LMOpSolve¶

solving of linear homogeneous Mahlerian equations¶

Calling sequence:¶

$\operatorname{LMOpSolve}(L, x, M, b)$

Parameters:¶

  • $L$, a linear Mahler operator, that is a polynomial in $M$ with rational functions coefficients in $x$
  • $x$, the variable
  • $M$, the name of the Mahler operator
  • $b$, the radix

Description:¶

  • $\operatorname{LMOpSolve}$ computes the general solution of the equation $Ly = 0$ in the difference ring of elements that can be called extended Puiseux series generated by the elements $\ln(1/x)^{\log_b(\lambda)} \times (\text{a Puiseux series in $x$})$, with $\lambda$ a nonzero algebraic number over the ground field.
  • It also computes the ramified rational solutions $u$ of the (non-linear) Mahlerian Riccati equation associated to $L$, or equivalently the right first order factors $M - u$ of $L$ with $u$ a ramified rational function.
  • Some optional keyword parameters are available to select the solutions type:
    • ${\mathit output} = {\mathit polynom}$ for polynomial solutions,
    • ${\mathit output} = {\mathit ratpoly}$ for rational functions solutions
    • ${\mathit output} = {\mathit ramratpoly}$ for ramified rational functions solutions
    • ${\mathit output} = {\mathit mhypergeom}$ for Mahlerian hypergeometric solutions,
    • ${\mathit output} = {\mathit series}$ for formal power series solutions,
    • ${\mathit output} = {\mathit laurentseries}$ for formal Laurent series solutions,
    • ${\mathit output} = {\mathit puiseuxseries}$ for formal Puiseux series solutions,
    • ${\mathit output} = {\mathit extendedpuiseuxseries}$ is the default,
    • ${\mathit output} = {\mathit riccati}$ for the solutions to the Riccati equation associated to $L$.
  • The solutions are by default expressed as parametrized expressions. The option ${\mathit free} =$ a name or symbol allows to choose the name of the parameters, which are then, for example, $C_1$, $C_2\ldots$ if the name $C$ is chosen. The default name is $\_C$.
  • Except for the case of the option ${\mathit output} = {\mathit riccati}$ (for which the equation is not linear), the option ${\mathit output\_format} = {\mathit basis}$provides a basis of solutions.
    • It is written as a list of algebraic expressions of the expected type.
    • There is one exception: with the option ${\mathit output} = {\mathit mhypergeom}$, the output is a set of lists. Each list provides a basis of a maximal vector space of hypergeometric solutions, corresponding to a similarity class with respect to the field of ramified rational fractions on the ground field.
  • The option ${\mathit field\_extension} = \ldots$ permits to choose where the possible $\lambda$ are searched for.
    • The default is ${\mathit field\_extension}= \{\}$, which computes the possible $\lambda$ in the field of coefficients of $L$.
    • Another choice is ${\mathit field\_extension}$ = a field extension à la Maple, that is a radalgnum or a set of radalgnum.
    • Last, the option ${\mathit field\_extension} = {\mathit splitting\_field}$ computes the possible $\lambda$ in the algebraic closure of the base field.
    • For both cases ${\mathit output} = {\mathit riccati}$ or ${\mathit output} = {\mathit mhypergeom}$, the choice ${\mathit field\_extension} = {\mathit splitting\_field}$ allows not only an extension that provides all possible $\lambda$, but also a computation whose termination and correction are guaranteed. Currently, the procedure uses the procedure absPrimdecGTZ of Singular, which has to be installed.

References:¶

  • Frédéric Chyzak, Thomas Dreyfus, Philippe Dumas, and Marc Mezzarobba (2018). Computing solutions of linear Mahler equations. Mathematics of Computation 87.314, pp. 2977–3021.
  • Roques Julien (2018). On the algebraic relations between Mahler functions. Trans. Amer. Math. Soc. 370.1, pp. 321--355.
  • Frédéric Chyzak, Thomas Dreyfus, Philippe Dumas, and Marc Mezzarobba. First-order factors of linear Mahler operators. In preparation.

Example:¶

In [1]:
libname := libname, FileTools:-JoinPath(["maple","lib","dcfun.mla"],base=homedir):
In [2]:
 with(dcfun):
Out[2]:

Let us consider the following linear Mahler operator.

In [3]:
 b := 3:
Out[3]:

$$$$

In [4]:
 L := 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Out[4]:

