{"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"],["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis_map_A013610 of delannoy"},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-360","-1116","-270","-1053","-1944","-729"],["-876","-3690","-2835","81","0","0"],["-36","-432","81","729","0","0"],["216","486","0","0","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-520","-2460","-4842","-9585","-11097","-5103","-729"],["-1212","-6066","-8613","-5994","-3402","-729","0"],["-180","-1296","-1377","243","0","0","0"],["216","486","0","0","0","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-160","-1344","-4572","-8532","-9153","-4374","-729"],["-336","-2376","-5778","-6075","-3402","-729","0"],["-144","-864","-1458","-486","0","0","0"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["72","180","-54","243","243"],["180","558","351","486","243"],["108","540","405","243","0"],["0","162","0","0","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["72","180","-54","243","243"],["180","558","351","486","243"],["108","540","405","243","0"],["0","162","0","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","0","0"],["-36","-36","-459","0","243"],["108","54","405","243","0"],["0","162","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["-264","-1656","-2646","405","1944","729"],["156","1368","2997","1377","0","0"],["684","1890","-81","-729","0","0"],["-216","-486","0","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-32","-168","252","2943","5751","3645","729"],["48","648","2862","5103","3402","729","0"],["144","864","1458","486","0","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"],["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","3"]]},"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":[["-18","-27"],["9","27"],["-18","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-36","-135","-81"],["24","72","27"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-8","-48","-81","-27"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["9"],["9"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","41472","601344","3545856","10917504","19298088","23225940","28934010","37023723","29229255","11160261","1594323"],["0","0","145152","2166912","13436928","44999712","90218124","119633274","124770537","112488345","70681653","23914845","3188646"],["0","0","165888","2623104","17356032","62495712","134509248","182933802","169116336","114968403","53675541","14348907","1594323"],["0","0","62208","1150848","8584704","33662304","75556476","99871542","79657101","40566663","12223143","1594323","0"],["0","0","0","93312","1119744","5248800","11967264","13345074","6377292","1062882","0","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","41472","601344","3545856","10917504","19298088","23225940","28934010","37023723","29229255","11160261","1594323"],["0","0","145152","2166912","13436928","44999712","90218124","119633274","124770537","112488345","70681653","23914845","3188646"],["0","0","165888","2623104","17356032","62495712","134509248","182933802","169116336","114968403","53675541","14348907","1594323"],["0","0","62208","1150848","8584704","33662304","75556476","99871542","79657101","40566663","12223143","1594323","0"],["0","0","0","93312","1119744","5248800","11967264","13345074","6377292","1062882","0","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","373248","6718464","51648192","220449600","568287576","901323936","859340097","463947993","129140163","14348907"],["0","0","0","1119744","20155392","154944576","661348800","1704862728","2703971808","2578020291","1391843979","387420489","43046721"],["0","0","0","1119744","20155392","154944576","661348800","1704862728","2703971808","2578020291","1391843979","387420489","43046721"],["0","0","0","373248","6718464","51648192","220449600","568287576","901323936","859340097","463947993","129140163","14348907"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","373248","8957952","91958976","530338752","1890985176","4311049392","6267283713","5619988575","2912828121","789189885","86093442"],["0","0","0","373248","13436928","172580544","1150117056","4536380376","11130500304","17083170945","15932069739","8480204037","2338871841","258280326"],["0","0","0","-1119744","-13436928","-34012224","268318656","2263230072","7525204560","13645810557","14076277767","7963643385","2281476213","258280326"],["0","0","0","-1866240","-31352832","-217930176","-792358848","-1518740280","-1096894224","1111243131","2836300617","2137987143","703096443","86093442"],["0","0","0","-746496","-13436928","-103296384","-440899200","-1136575152","-1802647872","-1718680194","-927895986","-258280326","-28697814","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-497664","-11570688","-118132992","-695780928","-2610753120","-6502475880","-10864779804","-12041921619","-8575863417","-3706800975","-875283327","-86093442"],["0","0","0","0","-2239488","-43670016","-370355328","-1787531328","-5393771856","-10522531800","-13267956006","-10517748831","-4950372915","-1248354909","-129140163"],["0","0","0","1492992","30233088","267058944","1346632128","4257196704","8719883928","11549275812","9589852845","4692092589","1219657095","