{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["0","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["-4/3"],["-1/3"]]},"terms":[{"coeff":{"coeffs":[["0","-1","0"],["0","-5/3","0"],["0","-2/3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["4/3","-4/3","1/3"],["4/3","-4/3","1/3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1)."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-10","-5"],["2","-10"],["2","-3"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["10","-15","5"],["4","-6","2"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0"],["5"],["1"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-4"],["3"],["1"]]},"terms":[{"coeff":{"coeffs":[["10","-2"],["2","4"],["-2","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","9","-3"],["-4","6","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["5","6","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["0","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["-4/3"],["-1/3"]]},"terms":[{"coeff":{"coeffs":[["0","-1","0"],["0","-5/3","0"],["0","-2/3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["4/3","-4/3","1/3"],["4/3","-4/3","1/3"]]},"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":[["0","-10"],["4","-11"],["2","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["8","-12","4"],["4","-6","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["5"],["6"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","0"],["10","-15","5"],["2","-3","1"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["50","-75","25"],["120","-180","60"],["92","-138","46"],["24","-36","12"],["2","-3","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["-240","480","-300","60"],["-768","1536","-960","192"],["-2812/3","5624/3","-3515/3","703/3"],["-552","1104","-690","138"],["-496/3","992/3","-620/3","124/3"],["-24","48","-30","6"],["-4/3","8/3","-5/3","1/3"]]},"deriv_order":0},{"coeff":{"coeffs":[["-480","1440","-1560","720","-120"],["-1056","3168","-3432","1584","-264"],["-2456/3","2456","-7982/3","1228","-614/3"],["-856/3","856","-2782/3","428","-214/3"],["-136/3","136","-442/3","68","-34/3"],["-8/3","8","-26/3","4","-2/3"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["1920","-5760","6240","-2880","480"],["5184","-15552","16848","-7776","1296"],["16160/3","-16160","52520/3","-8080","4040/3"],["8336/3","-8336","27092/3","-4168","2084/3"],["752","-2256","2444","-1128","188"],["304/3","-304","988/3","-152","76/3"],["16/3","-16","52/3","-8","4/3"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["2"],["1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","-1"],["2","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["2","-3","1"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["50","-75","25"],["40","-60","20"],["6","-9","3"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["500","-1500","1625","-750","125"],["1800","-5400","5850","-2700","450"],["2460","-7380","7995","-3690","615"],["1584","-4752","5148","-2376","396"],["492","-1476","1599","-738","123"],["72","-216","234","-108","18"],["4","-12","13","-6","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["-1200","3600","-3900","1800","-300"],["-5280","15840","-17160","7920","-1320"],["-28604/3","28604","-92963/3","14302","-7151/3"],["-9152","27456","-29744","13728","-2288"],["-5076","15228","-16497","7614","-1269"],["-1664","4992","-5408","2496","-416"],["-316","948","-1027","474","-79"],["-32","96","-104","48","-8"],["-4/3","4","-13/3","2","-1/3"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","1200","-3600","3900","-1800","300"],["-800","6880","-16040","15760","-6920","1120"],["-7360/3","42124/3","-84052/3","76183/3","-31906/3","5011/3"],["-8456/3","40688/3","-73442/3","62474/3","-25094/3","3830/3"],["-4576/3","20152/3","-34144/3","27742/3","-10780/3","1606/3"],["-1232/3","5168/3","-8420/3","6632/3","-2516/3","368/3"],["-160/3","652/3","-1036/3","799/3","-298/3","43/3"],["-8/3","32/3","-50/3","38/3","-14/3","2/3"]]},"deriv_order":1},{"coeff":{"coeffs":[["-800","3600","-6600","6300","-3300","900","-100"],["-7360/3","11040","-20240","19320","-10120","2760","-920/3"],["-8456/3","12684","-23254","22197","-11627","3171","-1057/3"],["-4576/3","6864","-12584","12012","-6292","1716","-572/3"],["-1232/3","1848","-3388","3234","-1694","462","-154/3"],["-160/3","240","-440","420","-220","60","-20/3"],["-8/3","12","-22","21","-11","3","-1/3"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["-1600","800","6800","-10600","5600","-1000"],["-12320/3","-13040/3","109960/3","-144020/3","71920/3","-12500/3"],["-2384","-64904/3","78612","-269942/3","42584","-21590/3"],["6664/3","-100420/3","262942/3","-271387/3","122308/3","-20107/3"],["10568/3","-75716/3","166382/3","-158891/3","68660/3","-11003/3"],["5488/3","-30904/3","61156/3","-55162/3","23032/3","-3610/3"],["1376/3","-6896/3","12776/3","-11060/3","4496/3","-692/3"],["56","-788/3","466","-1175/3","156","-71/3"],["8/3","-12","62/3","-17","20/3","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["3200","-14400","26400","-25200","13200","-3600","400"],["34240/3","-51360","94160","-89880","47080","-12840","4280/3"],["48544/3","-72816","133496","-127428","66748","-18204","6068/3"],["35216/3","-52824","96844","-92442","48422","-13206","4402/3"],["14080/3","-21120","38720","-36960","19360","-5280","1760/3"],["3104/3","-4656","8536","-8148","4268","-1164","388/3"],["352/3","-528","968","-924","484","-132","44/3"],["16/3","-24","44","-42","22","-6","2/3"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["2","-3","1"]]},"start_base":1}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-8","8","-2"],["-20/3","20/3","-5/3"],["-4/3","4/3","-1/3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","7"],["0","13/3"],["0","2/3"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["-4"],["-8/3"],["-1/3"]]},"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) D; so