Finite Series Representation of the Inverse Mittag-Leffler Function

Table 1

Coefficients of the inverse function for a power series.

𝑘	Coefficient 𝑏𝑘
1	1𝑎1
2	−𝑎2𝑎^31
3	1𝑎^512𝑎^22−𝑎1𝑎3
4	1𝑎^71−5𝑎^32+5𝑎1𝑎2𝑎3−𝑎^21𝑎4
5	1𝑎^9114𝑎^42−21𝑎1𝑎^22𝑎3+3𝑎^21𝑎^23+6𝑎^21𝑎2𝑎4−𝑎^31𝑎5
6	1𝑎1^11−42𝑎^52+84𝑎1𝑎^32𝑎3−28𝑎^21𝑎2𝑎^23−28𝑎^21𝑎^22𝑎4+7𝑎^31𝑎3𝑎4+7𝑎^31𝑎2𝑎5−𝑎^41𝑎6
7	1𝑎1^13(132𝑎^62−330𝑎1𝑎^42𝑎3+180𝑎^21𝑎^22𝑎^23−12𝑎^31𝑎^33+120𝑎^21𝑎^32𝑎4−72𝑎^31𝑎2𝑎3𝑎4+4𝑎^41𝑎^24−36𝑎^31𝑎^22𝑎5+8𝑎^41𝑎3𝑎5+8𝑎^41𝑎2𝑎6−𝑎^51𝑎7)
8	1𝑎1^15(−429𝑎^72+1287𝑎1𝑎^52𝑎3−990𝑎^21𝑎^32𝑎^23+165𝑎^31𝑎2𝑎^33−495𝑎^21𝑎^42𝑎4+495𝑎^31𝑎^22𝑎3𝑎4−45𝑎^41𝑎^23𝑎4−45𝑎^41𝑎2𝑎^24+165𝑎^31𝑎^32𝑎5−90𝑎^41𝑎2𝑎3𝑎5+9𝑎^51𝑎4𝑎5−45𝑎^41𝑎^22𝑎6+9𝑎^51𝑎3𝑎6+9𝑎^51𝑎2𝑎7−𝑎^61𝑎8)
9	1𝑎1^17(1430𝑎^82−5005𝑎1𝑎^62𝑎3+5005𝑎^21𝑎^42𝑎^23−1430𝑎^31𝑎^22𝑎^33+55𝑎^41𝑎^43+2002𝑎^21𝑎^52𝑎4−2860𝑎^31𝑎^32𝑎3𝑎4+660𝑎^41𝑎2𝑎^23𝑎4+330𝑎^41𝑎^22𝑎^24−55𝑎^51𝑎3𝑎^24−715𝑎^31𝑎^42𝑎5+660𝑎^41𝑎^22𝑎3𝑎5−55𝑎^51𝑎^23𝑎5−110𝑎^51𝑎2𝑎4𝑎5+5𝑎^61𝑎^25+220𝑎^41𝑎^32𝑎6−110𝑎^51𝑎2𝑎3𝑎6+10𝑎^61𝑎4𝑎6−55𝑎^51𝑎^22𝑎7+10𝑎^61𝑎3𝑎7+10𝑎^61𝑎2𝑎8−𝑎^71𝑎9)
10	1𝑎1^19(−4862𝑎^92+19448𝑎1𝑎^72𝑎3−24024𝑎^21𝑎^52𝑎^23+10010𝑎^31𝑎^32𝑎^33−1001𝑎^41𝑎2𝑎^43−8008𝑎^21𝑎^62𝑎4+15015𝑎^31𝑎^42𝑎3𝑎4−6006𝑎^41𝑎^22𝑎^23𝑎4+286𝑎^51𝑎^33𝑎4−2002𝑎^41𝑎^32𝑎^24+858𝑎^51𝑎2𝑎3𝑎^24−22𝑎^61𝑎^34+3003𝑎^31𝑎^52𝑎5−4004𝑎^41𝑎^32𝑎3𝑎5+858𝑎^51𝑎2𝑎^23𝑎5+858𝑎^51𝑎^22𝑎4𝑎5−132𝑎^61𝑎3𝑎4𝑎5−66𝑎^61𝑎2𝑎^25−1001𝑎^41𝑎^42𝑎6+858𝑎^51𝑎^22𝑎3𝑎6−66𝑎^61𝑎^23𝑎6−132𝑎^61𝑎2𝑎4𝑎6+11𝑎^71𝑎5𝑎6+286𝑎^51𝑎^32𝑎7−132𝑎^61𝑎2𝑎3𝑎7+11𝑎^71𝑎4𝑎7−66𝑎^61𝑎^22𝑎8+11𝑎^71𝑎3𝑎8+11𝑎^71𝑎2𝑎9−𝑎^81𝑎10)
