{"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"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis_map_A013611 of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["76","240","-256","0"],["244","944","-896","-2048"],["-176","-896","-2048","0"],["-192","-512","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["161","1192","3600","6656","4096"],["424","2560","3712","2048","0"],["-96","-320","-1024","0","0"],["-192","-512","0","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["85","952","3856","6656","4096"],["180","1616","4608","4096","0"],["80","576","1024","0","0"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["-76","-240","256"],["-60","16","256"],["80","256","0"],["64","0","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["-76","-240","256"],["-60","16","256"],["80","256","0"],["64","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"],["-28","-496","256"],["-112","256","0"],["64","0","0"]]},"terms":[{"coeff":{"coeffs":[["152","1088","1408","-2048"],["-20","-1200","-3200","2048"],["-400","-640","2048","0"],["192","512","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["15","88","-272","-2560","-4096"],["-20","-464","-2560","-4096","0"],["-80","-576","-1024","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"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"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":[["-4","0"],["-12","-32"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["5","36","64"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["4"],["4"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["-1520","-15744","-48896","-24576","65536"],["-2720","-24064","-56832","16384","131072"],["400","8320","49408","106496","65536"],["2880","25856","73728","65536","0"],["1280","9216","16384","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["-1520","-15744","-48896","-24576","65536"],["-2720","-24064","-56832","16384","131072"],["400","8320","49408","106496","65536"],["2880","25856","73728","65536","0"],["1280","9216","16384","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["80","576","1024"],["160","1152","2048"],["80","576","1024"]]},"step_quotients":[[{"coeff":{"coeffs":[["-160","-1792","-6656","-8192"],["-320","-3584","-13312","-16384"],["-160","-1792","-6656","-8192"]]},"deriv_order":0},{"coeff":{"coeffs":[["100","1440","7744","18432","16384"],["100","1440","7744","18432","16384"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["-100","-1440","-7744","-18432","-16384"],["-100","-1440","-7744","-18432","-16384"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["-16"],["16"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["24","96"],["-20","-64"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["5","36","64"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["2880","30976","110592","131072"],["8320","80384","253952","262144"],["8000","67840","176128","131072"],["2560","18432","32768","0"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["1600","23040","123904","294912","262144"],["4800","69120","371712","884736","786432"],["4800","69120","371712","884736","786432"],["1600","23040","123904","294912","262144"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0"],["6400","92160","495616","1179648","1048576"],["19200","276480","1486848","3538944","3145728"],["19200","276480","1486848","3538944","3145728"],["6400","92160","495616","1179648","1048576"]]},"deriv_order":0},{"coeff":{"coeffs":[["1600","35840","308224","1286144","2621440","2097152"],["-4800","-69120","-371712","-884736","-786432","0"],["-14400","-245760","-1668096","-5627904","-9437184","-6291456"],["-8000","-140800","-988160","-3457024","-6029312","-4194304"]]},"deriv_order":1},{"coeff":{"coeffs":[["2000","43200","387840","1852416","4964352","7077888","4194304"],["4000","86400","775680","3704832","9928704","14155776","8388608"],["2000","43200","387840","1852416","4964352","7077888","4194304"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["-4800","-94720","-740352","-2867200","-5505024","-4194304"],["-4800","-107520","-924672","-3858432","-7864320","-6291456"],["4800","69120","371712","884736","786432","0"],["4800","81920","556032","1875968","3145728","2097152"]]},"deriv_order":0},{"coeff":{"coeffs":[["-2000","-43200","-387840","-1852416","-4964352","-7077888","-4194304"],["-4000","-86400","-775680","-3704832","-9928704","-14155776","-8388608"],["-2000","-43200","-387840","-1852416","-4964352","-7077888","-4194304"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["80","576","1024"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","-4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-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":{"reason":"no parent theorem report with an interlacing conclusion","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 [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":{"reason":"no parent theorem report with an interlacing conclusion","status":"not_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"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"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_three_term","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_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":"not applicable: expected one nonzero lag term","route":"product_recurrence","success":false,"variant":"product_recurrence"},{"detail":"no theorem matched (weighted_renewal_window: support must satisfy offset = 1 + ell * t_degree; fixed_delay_predecessor_window: only P_(n-1) and one term w(n) t P_(n-L) are supported)","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":"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) = (1), V_n(x) = 0, W_n(x) = (1) + (4) x; open hypotheses: seed_degree_step: degrees 0 and 0 (exact); seed_strict_interlacing: the Bezout test fails or the degrees do not differ by one (exact); degree_step_positive: fails at n = 2: lambda_n = 0 = 0 there (exact); multipliers_vanish_at_zero: [x^0] c_2(n, x) = 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 = 0 (not a disproof): fails at n = 2: [x^0] W_n = 1 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 = 0 (not a disproof): fails at n = 2: [x^0] W_n = 1 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":"P_n = A_n(x) P_(n-1) + B_n(x) P_(n-2) for n >= 2, with A_n(x) = (1), B_n(x) = (1) + (4) x: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, -1/4] (so its coefficients have one sign); for n >= 1, P_(n-1) interlaces P_n (weakly, in the sense of Liu and Wang), and deg P_n - deg P_(n-1) is 0 or 1 for n >= 2","route":"liu_wang_three_term","success":true,"variant":"liu_wang_three_term"}],"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) + B_n(x) P_(n-2) for n >= 2, with A_n(x) = (1), B_n(x) = (1) + (4) x","every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, -1/4] (so its coefficients have one sign); for n >= 1, P_(n-1) interlaces P_n (weakly, in the sense of Liu and Wang), and deg P_n - deg P_(n-1) is 0 or 1 for n >= 2"],"theorem":"liu_wang_three_term: L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction"},"kind":"weak","orientation":"previous_interlaces_next"},"route":"liu_wang_three_term","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"}