129140163","0"],["0","0","0","995328","20901888","192595968","1021206528","3433974912","7611179904","11207027808","10815887232","6633978003","2463229035","502211745","43046721"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","-331776","-7962624","-85100544","-532998144","-2165864832","-5971034880","-11346855840","-14812323552","-13027213233","-7455054348","-2630632950","-516560652","-43046721"],["0","0","0","-663552","-15925248","-170201088","-1065996288","-4331729664","-11942069760","-22693711680","-29624647104","-26054426466","-14910108696","-5261265900","-1033121304","-86093442"],["0","0","0","-331776","-7962624","-85100544","-532998144","-2165864832","-5971034880","-11346855840","-14812323552","-13027213233","-7455054348","-2630632950","-516560652","-43046721"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","-497664","-10450944","-94618368","-480370176","-1484570592","-2813566752","-3046219812","-1351985904","550041435","856151451","330024861","43046721"],["0","-1492992","-32099328","-297292032","-1544406912","-4894610976","-9577275408","-10941307308","-5774637906","722228319","2310174027","961376769","129140163"],["0","-1492992","-33592320","-324165888","-1750999680","-5776409376","-11850425712","-14546603052","-9211998294","-1133563653","1793613375","903981141","129140163"],["0","-497664","-12690432","-134929152","-790259328","-2807268192","-6223292208","-8454163428","-6508026486","-2233646523","81310473","243931419","43046721"],["0","0","-746496","-13436928","-103296384","-440899200","-1136575152","-1802647872","-1718680194","-927895986","-258280326","-28697814","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","331776","7464960","73529856","414865152","1470083904","3360281760","4844379960","3947543748","985291614","-1120809069","-1076168025","-358722675","-43046721"],["0","663552","15925248","167961600","1022326272","3961374336","10154538432","17299939824","19102115304","12786470460","4392359865","310892985","-215233605","-43046721"],["0","331776","9455616","115333632","800057088","3512496960","10228231584","20066739768","26361599364","22617066078","12147146937","3850290045","645700815","43046721"],["0","0","995328","20901888","192595968","1021206528","3433974912","7611179904","11207027808","10815887232","6633978003","2463229035","502211745","43046721"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","-331776","-7962624","-85100544","-532998144","-2165864832","-5971034880","-11346855840","-14812323552","-13027213233","-7455054348","-2630632950","-516560652","-43046721"],["0","0","-663552","-15925248","-170201088","-1065996288","-4331729664","-11942069760","-22693711680","-29624647104","-26054426466","-14910108696","-5261265900","-1033121304","-86093442"],["0","0","-331776","-7962624","-85100544","-532998144","-2165864832","-5971034880","-11346855840","-14812323552","-13027213233","-7455054348","-2630632950","-516560652","-43046721"]]},"deriv_order":2}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0","0"],["12","18","-27"],["0","18","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-8","-12","54","54"],["0","-24","-72","-27"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","8","48","81","27"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","-62208","-891648","-5692032","-21555072","-52741692","-81172692","-67256811","-14526054","14880348","9565938","1594323"],["0","-279936","-3665088","-20225376","-62670672","-122791302","-155357919","-106052004","-2834352","40389516","20726199","3188646"],["0","-466560","-5681664","-27503712","-67394592","-88735338","-60229980","-5019165","38972340","36137988","12754584","1594323"],["0","-342144","-3934656","-17099424","-32997456","-20063538","20923029","39090438","28343520","10628820","1594323","0"],["0","-93312","-1026432","-4129056","-6718464","-1377810","6967782","5314410","1062882","0","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","41472","601344","3545856","10917504","19298088","23225940","28934010","37023723","29229255","11160261","1594323"],["0","145152","2166912","13436928","44999712","90218124","119633274","124770537","112488345","70681653","23914845","3188646"],["0","165888","2623104","17356032","62495712","134509248","182933802","169116336","114968403","53675541","14348907","1594323"],["0","62208","1150848","8584704","33662304","75556476","99871542","79657101","40566663","12223143","1594323","0"],["0","0","93312","1119744","5248800","11967264","13345074","6377292","1062882","0","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","373248","6718464","51648192","220449600","568287576","901323936","859340097","463947993","129140163","14348907"],["0","0","0","1119744","20155392","154944576","661348800","1704862728","2703971808","2578020291","1391843979","387420489","43046721"],["0","0","0","1119744","20155392","154944576","661348800","1704862728","2703971808","2578020291","1391843979","387420489","43046721"],["0","0","0","373248","6718464","51648192","220449600","568287576","901323936","859340097","463947993","129140163","14348907"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","497664","10450944","94618368","480370176","1484570592","2813566752","3046219812