{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","2","1"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","1"],["6","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"existing definition"},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-9","-3"],["-7","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["4","6","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["3"],["1"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","0"],["7","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-4","-6","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["3","1"],"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","2","1"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","1"],["6","2"]]},"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":[["-9","-3"],["-7","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["4","6","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["3"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["3"],["1"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["18","27","9"],["12","18","6"],["2","3","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["-24","-60","-48","-12"],["-38","-95","-76","-19"],["-16","-40","-32","-8"],["-2","-5","-4","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["96","288","312","144","24"],["56","168","182","84","14"],["8","24","26","12","2"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["-192","-576","-624","-288","-48"],["-112","-336","-364","-168","-28"],["-16","-48","-52","-24","-4"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["1"],["1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","0"],["-4","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["2","3","1"]]},"deriv_order":2,"shift":0}]},"quotient":[],"step_multiplier":{"coeffs":[["108","324","351","162","27"],["108","324","351","162","27"],["36","108","117","54","9"],["4","12","13","6","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["288","864","936","432","72"],["696","2088","2262","1044","174"],["620","1860","2015","930","155"],["260","780","845","390","65"],["52","156","169","78","13"],["4","12","13","6","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["-864","-2880","-3672","-2232","-648","-72"],["-1368","-4656","-6102","-3846","-1170","-138"],["-768","-2648","-3528","-2270","-708","-86"],["-184","-640","-862","-562","-178","-22"],["-16","-56","-76","-50","-16","-2"]]},"deriv_order":1},{"coeff":{"coeffs":[["288","1296","2376","2268","1188","324","36"],["264","1188","2178","2079","1089","297","33"],["80","360","660","630","330","90","10"],["8","36","66","63","33","9","1"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["1584","5184","6444","3780","1044","108"],["2460","8208","10479","6381","1857","207"],["1364","4608","5981","3723","1115","129"],["324","1104","1449","915","279","33"],["28","96","127","81","25","3"]]},"deriv_order":0},{"coeff":{"coeffs":[["-576","-2592","-4752","-4536","-2376","-648","-72"],["-528","-2376","-4356","-4158","-2178","-594","-66"],["-160","-720","-1320","-1260","-660","-180","-20"],["-16","-72","-132","-126","-66","-18","-2"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["1"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["1","2","1"],["1","2","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-15","-5"],["-6","-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["4"],["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 = P((1) x + (0)); 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":{"reason":"the parent proof claims no interlacing","status":"none"},"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":{"reason":"the parent proof claims no interlacing","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"P((1) x + (0))","kind":"preserver_image","operator":{"intercept":"0","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","2","1"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","1"],["6","2"]]},"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 + (0)), 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":"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,"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) = n + 2 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = n + 2 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","1"]},"numerator":["2","1"],"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 n + 2 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = 4 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":["10","4"]},"numerator":["2","4"],"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 = 1/20 (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^2 - 1/16)/(n^2 - 1/4) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of beta_n = (1/16 n^2 - 1/16)/(n^2 - 1/4) at n = 3 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["-1/4","0","1"],"method":{"kind":"taylor_shift","shifted_denominator":["35/4","6","1"],"shifted_numerator":["1/2","3/8","1/16"]},"numerator":["-1/16","0","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 + 1)) P_(n-1) - c_n (x + 1)^2 P_(n-2) for n >= 2, with a_n = (4 n + 2)/(n + 2), b_n = (2 n + 1)/(n + 2), c_n = (n - 1)/(n + 2); 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":[["1"],["3","1"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","2","1"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","1"],["6","2"]]},"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","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":"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":"Unresolved: 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 + 2)/(n + 2), b_n = (2 n + 1)/(n + 2), c_n = (n - 1)/(n + 2); 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); open hypotheses: root_region: the region contains the pole x = -1 of the coordinate change y = 1/(x + 1)","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) = ((6 n + 3)/(n + 2)) + ((2 n + 1)/(n + 2)) x, V_n(x) = 0, W_n(x) = ((-n + 1)/(n + 2)) + ((-2 n + 2)/(n + 2)) x + ((-n + 1)/(n + 2)) x^2; open hypotheses: lag_degree_step_nonnegative: fails at n = 2: mu_n = (-n + 1)/(n + 2) = -1/4 there (exact); multipliers_vanish_at_zero: [x^0] c_2(n, x) = -n + 1 is not the zero polynomial; lag_coefficient_nonpositive: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 1 (not a disproof): fails at n = 2: [x^1] W_n = -1/2 there (exact); derivative_or_lag_coefficient_negative: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 1 (not a disproof): fails at n = 2: [x^1] W_n = -1/2 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":"no lean_proved catalog entry equals the parent; 12 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A039692:lean_proved, oeis_catalog:A079638:lean_proved, oeis_catalog:A093560:lean_proved, oeis_catalog:A103327:lean_proved, oeis_catalog:A110441:lean_proved, oeis_catalog:A116412:lean_proved, oeis_catalog:A125662:lean_proved, oeis_catalog:A127893:lean_proved, oeis_catalog:A216182:lean_proved, oeis_catalog:A249139:lean_proved, oeis_catalog:A249253:lean_proved, oeis_catalog:A327997:lean_proved)","route":"attestation","success":false,"variant":"oeis_catalog"}],"conditional_on":[],"rows_checked":10,"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"not_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 + (0)) 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) = n + 2 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = n + 2 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","1"]},"numerator":["2","1"],"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 n + 2 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = 4 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":["10","4"]},"numerator":["2","4"],"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 = 1/20 (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^2 - 1/16)/(n^2 - 1/4) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of beta_n = (1/16 n^2 - 1/16)/(n^2 - 1/4) at n = 3 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["-1/4","0","1"],"method":{"kind":"taylor_shift","shifted_denominator":["35/4","6","1"],"shifted_numerator":["1/2","3/8","1/16"]},"numerator":["-1/16","0","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 + 1)) P_(n-1) - c_n (x + 1)^2 P_(n-2) for n >= 2, with a_n = (4 n + 2)/(n + 2), b_n = (2 n + 1)/(n + 2), c_n = (n - 1)/(n + 2); 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":[["1"],["3","1"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","2","1"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","1"],["6","2"]]},"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","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"}},"kind":"definition_equality"}