","-271920","815760","-815760","271920"],["0","0","0","-181264","543792","-543792","181264"],["0","0","0","-81864","245592","-245592","81864"],["0","0","0","-25350","76050","-76050","25350"],["0","0","0","-5311","15933","-15933","5311"],["0","0","0","-720","2160","-2160","720"],["0","0","0","-57","171","-171","57"],["0","0","0","-2","6","-6","2"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","1958400","-2995200","115200","921600"],["0","0","9392640","-13585920","-1006080","5199360"],["0","0","20604832","-27896896","-6020704","13312768"],["0","0","27364368","-34215104","-13662896","20513632"],["0","0","24536048","-27826480","-17955184","21245616"],["0","0","15678392","-15727808","-15579560","15628976"],["0","0","7341954","-6278044","-9469774","8405864"],["0","0","2549893","-1751900","-4145879","3347886"],["0","0","656468","-325313","-1318778","987623"],["0","0","123669","-34180","-302647","213158"],["0","0","16561","-404","-48875","32718"],["0","0","1492","398","-5272","3382"],["0","0","81","49","-341","211"],["0","0","2","2","-10","6"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","576000","-1728000","1728000","-576000"],["0","0","0","2688000","-8064000","8064000","-2688000"],["0","0","0","5708320","-17124960","17124960","-5708320"],["0","0","0","7292272","-21876816","21876816","-7292272"],["0","0","0","6239424","-18718272","18718272","-6239424"],["0","0","0","3765704","-11297112","11297112","-3765704"],["0","0","0","1643282","-4929846","4929846","-1643282"],["0","0","0","522243","-1566729","1566729","-522243"],["0","0","0","119921","-359763","359763","-119921"],["0","0","0","19397","-58191","58191","-19397"],["0","0","0","2097","-6291","6291","-2097"],["0","0","0","136","-408","408","-136"],["0","0","0","4","-12","12","-4"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["1"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["84","-168","84"],["157","-314","157"],["94","-188","94"],["23","-46","23"],["2","-4","2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-304","-256"],["-420","-384"],["-214","-208"],["-48","-48"],["-4","-4"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["150"],["215"],["112"],["25"],["2"]]},"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 = P((1) x + (1)); 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 (ThreeTermRecurrence) proves P_(m-1) strictly interlacing P_m for every m >= 1; T = P((1) x + (1)) 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":["Polynomial.Splits.comp_of_degree_le_one"],"lift_summary":{"hypotheses":[{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":0,"input_interlacing":{"interlacing":{"from_index":1,"justification":{"premises":["the parent theorem (ThreeTermRecurrence) proves P_(m-1) strictly interlacing P_m for every m >= 1; T = P((1) x + (1)) 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":"P((1) x + (1))","kind":"preserver_image","operator":{"intercept":"1","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["8"],["18","12"]],"recurrence":{"denominator":{"coeffs":[["0"],["3"],["10"],["9"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","-12","6"],["7","-14","7"],["-4","8","-4"],["-7","14","-7"],["-2","4","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["12","0"],["22","16"],["16","16"],["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (1)), for every n >= 0"},"parent_proof":{"route":"three_term_recurrence","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, -1]","input_root_region":{"upper":{"closed":true,"value":"-1"}},"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":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically"},"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 simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)"},"hypotheses":[{"evidence":"f(n) = 2 n^4 + 9 n^3 + 10 n^2 + 3 n != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = 2 n^4 + 9 n^3 + 10 n^2 + 3 n 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":["150","215","112","25","2"]},"numerator":["0","3","10","9","2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 2","status":{"kind":"verified"}},{"evidence":"f(n) a_n = 8 n^4 + 32 n^3 + 38 n^2 + 12 n != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = 8 n^4 + 32 n^3 + 38 n^2 + 12 n at n = 2 + u have one sign)","name":"leading_coefficient_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["560","804","422","96","8"]},"numerator":["0","12","38","32","8"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"a_n != 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"Q_0 has degree 0, Q_1 degree 1; simple real roots and strict interlacing checked exactly","name":"seeds_interlace","statement":"Q_0 and Q_1 have simple real roots, degrees e and e + 1, and Q_0 strictly interlaces Q_1","status":{"kind":"verified"}},{"evidence":"beta_2 = 1/25 (exact)","name":"first_coefficient_positive","statement":"beta_2 = c_2 L_0 / (a_2 L_1) > 0","status":{"kind":"verified"}},{"evidence":"beta_n = (1/16 n^8 + 1/4 n^7 + 5/64 n^6 - 41/64 n^5 - 25/64 n^4 + 37/64 n^3 + 1/4 n^2 - 3/16 n)/(n^8 + 4 n^7 + 7/2 n^6 - 7/2 n^5 - 91/16 n^4 - 7/8 n^3 + 19/16 n^2 + 3/8 n) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of beta_n = (1/16 n^8 + 1/4 n^7 + 5/64 n^6 - 41/64 n^5 - 25/64 n^4 + 37/64 n^3 + 1/4 n^2 - 3/16 n)/(n^8 + 4 n^7 + 7/2 n^6 - 7/2 n^5 - 91/16 n^4 - 7/8 n^3 + 19/16 n^2 + 3/8 n) at n = 3 