$$-6334 x^{252}+42294 x^{246}+121172 x^{240}+152212 x^{236}+163326 x^{234}+160628 x^{232}+173166 x^{230}+177368 x^{228}+194216 x^{226}+187264 x^{224}+187112 x^{222}+171000 x^{220}+145672 x^{218}+120516 x^{216}+97378 x^{214}+90740 x^{212}+79102 x^{210}+74514 x^{208}+70758 x^{206}+59286 x^{202}+46038 x^{198}+49974 x^{194}+87078 x^{190}+109950 x^{186}+121582 x^{182}+124408 x^{178}+148756 x^{174}+181100 x^{170}+201520 x^{166}+206488 x^{162}+196408 x^{158}+177436 x^{154}+136324 x^{150}+80788 x^{146}+19664 x^{142}-29812 x^{138}-70380 x^{134}-99492 x^{130}-118728 x^{126}-121288 x^{122}-114124 x^{118}-104716 x^{114}-99724 x^{110}-98448 x^{106}-125788 x^{123}-114392 x^{117}-107708 x^{115}-105944 x^{113}-102332 x^{111}-101528 x^{109}-100252 x^{107}-98720 x^{105}-95944 x^{103}-93552 x^{101}-89884 x^{99}-86982 x^{97}-85484 x^{95}-84786 x^{93}-84178 x^{91}-85178 x^{89}-85330 x^{87}-86330 x^{85}-86226 x^{83}-84970 x^{79}-81354 x^{77}-77986 x^{75}-73750 x^{73}-68370 x^{71}-61270 x^{69}-54290 x^{67}-38234 x^{63}-29642 x^{61}-22698 x^{59}-8660 x^{55}-2966 x^{53}+\left(-216 x^{272}-216 x^{271}-300 x^{270}-336 x^{269}-504 x^{268}-560 x^{267}-572 x^{266}-612 x^{265}-740 x^{264}-634 x^{263}-632 x^{262}-618 x^{261}-666 x^{260}-624 x^{259}-542 x^{258}-587 x^{257}-608 x^{256}-620 x^{255}-1791 x^{254}-1871 x^{253}-2358 x^{252}-2202 x^{251}-3065 x^{250}-3337 x^{249}-3843 x^{248}-3690 x^{247}-4537 x^{246}-4001 x^{245}-4505 x^{244}-4026 x^{243}-980 x^{242}-209 x^{241}+463 x^{240}+5556 x^{239}+5006 x^{238}+1207 x^{237}+566 x^{236}+2154 x^{235}+1642 x^{234}+1930 x^{233}+146 x^{232}+510 x^{231}+3128 x^{230}+1114 x^{229}+26 x^{228}+574 x^{227}+2588 x^{226}+2786 x^{225}+2372 x^{224}+2374 x^{223}+3152 x^{222}+1230 x^{221}+1128 x^{220}+2530 x^{219}+2750 x^{218}+1842 x^{217}+1282 x^{216}-1604 x^{215}-916 x^{214}-2028 x^{213}-5815 x^{212}-5320 x^{211}-1232 x^{210}-1398 x^{209}-2161 x^{208}-1446 x^{207}-1281 x^{206}-632 x^{205}-1031 x^{204}-4611 x^{203}-1909 x^{202}-1036 x^{201}-313 x^{200}-2873 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x^{120}-2 x^{117}\right) M^{4}-44 x^{5}-46038 x^{65}+38 x^{52}-5160 x^{54}+3110 x^{22}+6698 x^{48}-32960 x^{62}-17772 x^{58}+9586 x^{46}+10618 x^{42}+4594 x^{30}+4368 x^{28}+494 x^{14}+7502 x^{36}+5346 x^{33}+3888 x^{25}+8842 x^{38}+3720 x^{24}+4138 x^{26}+1096 x^{17}-15134 x^{57}+10930 x^{44}+4706 x^{49}+9630 x^{41}+128814 x^{337}+133080 x^{336}+145486 x^{335}+149800 x^{334}+163086 x^{333}+166120 x^{332}+176678 x^{331}+182680 x^{330}+196734 x^{329}+200920 x^{328}+212090 x^{327}+218168 x^{326}+232336 x^{325}+236596 x^{324}+247778 x^{323}+253874 x^{322}+267856 x^{321}+272052 x^{320}+283138 x^{319}+282098 x^{318}+293712 x^{317}+294356 x^{316}+304234 x^{315}+303186 x^{314}+314788 x^{313}+312524 x^{312}+321434 x^{311}+318934 x^{310}+283768 x^{309}+266076 x^{308}+251844 x^{307}+201936 x^{305}+154308 x^{303}+93200 x^{301}-31644 x^{299}-87040 x^{297}-175884 x^{295}-151600 x^{293}-190392 x^{291}-154598 x^{289}-135232 x^{287}-72614 x^{285}-54168 x^{283}+23568 x^{386}+27288 x^{384}+30104 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x^{210}+7622 x^{209}+6228 x^{208}+9145 x^{207}-1119 x^{206}-2630 x^{205}-4612 x^{204}-5467 x^{203}-10753 x^{202}-6914 x^{201}-5259 x^{200}-1653 x^{199}-4497 x^{198}+4541 x^{197}+7701 x^{196}+14143 x^{195}+18443 x^{194}+22341 x^{193}+18305 x^{192}+26395 x^{191}+24171 x^{190}+27713 x^{189}+27725 x^{188}+32739 x^{187}+29163 x^{186}+35811 x^{185}+31427 x^{184}+33203 x^{183}+32352 x^{182}+37293 x^{181}+35775 x^{180}+43464 x^{179}+41940 x^{178}+46885 x^{177}+49431 x^{176}+52366 x^{175}+46536 x^{174}+52138 x^{173}+50765 