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the first 10 parent rows pass the exact check (evidence only)","status":"open"},"root_transfer":{"input_conclusion":"every Q_n, n >= 1, 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":"open"},"notes":[],"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: coefficients must be index-independent; fixed_delay_predecessor_window: derivatives and alternating signs are unsupported)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: the theorem needs at least two seed rows","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"HypothesisFailed: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 2, with a_n(x) = (2) + (n - 2) x and b_n(x) = (0) + (2) x + (-2) x^2; open hypotheses: degree_step_positive: fails at n = 2: lambda_n = -n + 2 = 0 there (exact)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"not applicable: the theorem needs two seed rows","route":"liu_wang_derivative_lag","success":false,"variant":"liu_wang_derivative_lag"},{"detail":"HypothesisFailed: P_n = (alpha_n x + beta_n) P_(n-1) + kappa_n (x - L)(x - U) P'_(n-1) for n >= 2, with L = 0, U = 1, alpha_n = n - 2, beta_n = 2, kappa_n = -2; open hypotheses: slope_positive: fails at n = 2: a_n = -1/2 n + 1 = 0 there (exact); upper_endpoint_threshold: no hit before n = 2; E(2) = e(2) + 0 = -1 < 0 with no hit in between (exact); when a_n > 0 for every n <= 2, the criterion puts a root of P_2 outside the interval","route":"ma_wang_interval","success":false,"variant":"ma_wang_interval"},{"detail":"no lean_proved catalog entry equals the parent; 68 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A003506:lean_proved, oeis_catalog:A008277:lean_proved, oeis_catalog:A008278:lean_proved, oeis_catalog:A008288:lean_proved, oeis_catalog:A008292:lean_proved, oeis_catalog:A019538:lean_proved, oeis_catalog:A026729:lean_proved, oeis_catalog:A035607:lean_proved, oeis_catalog:A039755:lean_proved, oeis_catalog:A046716:lean_proved, oeis_catalog:A048854:lean_proved, oeis_catalog:A052553:lean_proved, oeis_catalog:A056242:lean_proved, oeis_catalog:A059110:lean_proved, oeis_catalog:A060187:lean_proved, oeis_catalog:A062145:lean_proved, oeis_catalog:A063007:lean_proved, oeis_catalog:A063967:lean_proved, oeis_catalog:A071951:lean_proved, oeis_catalog:A078812:lean_proved, oeis_catalog:A085478:lean_proved, oeis_catalog:A086645:lean_proved, oeis_catalog:A089627:lean_proved, oeis_catalog:A091042:lean_proved, oeis_catalog:A098158:lean_proved, oeis_catalog:A098172:lean_proved, oeis_catalog:A099097:lean_proved, oeis_catalog:A104684:lean_proved, oeis_catalog:A105278:lean_proved, oeis_catalog:A106800:lean_proved, oeis_catalog:A111125:lean_proved, oeis_catalog:A114656:lean_proved, oeis_catalog:A122542:lean_proved, oeis_catalog:A123160:lean_proved, oeis_catalog:A127648:lean_proved, oeis_catalog:A131689:lean_proved, oeis_catalog:A134991:lean_proved, oeis_catalog:A142075:lean_proved, oeis_catalog:A142459:lean_proved, oeis_catalog:A143395:lean_proved, oeis_catalog:A154602:lean_proved, oeis_catalog:A156289:lean_proved, oeis_catalog:A158909:lean_proved, oeis_catalog:A166973:lean_proved, oeis_catalog:A173018:lean_proved, oeis_catalog:A201637:lean_proved, oeis_catalog:A201730:lean_proved, oeis_catalog:A219836:lean_proved, oeis_catalog:A225117:lean_proved, oeis_catalog:A225466:lean_proved, oeis_catalog:A225467:lean_proved, oeis_catalog:A225468:lean_proved, oeis_catalog:A225469:lean_proved, oeis_catalog:A225472:lean_proved, oeis_catalog:A225473:lean_proved, oeis_catalog:A235706:lean_proved, oeis_catalog:A253283:lean_proved, oeis_catalog:A259454:lean_proved, oeis_catalog:A269939:lean_proved, oeis_catalog:A269945:lean_proved, oeis_catalog:A271697:lean_proved, oeis_catalog:A271703:lean_proved, oeis_catalog:A271705:lean_proved, oeis_catalog:A286724:lean_proved, oeis_catalog:A292219:lean_proved, oeis_catalog:A321331:lean_proved, oeis_catalog:A334218:lean_proved, oeis_catalog:A390433: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 >= 1, 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":"Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (2) theta + (1) theta^2 for every n >= 1 is certified; the lift is Open (open: no route proved that every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"preserver_image"}