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["0","3/8","19/16","-7/8","-91/16","-7/2","7/2","4","1"],"method":{"kind":"taylor_shift","shifted_denominator":["33075/2","327705/8","701083/16","211631/8","157829/16","4655/2","679/2","28","1"],"shifted_numerator":["3375/4","35175/16","39325/16","98539/64","37835/64","9121/64","1349/64","7/4","1/16"]},"numerator":["0","-3/16","1/4","37/64","-25/64","-41/64","5/64","1/4","1/16"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"beta_n = c_n / (a_n a_(n-1)) > 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"seeds checked exactly; positive-ratio induction at t = 0 with g(n) = (1/4 n + 3/4)/(n + 3/2) (the exact ratio r_n): g(n) > 0 for n >= 2, r_2 >= g(2) exactly, and (t - alpha_n) - beta_n/g(n-1) - g(n) >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of (t - alpha_n) - beta_n/g(n-1) - g(n) at n = 3 + u have one sign)","name":"root_upper_bound","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["0","0","27/16","405/32","1953/64","163/128","-995/8","-29899/128","-9489/64","2185/32","2947/16","1119/8","221/4","23/2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["1406514375/16","22169345625/64","80562804525/128","89489570265/128","4244364843/8","9312801091/32","15235305649/128","4720514741/128","556825065/64","49776817/32","3320041/16","160239/8","5291/4","107/2","1"],"shifted_numerator":[]},"numerator":[],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"every root of Q_n is < 0 for every n >= 0 (region (-inf, 0) for P)","status":{"kind":"verified"}},{"evidence":"on the reflected rows (-1)^(deg) p_n(-y): seeds checked exactly; positive-ratio induction at t = 1 with g(n) = (1/4 n^2 + 5/4 n + 3/2)/(n^2 + 5/2 n + 3/2) (the exact ratio r_n): g(n) > 0 for n >= 2, r_2 >= g(2) exactly, and (t - alpha_n) - beta_n/g(n-1) - g(n) >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of (t - alpha_n) - beta_n/g(n-1) - g(n) at n = 3 + u have one sign)","name":"root_lower_bound","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["0","0","27/16","513/32","3681/64","9595/128","-1461/16","-7697/16","-11837/16","-59115/128","11039/64","18449/32","8307/16","2095/8","317/4","27/2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["1406514375","24982374375/4","206870814675/16","1060337125845/64","1883036766453/128","1228787539993/128","19052636337/4","58665471353/32","70817825657/128","16815432069/128","1565736761/64","113152769/32","6222489/16","251743/8","7067/4","123/2","1"],"shifted_numerator":[]},"numerator":[],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"every root of Q_n is > -1 for every n >= 0 (region (-inf, 0) for P)","status":{"kind":"verified"}}],"lean_replay":"not_attempted","normal_form":"P_n = (a_n + b_n (x - 1)) P_(n-1) - c_n (x - 1)^2 P_(n-2) for n >= 2, with a_n = (4 n^4 + 16 n^3 + 19 n^2 + 6 n)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n), b_n = (2 n^4 + 8 n^3 + 8 n^2)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n), c_n = (n^4 + 7/2 n^3 + 2 n^2 - 7/2 n - 3)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n); Q_n(y) = y^(d_n) P_n(1/y + 1), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2)","recurrence":{"first_index":0,"initial_polynomials":[["8"],["18","12"]],"recurrence":{"denominator":{"coeffs":[["0"],["3"],["10"],["9"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","-12","6"],["7","-14","7"],["-4","8","-4"],["-7","14","-7"],["-2","4","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["12","0"],["22","16"],["16","16"],["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","requested_root_region":{"upper":{"closed":false,"value":"0"}},"schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"},"parent_proof":{"attempts":[{"detail":"the leader is not a nonzero constant","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: LHS must be a nonzero constant; fixed_delay_predecessor_window: LHS must be a nonzero constant)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"P_n = (a_n + b_n (x - 1)) P_(n-1) - c_n (x - 1)^2 P_(n-2) for n >= 2, with a_n = (4 n^4 + 16 n^3 + 19 n^2 + 6 n)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n), b_n = (2 n^4 + 8 n^3 + 8 n^2)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n), c_n = (n^4 + 7/2 n^3 + 2 n^2 - 7/2 n - 3)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n); Q_n(y) = y^(d_n) P_n(1/y + 1), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2): every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)","route":"three_term_recurrence","success":true,"variant":"three_term_recurrence"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = (a_n + b_n (x - 1)) P_(n-1) - c_n (x - 1)^2 P_(n-2) for n >= 2, with a_n = (4 n^4 + 16 n^3 + 19 n^2 + 6 n)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n), b_n = (2 n^4 + 8 n^3 + 8 n^2)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n), c_n = (n^4 + 7/2 n^3 + 2 n^2 - 7/2 n - 3)/(n^4 + 9/2 n^3 + 5 n^2 + 3/2 n); Q_n(y) = y^(d_n) P_n(1/y + 1), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2)","every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)"],"theorem":"three_term_recurrence: three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction"},"kind":"strict","orientation":"previous_interlaces_next"},"route":"three_term_recurrence","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 = P((1) x + (1)) 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, -1]"},"kind":"preserver_image"}