x^{172}+53404 x^{171}+49967 x^{170}+53960 x^{169}+52307 x^{168}+56859 x^{167}+51243 x^{166}+53392 x^{165}+51837 x^{164}+55761 x^{163}+51753 x^{162}+56363 x^{161}+54927 x^{160}+57959 x^{159}+52854 x^{158}+55367 x^{157}+50419 x^{156}+52957 x^{155}+46132 x^{154}+46583 x^{153}+41262 x^{152}+43431 x^{151}+36656 x^{150}+37240 x^{149}+31622 x^{148}+32417 x^{147}+25780 x^{146}+26406 x^{145}+21636 x^{144}+22838 x^{143}+16186 x^{142}+15614 x^{141}+10377 x^{140}+11114 x^{139}+5068 x^{138}+4224 x^{137}-1405 x^{136}-2164 x^{135}-9345 x^{134}-10760 x^{133}-15513 x^{132}-15636 x^{131}-21015 x^{130}-19244 x^{129}-23170 x^{128}-21538 x^{127}-24343 x^{126}-22375 x^{125}-25736 x^{124}-24422 x^{123}-27168 x^{122}-24965 x^{121}-26806 x^{120}-25671 x^{119}-28316 x^{118}-26803 x^{117}-28822 x^{116}-27729 x^{115}-29336 x^{114}-27582 x^{113}-29210 x^{112}-28659 x^{111}-30058 x^{110}-28124 x^{109}-28728 x^{108}-27888 x^{107}-28986 x^{106}-27580 x^{105}-28248 x^{104}-27610 x^{103}-28434 x^{102}-27167 x^{101}-27958 x^{100}-27834 x^{99}-28817 x^{98}-27703 x^{97}-28230 x^{96}-28096 x^{95}-28983 x^{94}-28007 x^{93}-28395 x^{92}-28122 x^{91}-28623 x^{90}-27548 x^{89}-27859 x^{88}-27850 x^{87}-28411 x^{86}-27436 x^{85}-27611 x^{84}-27602 x^{83}-28147 x^{82}-27300 x^{81}-27751 x^{80}-27890 x^{79}-28435 x^{78}-27452 x^{77}-27415 x^{76}-26330 x^{75}-26659 x^{74}-25148 x^{73}-24559 x^{72}-23498 x^{71}-23731 x^{70}-22172 x^{69}-21535 x^{68}-20322 x^{67}-20139 x^{66}-18560 x^{65}-17807 x^{64}-16546 x^{63}-16363 x^{62}-14748 x^{61}-13847 x^{60}-12552 x^{59}-12315 x^{58}-10776 x^{57}-9823 x^{56}-8476 x^{55}-8039 x^{54}-6238 x^{53}-5171 x^{52}-3956 x^{51}-3687 x^{50}-2416 x^{49}-2503 x^{48}-1535 x^{47}-1821 x^{46}-1232 x^{45}-1334 x^{44}-501 x^{43}-769 x^{42}-221 x^{41}-448 x^{40}-67 x^{39}-324 x^{38}+139 x^{37}-38 x^{36}+367 x^{35}+94 x^{34}+343 x^{33}+117 x^{32}+443 x^{31}+280 x^{30}+477 x^{29}+203 x^{28}+297 x^{27}+73 x^{26}+209 x^{25}+67 x^{24}+177 x^{23}+x^{22}+69 x^{21}-37 x^{20}+105 x^{19}+57 x^{18}+165 x^{17}+97 x^{16}+173 x^{15}+124 x^{14}+209 x^{13}+175 x^{12}+215 x^{11}+130 x^{10}+119 x^{9}+60 x^{8}+81 x^{7}+62 x^{6}+66 x^{5}+32 x^{4}+19 x^{3}-2\right) M^{2}+3982 x^{27}+238234 x^{306}+182546 x^{304}+136218 x^{302}+15338 x^{300}-77850 x^{298}-150502 x^{296}-160530 x^{294}-187882 x^{292}-170434 x^{290}-146686 x^{288}-94328 x^{286}-65070 x^{284}-63452 x^{282}-75864 x^{280}-63268 x^{278}-52296 x^{276}-28260 x^{274}-64136 x^{272}-37368 x^{270}-50494 x^{268}-29448 x^{266}-34882 x^{264}-3146 x^{262}+3782 x^{260}+19678 x^{258}+9146 x^{256}+8078 x^{254}-5022 x^{250}+3890 x^{248}+72596 x^{244}+110334 x^{242}+144750 x^{238}+1598 x^{18}+610 x^{15}+1858 x^{19}-24 x^{4}+5902 x^{34}-36 x^{6}+2 x^{11}+208 x^{13}-40 x^{7}+39158 x^{195}+85178 x^{189}+102706 x^{187}+104586 x^{185}+114550 x^{183}+115900 x^{181}+119662 x^{179}+123964 x^{177}+132726 x^{175}+151740 x^{173}+165642 x^{171}+183226 x^{169}+189002 x^{167}+198682 x^{165}+201138 x^{163}+201482 x^{161}+193194 x^{159}+188198 x^{157}+175482 x^{155}+162518 x^{153}+144562 x^{151}+118358 x^{149}+90346 x^{147}+60842 x^{145}+30026 x^{143}+4226 x^{141}-22610 x^{139}-44462 x^{137}-64814 x^{135}-82576 x^{133}-96158 x^{131}-108556 x^{129}-120148 x^{127}-126340 x^{125}-124280 x^{121}-119596 x^{119}-4 x^{2}-16 x^{3}+159236 x^{152}+203884 x^{160}+164888 x^{172}+131970 x^{176}+121906 x^{180}+115270 x^{184}+101058 x^{188}+65698 x^{192}+40834 x^{196}+50162 x^{200}+66162 x^{204}-24 x^{8}-44 x^{9}-36 x^{10}+144 x^{12}-4304 x^{140}+110504 x^{148}+204336 x^{164}+\left(216 x^{380}+216 x^{379}+300 x^{378}+336 x^{377}+504 x^{376}+560 x^{375}+572 x^{374}+612 x^{373}+740 