11	1𝑎1^21(16796𝑎2^10−75582𝑎1𝑎^82𝑎3+111384𝑎^21𝑎^62𝑎^23−61880𝑎^31𝑎^42𝑎^33+10920𝑎^41𝑎^22𝑎^43−273𝑎^51𝑎^53+31824𝑎^21𝑎^72𝑎4−74256𝑎^31𝑎^52𝑎3𝑎4+43680𝑎^41𝑎^32𝑎^23𝑎4−5460𝑎^51𝑎2𝑎^33𝑎4+10920𝑎^41𝑎^42𝑎^24+−8190𝑎^51𝑎^22𝑎3𝑎^24+546𝑎^61𝑎^23𝑎^24+364𝑎^61𝑎2𝑎^34−12376𝑎^31𝑎^62𝑎5+21840𝑎^41𝑎^42𝑎3𝑎5+−8190𝑎^51𝑎^22𝑎^23𝑎5+364𝑎^61𝑎^33𝑎5−5460𝑎^51𝑎^32𝑎4𝑎5+2184𝑎^61𝑎2𝑎3𝑎4𝑎5−78𝑎^71𝑎^24𝑎5+546𝑎^61𝑎^22𝑎^25−78𝑎^71𝑎3𝑎^25+4368𝑎^41𝑎^52𝑎6−5460𝑎^51𝑎^32𝑎3𝑎6+1092𝑎^61𝑎2𝑎^23𝑎6+1092𝑎^61𝑎^22𝑎4𝑎6−156𝑎^71𝑎3𝑎4𝑎6−156𝑎^71𝑎2𝑎5𝑎6+6𝑎^81𝑎^26−1365𝑎^51𝑎^42𝑎7+1092𝑎^61𝑎^22𝑎3𝑎7+−78𝑎^71𝑎^23𝑎7−156𝑎^71𝑎2𝑎4𝑎7+12𝑎^81𝑎5𝑎7+364𝑎^61𝑎^32𝑎8−156𝑎^71𝑎2𝑎3𝑎8+12𝑎^81𝑎4𝑎8+−78𝑎^71𝑎^22𝑎9+12𝑎^81𝑎3𝑎9+12𝑎^81𝑎2𝑎10−𝑎^91𝑎11)
12	1𝑎1^23(−58786𝑎2^11+293930𝑎1𝑎^92𝑎3−503880𝑎^21𝑎^72𝑎^23+352716𝑎^31𝑎^52𝑎^33−92820𝑎^41𝑎^32𝑎^43+6188𝑎^51𝑎2𝑎^53+−125970𝑎^21𝑎^82𝑎4+352716𝑎^31𝑎^62𝑎3𝑎4−278460𝑎^41𝑎^42𝑎^23𝑎4+61880𝑎^51𝑎^22𝑎^33𝑎4−1820𝑎^61𝑎^43𝑎4+−55692𝑎^41𝑎^52𝑎^24+61880𝑎^51𝑎^32𝑎3𝑎^24−10920𝑎^61𝑎2𝑎^23𝑎^24−3640𝑎^61𝑎^22𝑎^34+455𝑎^71𝑎3𝑎^34+50388𝑎^31𝑎^72𝑎5−111384𝑎^41𝑎^52𝑎3𝑎5+61880𝑎^51𝑎^32𝑎^23𝑎5−7280𝑎^61𝑎2𝑎^33𝑎5+30940𝑎^51𝑎^42𝑎4𝑎5+−21840𝑎^61𝑎^22𝑎3𝑎4𝑎5+1365𝑎^71𝑎^23𝑎4𝑎5+1365𝑎^71𝑎2𝑎^24𝑎5−3640𝑎^61𝑎^32𝑎^25+1365𝑎^71𝑎2𝑎3𝑎^25+−91𝑎^81𝑎4𝑎^25−18564𝑎^41𝑎^62𝑎6+30940𝑎^51𝑎^42𝑎3𝑎6−10920𝑎^61𝑎^22𝑎^23𝑎6+455𝑎^71𝑎^33𝑎6+−7280𝑎^61𝑎^32𝑎4𝑎6+2730𝑎^71𝑎2𝑎3𝑎4𝑎6−91𝑎^81𝑎^24𝑎6+1365𝑎^71𝑎^22𝑎5𝑎6−182𝑎^81𝑎3𝑎5𝑎6+−91𝑎^81𝑎2𝑎^26+6188𝑎^51𝑎^52𝑎7−7280𝑎^61𝑎^32𝑎3𝑎7+1365𝑎^71𝑎2𝑎^23𝑎7+1365𝑎^71𝑎^22𝑎4𝑎7+−182𝑎^81𝑎3𝑎4𝑎7−182𝑎^81𝑎2𝑎5𝑎7+13𝑎^91𝑎6𝑎7−1820𝑎^61𝑎^42𝑎8+1365𝑎^71𝑎^22𝑎3𝑎8−91𝑎^81𝑎^23𝑎8+−182𝑎^81𝑎2𝑎4𝑎8+13𝑎^91𝑎5𝑎8+455𝑎^71𝑎^32𝑎9−182𝑎^81𝑎2𝑎3𝑎9+13𝑎^91𝑎4𝑎9−91𝑎^81𝑎^22𝑎10+13𝑎^91𝑎3𝑎10+13𝑎^91𝑎2𝑎11−𝑎1^10𝑎12)