","1351985904","-550041435","-856151451","-330024861","-43046721"],["0","0","0","1492992","32099328","297292032","1544406912","4894610976","9577275408","10941307308","5774637906","-722228319","-2310174027","-961376769","-129140163"],["0","0","0","1492992","33592320","324165888","1750999680","5776409376","11850425712","14546603052","9211998294","1133563653","-1793613375","-903981141","-129140163"],["0","0","0","497664","12690432","134929152","790259328","2807268192","6223292208","8454163428","6508026486","2233646523","-81310473","-243931419","-43046721"],["0","0","0","0","746496","13436928","103296384","440899200","1136575152","1802647872","1718680194","927895986","258280326","28697814","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-331776","-7464960","-73529856","-414865152","-1470083904","-3360281760","-4844379960","-3947543748","-985291614","1120809069","1076168025","358722675","43046721"],["0","0","0","-663552","-15925248","-167961600","-1022326272","-3961374336","-10154538432","-17299939824","-19102115304","-12786470460","-4392359865","-310892985","215233605","43046721"],["0","0","0","-331776","-9455616","-115333632","-800057088","-3512496960","-10228231584","-20066739768","-26361599364","-22617066078","-12147146937","-3850290045","-645700815","-43046721"],["0","0","0","0","-995328","-20901888","-192595968","-1021206528","-3433974912","-7611179904","-11207027808","-10815887232","-6633978003","-2463229035","-502211745","-43046721"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","331776","7962624","85100544","532998144","2165864832","5971034880","11346855840","14812323552","13027213233","7455054348","2630632950","516560652","43046721"],["0","0","0","0","663552","15925248","170201088","1065996288","4331729664","11942069760","22693711680","29624647104","26054426466","14910108696","5261265900","1033121304","86093442"],["0","0","0","0","331776","7962624","85100544","532998144","2165864832","5971034880","11346855840","14812323552","13027213233","7455054348","2630632950","516560652","43046721"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","995328","21275136","198194688","1052699328","3499479936","7518118680","10403489016","8971255521","4519905705","1200525219","129140163"],["0","0","2985984","64571904","608020992","3261394368","10939339008","23690931192","33013114920","28632446757","14487613101","3859855983","416118303"],["0","0","2985984","66064896","634894848","3467987136","11821137408","25964081496","36618410664","32069807145","16343405073","4376416635","473513931"],["0","0","995328","23514624","238505472","1362588480","4822177536","10927844136","15811432632","14127296103","7303593663","1975366197","215233605"],["0","0","0","746496","13436928","103296384","440899200","1136575152","1802647872","1718680194","927895986","258280326","28697814"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","-663552","-15427584","-158630400","-947863296","-3635948736","-9331316640","-16191235800","-18759867300","-14012504847","-6334245279","-1554464925","-157837977","0"],["0","0","-1327104","-31850496","-338162688","-2088322560","-8293104000","-22096608192","-39993651504","-48726762408","-38840896926","-19302468561","-5572158885","-817887699","-43046721"],["0","0","-663552","-17418240","-200434176","-1333055232","-5678361792","-16199266464","-31413595608","-41173922916","-35644279311","-19602201285","-6480922995","-1162261467","-86093442"],["0","0","0","-995328","-20901888","-192595968","-1021206528","-3433974912","-7611179904","-11207027808","-10815887232","-6633978003","-2463229035","-502211745","-43046721"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","331776","7962624","85100544","532998144","2165864832","5971034880","11346855840","14812323552","13027213233","7455054348","2630632950","516560652","43046721"],["0","0","0","663552","15925248","170201088","1065996288","4331729664","11942069760","22693711680","29624647104","26054426466","14910108696","5261265900","1033121304","86093442"],["0","0","0","331776","7962624","85100544","532998144","2165864832","5971034880","11346855840","14812323552","13027213233","7455054348","2630632950","516560652","43046721"]]},"deriv_order":2}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","0","5184","62208","291600","664848","741393","354294","59049"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","-3"]]},"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"],["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","3"]]},"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: coefficient of P_(n-1) must be 1 after normalization; fixed_delay_predecessor_window: coefficient of P_(n-1) must be 1 (+ w(n) t) after normalization)","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) = (2) + (3) x, V_n(x) = 0, W_n(x) = (1) + (3) x; open hypotheses: 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) = (2) + (3) x, B_n(x) = (1) + (3) x: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, -1/3] (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) = (2) + (3) x, B_n(x) = (1) + (3) x","every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, -1/3] (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"}