x^{372}+634 x^{371}+632 x^{370}+618 x^{369}+666 x^{368}+624 x^{367}+542 x^{366}+587 x^{365}+608 x^{364}+620 x^{363}+495 x^{362}+575 x^{361}+558 x^{360}+618 x^{359}+473 x^{358}+577 x^{357}+579 x^{356}+666 x^{355}+505 x^{354}+377 x^{353}+353 x^{352}+366 x^{351}+164 x^{350}+89 x^{349}+9 x^{348}+88 x^{347}-122 x^{346}-155 x^{345}-68 x^{344}+30 x^{343}-128 x^{342}-80 x^{341}-58 x^{340}+120 x^{339}-97 x^{338}-22 x^{337}-54 x^{336}+167 x^{335}-85 x^{334}+28 x^{333}-52 x^{332}+189 x^{331}-87 x^{330}+7 x^{329}-100 x^{328}+157 x^{327}-103 x^{326}+17 x^{325}-100 x^{324}+162 x^{323}-103 x^{322}+17 x^{321}-94 x^{320}+172 x^{319}-95 x^{318}+20 x^{317}-96 x^{316}+172 x^{315}-96 x^{314}+20 x^{313}-96 x^{312}+172 x^{311}-96 x^{310}+20 x^{309}-7872 x^{308}-7604 x^{307}-10896 x^{306}-9484 x^{305}-15648 x^{304}-16388 x^{303}-19680 x^{302}-18124 x^{301}-24288 x^{300}-20060 x^{299}-23496 x^{298}-19840 x^{297}-3504 x^{296}+2116 x^{295}+5456 x^{294}+37552 x^{293}+32196 x^{292}+10848 x^{291}+490 x^{290}+11788 x^{289}+4218 x^{288}+934 x^{287}-16128 x^{286}-14654 x^{285}-12949 x^{284}-22644 x^{283}-26596 x^{282}-29197 x^{281}-24289 x^{280}-23702 x^{279}-20262 x^{278}-21059 x^{277}-19343 x^{276}-19697 x^{275}-18774 x^{274}-19587 x^{273}-16519 x^{272}-16755 x^{271}-14418 x^{270}-7256 x^{269}-3175 x^{268}-1243 x^{267}+10664 x^{266}+9594 x^{265}+2453 x^{264}-2288 x^{263}+1726 x^{262}-1908 x^{261}-5144 x^{260}-12414 x^{259}-12464 x^{258}-16273 x^{257}-19134 x^{256}-19146 x^{255}-17889 x^{254}-18413 x^{253}-17100 x^{252}-14332 x^{251}-11707 x^{250}-11367 x^{249}-6473 x^{248}-6132 x^{247}-6291 x^{246}-6835 x^{245}-3767 x^{244}-2816 x^{243}-1410 x^{242}-2959 x^{241}-1771 x^{240}-3912 x^{239}-2680 x^{238}-3607 x^{237}+232 x^{236}-1710 x^{235}-824 x^{234}+142 x^{233}+4320 x^{232}+4994 x^{231}+11244 x^{230}+9294 x^{229}+8576 x^{228}+7602 x^{227}+10716 x^{226}+9546 x^{225}+9616 x^{224}+7426 x^{223}+7616 x^{222}+1930 x^{221}+2848 x^{220}+3654 x^{219}+5192 x^{218}+1186 x^{217}+1320 x^{216}+726 x^{215}+3244 x^{214}+1766 x^{213}+4750 x^{212}+4206 x^{211}+4876 x^{210}+1258 x^{209}+3674 x^{208}+3782 x^{207}+3570 x^{206}-202 x^{205}-326 x^{204}-3888 x^{203}-2538 x^{202}-2718 x^{201}-2152 x^{200}-3632 x^{199}-3382 x^{198}-4546 x^{197}-3308 x^{196}-2740 x^{195}+100 x^{194}-1086 x^{193}-1476 x^{192}-1838 x^{191}+228 x^{190}-328 x^{189}+172 x^{188}-580 x^{187}-30 x^{186}-2091 x^{185}-1704 x^{184}-1404 x^{183}+844 x^{182}-953 x^{181}-836 x^{180}+449 x^{179}+2624 x^{178}+2443 x^{177}+5142 x^{176}+5085 x^{175}+4636 x^{174}+3717 x^{173}+4770 x^{172}+5243 x^{171}+5122 x^{170}+3673 x^{169}+3216 x^{168}+1397 x^{167}+1642 x^{166}+1723 x^{165}+1756 x^{164}+829 x^{163}+578 x^{162}+5 x^{161}+566 x^{160}+875 x^{159}+2061 x^{158}+1525 x^{157}+1158 x^{156}+1023 x^{155}+1911 x^{154}+1957 x^{153}+1759 x^{152}+1477 x^{151}+1375 x^{150}+342 x^{149}+345 x^{148}+1059 x^{147}+894 x^{146}+286 x^{145}-307 x^{144}-159 x^{143}-162 x^{142}-232 x^{141}-601 x^{140}-359 x^{139}-1060 x^{138}-1324 x^{137}-1929 x^{136}-1865 x^{135}-2558 x^{134}-2882 x^{133}-3537 x^{132}-3400 x^{131}-3844 x^{130}-3552 x^{129}-4054 x^{128}-3616 x^{127}-3670 x^{126}-3358 x^{125}-3810 x^{124}-3480 x^{123}-3624 x^{122}-3392 x^{121}-3758 x^{120}-3383 x^{119}-3474 x^{118}-3324 x^{117}-3819 x^{116}-3527 x^{115}-3618 x^{114}-3400 x^{113}-3639 x^{112}-2747 x^{111}-2652 x^{110}-2080 x^{109}-1839 x^{108}-791 x^{107}-594 x^{106}-196 x^{105}-96 x^{104}+619 x^{103}+324 x^{102}+551 x^{101}+324 x^{100}+457 x^{99}+72 x^{98}+173 x^{97}+120 x^{96}+334 x^{95}+48 x^{94}+35 x^{93}-48 x^{92}+124 x^{91}-54 x^{90}-28 x^{89}-48 x^{88}+124 x^{87}-48 x^{86}-28 x^{85}-48 