the candidate rows satisfy the input rule at every base n >= 1 (the exceptional bases checked exactly), and agreement on the 2 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":{"reason":"the parent proof claims no interlacing","status":"none"},"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 [0, 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":{"reason":"the parent proof claims no interlacing","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"phi(n, D) = (1) D","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["-4/3"],["-1/3"]]},"terms":[{"coeff":{"coeffs":[["0","-1","0"],["0","-5/3","0"],["0","-2/3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["4/3","-4/3","1/3"],["4/3","-4/3","1/3"]]},"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) D, for every n >= 0"},"parent_proof":{"route":"three_term_recurrence_whole_line","statement":"every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; Q_(n-1) strictly interlaces Q_n in y for n >= 1","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, +inf)","input_root_region":{},"parent_hypothesis":"every parent row used is real-rooted (anywhere on the line)","parent_root_region":{},"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,"simple_roots":true,"statement":"every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; Q_(n-1) strictly interlaces Q_n in y for n >= 1"},"hypotheses":[{"evidence":"f(n) = -1/3 n^2 - 4/3 n != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = -1/3 n^2 - 4/3 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":["-4","-8/3","-1/3"]},"numerator":["0","-4/3","-1/3"],"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 = -4/3 n^2 - 10/3 n - 2 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = -4/3 n^2 - 10/3 n - 2 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":["-14","-26/3","-4/3"]},"numerator":["-2","-10/3","-4/3"],"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 = 1/28 (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^4 + 3/16 n^3 - 1/16 n^2 - 3/16 n)/(n^4 + 3 n^3 + 11/4 n^2 + 3/4 n) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of beta_n = (1/16 n^4 + 3/16 n^3 - 1/16 n^2 - 3/16 n)/(n^4 + 3 n^3 + 11/4 n^2 + 3/4 n) at n = 3 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["0","3/4","11/4","3","1"],"method":{"kind":"taylor_shift","shifted_denominator":["189","825/4","335/4","15","1"],"shifted_numerator":["9","45/4","5","15/16","1/16"]},"numerator":["0","-3/16","-1/16","3/16","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"}}],"lean_replay":"not_attempted","normal_form":"P_n = (a_n + b_n (x - 2)) P_(n-1) - c_n (x - 2)^2 P_(n-2) for n >= 2, with a_n = (4 n^2 + 10 n + 6)/(n^2 + 4 n), b_n = (2 n^2 + 5 n + 3)/(n^2 + 4 n), c_n = (n^2 + n)/(n^2 + 4 n); Q_n(y) = y^(d_n) P_n(1/y + 2), 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":[["1"],["0","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["-4/3"],["-1/3"]]},"terms":[{"coeff":{"coeffs":[["0","-1","0"],["0","-5/3","0"],["0","-2/3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["4/3","-4/3","1/3"],["4/3","-4/3","1/3"]]},"deriv_order":0,"offset":2,"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","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"},"parent_proof":{"attempts":[],"conditional_on":[],"failing_row_index":2,"rows_checked":10,"statement":"parent row P_2 is nonzero and not real-rooted with coefficients of one sign; this parent gives no proof","status":"refuted"},"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) D for every n >= 0, the side conditions hold for every n, and every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; Q_(n-1) strictly interlaces Q_n in y for n >= 1; so every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, +inf)","whole_line_parent_proof":{"conclusion":{"first_index":0,"simple_roots":true,"statement":"every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; Q_(n-1) strictly interlaces Q_n in y for n >= 1"},"hypotheses":[{"evidence":"f(n) = -1/3 n^2 - 4/3 n != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = -1/3 n^2 - 4/3 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":["-4","-8/3","-1/3"]},"numerator":["0","-4/3","-1/3"],"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 = -4/3 n^2 - 10/3 n - 2 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = -4/3 n^2 - 10/3 n - 2 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":["-14","-26/3","-4/3"]},"numerator":["-2","-10/3","-4/3"],"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 = 1/28 (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^4 + 3/16 n^3 - 1/16 n^2 - 3/16 n)/(n^4 + 3 n^3 + 11/4 n^2 + 3/4 n) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of beta_n = (1/16 n^4 + 3/16 n^3 - 1/16 n^2 - 3/16 n)/(n^4 + 3 n^3 + 11/4 n^2 + 3/4 n) at n = 3 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["0","3/4","11/4","3","1"],"method":{"kind":"taylor_shift","shifted_denominator":["189","825/4","335/4","15","1"],"shifted_numerator":["9","45/4","5","15/16","1/16"]},"numerator":["0","-3/16","-1/16","3/16","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"}}],"lean_replay":"not_attempted","normal_form":"P_n = (a_n + b_n (x - 2)) P_(n-1) - c_n (x - 2)^2 P_(n-2) for n >= 2, with a_n = (4 n^2 + 10 n + 6)/(n^2 + 4 n), b_n = (2 n^2 + 5 n + 3)/(n^2 + 4 n), c_n = (n^2 + n)/(n^2 + 4 n); Q_n(y) = y^(d_n) P_n(1/y + 2), 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":[["1"],["0","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["-4/3"],["-1/3"]]},"terms":[{"coeff":{"coeffs":[["0","-1","0"],["0","-5/3","0"],["0","-2/3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["4/3","-4/3","1/3"],["4/3","-4/3","1/3"]]},"deriv_order":0,"offset":2,"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","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"}},"kind":"preserver_image"}