x^{84}+124 x^{83}-48 x^{82}-28 x^{81}-48 x^{80}+124 x^{79}-48 x^{78}-28 x^{77}-48 x^{76}+124 x^{75}-48 x^{74}-28 x^{73}-48 x^{72}+124 x^{71}-48 x^{70}-28 x^{69}-54 x^{68}+114 x^{67}-56 x^{66}-31 x^{65}-52 x^{64}+114 x^{63}-91 x^{62}-79 x^{61}-100 x^{60}+90 x^{59}-31 x^{58}+181 x^{57}+222 x^{56}+530 x^{55}+569 x^{54}+833 x^{53}+908 x^{52}+1158 x^{51}+1150 x^{50}+1303 x^{49}+1214 x^{48}+1407 x^{47}+1290 x^{46}+1249 x^{45}+1130 x^{44}+1281 x^{43}+1222 x^{42}+1214 x^{41}+1132 x^{40}+1243 x^{39}+1169 x^{38}+1142 x^{37}+1098 x^{36}+1257 x^{35}+1217 x^{34}+1190 x^{33}+1124 x^{32}+1197 x^{31}+957 x^{30}+868 x^{29}+684 x^{28}+597 x^{27}+305 x^{26}+182 x^{25}+56 x^{24}+16 x^{23}-165 x^{22}-124 x^{21}-193 x^{20}-124 x^{19}-111 x^{18}-40 x^{17}-67 x^{16}-56 x^{15}-70 x^{14}-32 x^{13}-21 x^{12}+2 x^{9}\right) M^{3}-84772 x^{86}-83620 x^{90}-83608 x^{94}-87156 x^{98}-93420 x^{102}-64176 x^{70}-74500 x^{74}-85572 x^{82}+4570 x^{31}+4146 x^{29}+3182 x^{23}+652 x^{51}+3496 x^{50}+7084 x^{47}+2664 x^{21}+9374 x^{45}+10662 x^{43}+9006 x^{39}+7674 x^{37}+6318 x^{35}-70216 x^{72}-49152 x^{66}-41552 x^{64}-25784 x^{60}-11464 x^{56}-85904 x^{84}-81732 x^{78}-57096 x^{68}+50720 x^{144}-84642 x^{132}-109826 x^{128}-125432 x^{124}-119888 x^{120}-109200 x^{116}-102192 x^{112}-99576 x^{108}-96168 x^{104}-90790 x^{100}-85822 x^{96}-83982 x^{92}-84944 x^{88}-85200 x^{80}-79360 x^{76}+190208 x^{168}+187012 x^{156}-51810 x^{136}+\left(-2592 x^{401}-2592 x^{400}-3600 x^{399}-4464 x^{398}-6912 x^{397}-7248 x^{396}-9552 x^{395}-10368 x^{394}-13320 x^{393}-15816 x^{392}-18048 x^{391}-19704 x^{390}-22584 x^{389}-24744 x^{388}-27200 x^{387}-29504 x^{386}-32280 x^{385}-34776 x^{384}-36656 x^{383}-38912 x^{382}-41464 x^{381}-43672 x^{380}-45584 x^{379}-47728 x^{378}-49848 x^{377}-52048 x^{376}-53816 x^{375}-53644 x^{374}-55824 x^{373}-57104 x^{372}-58236 x^{371}-58192 x^{370}-60104 x^{369}-60322 x^{368}-61552 x^{367}-61044 x^{366}-60458 x^{365}-60682 x^{364}-60964 x^{363}-60368 x^{362}-60170 x^{361}-60098 x^{360}-59524 x^{359}-58996 x^{358}-58414 x^{357}-58828 x^{356}-58352 x^{355}-58000 x^{354}-57608 x^{353}-57938 x^{352}-57584 x^{351}-57623 x^{350}-57270 x^{349}-57706 x^{348}-72647 x^{347}-72579 x^{346}-78214 x^{345}-83700 x^{344}-97857 x^{343}-99723 x^{342}-112943 x^{341}-118066 x^{340}-135237 x^{339}-145202 x^{338}-158087 x^{337}-166326 x^{336}-181505 x^{335}-189674 x^{334}-203311 x^{333}-213050 x^{332}-227749 x^{331}-236966 x^{330}-251561 x^{329}-261102 x^{328}-276185 x^{327}-285536 x^{326}-300147 x^{325}-309678 x^{324}-324814 x^{323}-334134 x^{322}-348787 x^{321}-350521 x^{320}-365710 x^{319}-371974 x^{318}-384339 x^{317}-386569 x^{316}-400766 x^{315}-403230 x^{314}-415815 x^{313}-416497 x^{312}-394856 x^{311}-397498 x^{310}-396159 x^{309}-379479 x^{308}-357906 x^{307}-356058 x^{306}-335500 x^{305}-318797 x^{304}-290514 x^{303}-195637 x^{302}-176044 x^{301}-123165 x^{300}-65758 x^{299}+23963 x^{298}+54868 x^{297}+120287 x^{296}+134582 x^{295}+153171 x^{294}+175710 x^{293}+188635 x^{292}+180070 x^{291}+178261 x^{290}+165296 x^{289}+134467 x^{288}+114219 x^{287}+86799 x^{286}+59540 x^{285}+79784 x^{284}+52623 x^{283}+46051 x^{282}+41022 x^{281}+59594 x^{280}+39303 x^{279}+48724 x^{278}+27040 x^{277}+21710 x^{276}+58352 x^{275}+46848 x^{274}+39676 x^{273}+40608 x^{272}+58486 x^{271}+37360 x^{270}+47355 x^{269}+29798 x^{268}+38542 x^{267}+24803 x^{266}+22223 x^{265}+4630 x^{264}+820 x^{263}-22823 x^{262}-32361 x^{261}-47067 x^{260}-47042 x^{259}-53035 x^{258}-41884 x^{257}-46731 x^{256}-34118 x^{255}-30579 x^{254}-12664 x^{253}-5795 x^{252}+9060 x^{251}+7197 x^{250}+8484 x^{249}-20115 x^{248}-17120 x^{247}-38079 x^{246}-52894 x^{245}-88765 x^{244}-101328 x^{243}-131220 x^{242}-138236 x^{241}-154773 x^{240}-168543 x^{239}-182620 x^{238}-183324 x^{237}-194393 x^{236}-196331 x^{235}-196196 x^{234}-195146 x^{233}-198345 x^{232}-197799 x^{231}-211764 x^{230}-209390 x^{229}-218693 x^{228}-221465 x^{227}-234526 x^{226}-234806 x^{225}-244930 x^{224}-240091 x^{223}-239778 x^{222}-216415 x^{221}-216998 x^{220}-199303 x^{219}-187472 x^{218}-158601 x^{217}-148262 x^{216}-125128 x^{215}-119242 x^{214}-105553 x^{213}-99827 x^{212}-88596 x^{211}-89554 x^{210}-80955 x^{209}-83525 x^{208}-83628 x^{207}-86607 x^{206}-81769 x^{205}-82565 x^{204}-74363 x^{203}-75455 x^{202}-66017 x^{201}-63813 x^{200}-54371 x^{199}-51511 x^{198}-42069 x^{197}-44117 x^{196}-44515 x^{195}-62919 x^{194}-60221 x^{193}-73357 x^{192}-82899 x^{191}-105043 x^{190}-110949 x^{189}-129095 x^{188}-133363 x^{187}-143339 x^{186}-147853 x^{185}-156187 x^{184}-155803 x^{183}-161625 x^{182}-158995 x^{181}-158371 x^{180}-157810 x^{179}-161383 x^{178}-162431 x^{177}-174194 x^{176}-176110 x^{175}-186067 x^{174}-193013 x^{173}-206768 x^{172}-213430 x^{171}-226578 x^{170}-230687 x^{169}-237900 x^{168}-240819 x^{167}-248666 x^{166}-249011 x^{165}-254144 x^{164}-253651 x^{163}-256250 x^{162}-252211 x^{161}-252248 x^{160}-244203 x^{159}-243798 x^{158}-236043 x^{157}-233840 x^{156}-225535 x^{155}-223522 x^{154}-213015 x^{153}-206782 x^{152}-192755 x^{151}-180246 x^{150}-163125 x^{149}-151038 x^{148}-132483 x^{147}-116298 x^{146}-95737 x^{145}-78434 x^{144}-58653 x^{143}-42498 x^{142}-25325 x^{141}-10522 x^{140}+6735 x^{139}+20470 x^{138}+36609 x^{137}+49468 x^{136}+63687 x^{135}+74413 x^{134}+87087 x^{133}+94012 x^{132}+105222 x^{131}+112809 x^{130}+123463 x^{129}+130382 x^{128}+140552 x^{127}+145661 x^{126}+152516 x^{125}+154188 x^{124}+154860 x^{123}+154536 x^{122}+156276 x^{121}+153972 x^{120}+152232 x^{119}+148836 x^{118}+145968 x^{117}+142434 x^{116}+139964 x^{115}+137784 x^{114}+136222 x^{113}+133794 x^{112}+132732 x^{111}+131840 x^{110}+131682 x^{109}+130962 x^{108}+130394 x^{107}+129520 x^{106}+128610 x^{105}+127114 x^{104}+126006 x^{103}+124368 x^{102}+122708 x^{101}+120428 x^{100}+118486 x^{99}+116837 x^{98}+115566 x^{97}+114460 x^{96}+113395 x^{95}+112473 x^{94}+111950 x^{93}+111582 x^{92}+111241 x^{91}+111177 x^{90}+111217 x^{89}+111272 x^{88}+111353 x^{87}+111390 x^{86}+111601 x^{85}+111656 x^{84}+112041 x^{83}+112142 x^{82}+112369 x^{81}+111648 x^{80}+110945 x^{79}+108574 x^{78}+107373 x^{77}+105240 x^{76}+103441 x^{75}+100754 x^{74}+98817 x^{73}+95360 x^{72}+92435 x^{71}+88334 x^{70}+84217 x^{69}+79606 x^{68}+75587 x^{67}+70702 x^{66}+65969 x^{65}+60726 x^{64}+55827 x^{63}+50726 x^{62}+45917 x^{61}+41110 x^{60}+36569 x^{59}+31802 x^{58}+27341 x^{57}+22548 x^{56}+18239 x^{55}+13906 x^{54}+10324 x^{53}+6658 x^{52}+4607 x^{51}+1521 x^{50}-800 x^{49}-3558 x^{48}-5455 x^{47}-7989 x^{46}-9196 x^{45}-10821 x^{44}-11209 x^{43}-11265 x^{42}-11101 x^{41}-11445 x^{40}-10865 x^{39}-10389 x^{38}-9505 x^{37}-8753 x^{36}-7907 x^{35}-7277 x^{34}-6725 x^{33}-6305 x^{32}-5751 x^{31}-5469 x^{30}-5251 x^{29}-5179 x^{28}-5055 x^{27}-4898 x^{26}-4689 x^{25}-4427 x^{24}-4110 x^{23}-3818 x^{22}-3417 x^{21}-2959 x^{20}-2442 x^{19}-1942 x^{18}-1535 x^{17}-1175 x^{16}-950 x^{15}-670 x^{14}-443 x^{13}-271 x^{12}-176 x^{11}-60 x^{10}-47 x^{9}+2 x^{8}-2 x^{7}+8 x^{6}-4 x^{5}+2 x^{4}-2 x^{3}+4 x^{2}+2\right) M +2592 x^{399}+3456 x^{398}+4752 x^{397}+5184 x^{396}+7776 x^{395}+8640 x^{394}+10944 x^{393}+11712 x^{392}+14808 x^{391}+16272 x^{390}+18720 x^{389}+19704 x^{388}+22296 x^{387}+25728 x^{385}$$

In [5]:
 r, d := degree(L, M), degree(L, x);
Out[5]:

$$4, 401$$

First, let us find the polynomial solutions of the equation $Ly = 0$.

In [6]:
 LMOpSolve(L, x, M, b, 'free' = K, 'output' = 'polynom');
Out[6]:

$$K_{1} \left(x^{2}+x +2\right)$$

There is a line of polynomial generated by $x^2 + x + 2$.

Next, we search for the ramified rational functions and we request a basis.

In [7]:
 LMOpSolve(L, x, M, b, 'output' = 'ramratpoly', 'output_format' = 'basis');
Out[7]:

$$\left[\frac{1-3 x}{x^{2}}, x^{2}+x +2\right]$$

We find a plane of rational functions.

We continue with the Mahlerian hypergeometric functions.

In [8]:
 LMOpSolve(L, x, M, b, 'output' = 'mhypergeom', 'output_format' = 'basis');
Out[8]:

$$\left\{\left[\frac{\overset{\infty}{\underset{\textit{\_k2} =0}{\textcolor{gray}{\prod}}}\! \frac{1}{2 x^{2 \,3^{\textit{\_k2}}}+1}}{\sqrt{x}}\right], \left[\ln \! \left(\frac{1}{x}\right)^{\frac{\ln \left(2\right)}{\ln \left(3\right)}} x \left(\overset{\infty}{\underset{\textit{\_k3} =0}{\textcolor{gray}{\prod}}}\! \frac{1-x^{3^{\textit{\_k3}}}}{1+3 x^{3^{\textit{\_k3}}}}\right)\right], \left[\frac{1-3 x}{x^{2}}, \frac{1}{2} x^{2}+\frac{1}{2} x +1\right]\right\}$$

We find the union of two lines and a plane. These vector spaces are in direct sum. Obviously the rational solutions are hypergeometric solutions.

Let us now look for the formal Laurent series.

In [9]:
 LMOpSolve(L, x, M, b, 'free = K', 'output' = 'laurentseries');
Out[9]:

$$K_{1} x^{-2}-3 K_{1} x^{-1}+2 K_{2}+K_{2} x +K_{2} x^{2}+\mathrm{O}\! \left(x^{7}\right)$$

We obtain the series expansions corresponding to the rational solutions.

Let us search in a larger space, namely the space of Puiseux series.

In [10]:
 LMOpSolve(L, x, M, b, 'free' = K, 'output' = 'puiseuxseries');
Out[10]:

$$\frac{K_{1,1}}{x^{2}}-\frac{3 K_{1,1}}{x}+\frac{K_{1,2}}{\sqrt{x}}+2 K_{1,3}+K_{1,3} x -2 K_{1,2} x^{\frac{3}{2}}+K_{1,3} x^{2}+4 K_{1,2} x^{\frac{7}{2}}-10 K_{1,2} x^{\frac{11}{2}}+O\! \left(x^{\frac{13}{2}}\right)$$

As there are three 'constants' $K_{1,1}$, $K_{1,2}$ and $K_{1,3}$, the space of Puiseux series solutions has dimension $3$.

Note that even if we reduce the accuracy of the expansions, we always obtain three constants.

In [11]:
 Order := 0;
Out[11]:

$$0$$

In [12]:
 LMOpSolve(L, x, M, b, 'free' = K, 'output' = 'puiseuxseries');
Out[12]:

$$\frac{K_{1,1}}{x^{2}}-\frac{3 K_{1,1}}{x}+\frac{K_{1,2}}{\sqrt{x}}+2 K_{1,3}+K_{1,3} x +O\! \left(x^{\frac{3}{2}}\right)$$

In [13]:
 Order := 6:
Out[13]:

$$$$

Now let us search for 'extended' Puiseux series solutions.

In [14]:
 res := LMOpSolve(L, x, M, b, 'free' = K);
Out[14]:

$$\frac{K_{1,1}}{x^{2}}-\frac{3 K_{1,1}}{x}+\frac{K_{1,2}}{\sqrt{x}}+2 K_{1,3}+K_{1,3} x -2 K_{1,2} x^{\frac{3}{2}}+K_{1,3} x^{2}+4 K_{1,2} x^{\frac{7}{2}}-10 K_{1,2} x^{\frac{11}{2}}+O\! \left(x^{\frac{13}{2}}\right)+\ln \! \left(\frac{1}{x}\right)^{\frac{\ln \left(2\right)}{\ln \left(3\right)}} \left(K_{2,1} x -4 K_{2,1} x^{2}+12 K_{2,1} x^{3}-40 K_{2,1} x^{4}+124 K_{2,1} x^{5}-372 K_{2,1} x^{6}+\mathrm{O}\! \left(x^{7}\right)\right)$$

In [15]:
 indets(res, name);
Out[15]:

$$\{x, K_{1,1}, K_{1,2}, K_{1,3}, K_{2,1}\}$$

Then, the space of solutions has dimension $4$.

We can consider a basis of 'extended' Puiseux series solutions.

In [16]:
 LMOpSolve(L, x, M, b, 'output_format' = 'basis');
Out[16]:

$$\mathrm{table}\left(\left[1=\left[2, \left[x^{-4}-3 x^{-2}+\mathrm{O}\! \left(x^{13}\right), x^{-1}-2 x^{3}+4 x^{7}-10 x^{11}+\mathrm{O}\! \left(x^{13}\right), 2+x^{2}+x^{4}+\mathrm{O}\! \left(x^{13}\right)\right], 2\right], 2=\left[1, \left[x -4 x^{2}+12 x^{3}-40 x^{4}+124 x^{5}-372 x^{6}+\mathrm{O}\! \left(x^{7}\right)\right], 1\right], \mathit{roots} =\left\{1, 2\right\}\right]\right)$$

For each $\lambda$ in the set associated to the key $\mathit roots$, the first element of the list provides a suitable ramification order. For example for $\lambda = 1$, the order of ramification is $q = 2$ and this make the link with the expression we obtained with the penultimate command.

Last let us deal with the nonlinear Riccati equation associated with the linear operator $L$.

In [17]:
 LMOpSolve(L, x, M, b, 'output' = 'riccati', 'free' = K);
Out[17]:

$$\left\{\frac{2 x^{2}+1}{x}, \frac{x^{12} K_{2}+K_{2} x^{9}+2 x^{6} K_{2}-3 x^{3} K_{1}+K_{1}}{x^{4} \left(x^{4} K_{2}+K_{2} x^{3}+2 K_{2} x^{2}-3 K_{1} x +K_{1}\right)}, -\frac{2 x^{2} \left(1+3 x \right)}{-1+x}\right\}$$

We find two isolated solutions $\frac{2 x^{2}+1}{x}$, $-\frac{2 x^{2} (1+3 x )}{x -1}$ and the image of a projective line by the map $$(K_1:K_2)\mapsto \frac{x^{12} K_{2}+x^{9} K_{2}+2 x^{6} K_{2}-3 x^{3} K_{1}+K_{1}}{x^{4} (x^{4} K_{2}+x^{3} K_{2}+2 x^{2} K_{2}-3 x K_{1}+K_{1})}.$$

We can verify that we have actual solutions by Euclidean division.

In [18]:
 u := (2*x^2 + 1)/x;
Out[18]:

$$\frac{2 x^{2}+1}{x}$$

In [19]:
 Q, R := LMOpRightEuclideanDivision(L, M - u, x, M, b):
Out[19]:

$$$$

In [20]:
 R;
Out[20]:

$$0$$