13	1𝑎1^25(208012𝑎2^12−1144066𝑎1𝑎2^10𝑎3+2238390𝑎^21𝑎^82𝑎^23−1899240𝑎^31𝑎^62𝑎^33+678300𝑎^41𝑎^42𝑎^43+−81396𝑎^51𝑎^22𝑎^53+1428𝑎^61𝑎^63+497420𝑎^21𝑎^92𝑎4−1627920𝑎^31𝑎^72𝑎3𝑎4+1627920𝑎^41𝑎^52𝑎^23𝑎4+−542640𝑎^51𝑎^32𝑎^33𝑎4+42840𝑎^61𝑎2𝑎^43𝑎4+271320𝑎^41𝑎^62𝑎^24−406980𝑎^51𝑎^42𝑎3𝑎^24+128520𝑎^61𝑎^22𝑎^23𝑎^24−4760𝑎^71𝑎^33𝑎^24+28560𝑎^61𝑎^32𝑎^34−9520𝑎^71𝑎2𝑎3𝑎^34+140𝑎^81𝑎^44+−203490𝑎^31𝑎^82𝑎5+542640𝑎^41𝑎^62𝑎3𝑎5−406980𝑎^51𝑎^42𝑎^23𝑎5+85680𝑎^61𝑎^22𝑎^33𝑎5−2380𝑎^71𝑎^43𝑎5+−162792𝑎^51𝑎^52𝑎4𝑎5+171360𝑎^61𝑎^32𝑎3𝑎4𝑎5−28560𝑎^71𝑎2𝑎^23𝑎4𝑎5−14280𝑎^71𝑎^22𝑎^24𝑎5+1680𝑎^81𝑎3𝑎^24𝑎5+21420𝑎^61𝑎^42𝑎^25−14280𝑎^71𝑎^22𝑎3𝑎^25+840𝑎^81𝑎^23𝑎^25+1680𝑎^81𝑎2𝑎4𝑎^25−35𝑎^91𝑎^35+77520𝑎^41𝑎^72𝑎6−162792𝑎^51𝑎^52𝑎3𝑎6+85680𝑎^61𝑎^32𝑎^23𝑎6−9520𝑎^71𝑎2𝑎^33𝑎6+42840𝑎^61𝑎^42𝑎4𝑎6+−28560𝑎^71𝑎^22𝑎3𝑎4𝑎6+1680𝑎^81𝑎^23𝑎4𝑎6+1680𝑎^81𝑎2𝑎^24𝑎6−9520𝑎^71𝑎^32𝑎5𝑎6+3360𝑎^81𝑎2𝑎3𝑎5𝑎6+−210𝑎^91𝑎4𝑎5𝑎6+840𝑎^81𝑎^22𝑎^26−105𝑎^91𝑎3𝑎^26−27132𝑎^51𝑎^62𝑎7+42840𝑎^61𝑎^42𝑎3𝑎7−14280𝑎^71𝑎^22𝑎^23𝑎7+560𝑎^81𝑎^33𝑎7−9520𝑎^71𝑎^32𝑎4𝑎7+3360𝑎^81𝑎2𝑎3𝑎4𝑎7−105𝑎^91𝑎^24𝑎7+1680𝑎^81𝑎^22𝑎5𝑎7−210𝑎^91𝑎3𝑎5𝑎7+−210𝑎^91𝑎2𝑎6𝑎7+7𝑎1^10𝑎^27+8568𝑎^61𝑎^52𝑎8−9520𝑎^71𝑎^32𝑎3𝑎8+1680𝑎^81𝑎2𝑎^23𝑎8+1680𝑎^81𝑎^22𝑎4𝑎8+−210𝑎^91𝑎3𝑎4𝑎8−210𝑎^91𝑎2𝑎5𝑎8+14𝑎1^10𝑎6𝑎8−2380𝑎^71𝑎^42𝑎9+1680𝑎^81𝑎^22𝑎3𝑎9−105𝑎^91𝑎^23𝑎9+−210𝑎^91𝑎2𝑎4𝑎9+14𝑎1^10𝑎5𝑎9+560𝑎^81𝑎^32𝑎10−210𝑎^91𝑎2𝑎3𝑎10+14𝑎1^10𝑎4𝑎10−105𝑎^91𝑎^22𝑎11+14𝑎1^10𝑎3𝑎11+14𝑎1^10𝑎2𝑎12−𝑎1^11𝑎13)
14	1𝑎1^27(−742900𝑎2^13+4457400𝑎1𝑎2^11𝑎3−9806280𝑎^21𝑎^92𝑎^23+9806280𝑎^31𝑎^72𝑎^33−4476780𝑎^41𝑎^52𝑎^43+813960𝑎^51𝑎^32𝑎^53−38760𝑎^61𝑎2𝑎^63−1961256𝑎^21𝑎2^10𝑎4+7354710𝑎^31𝑎^82𝑎3𝑎4−8953560𝑎^41𝑎^62𝑎^23𝑎4+4069800𝑎^51𝑎^42𝑎^33𝑎4−581400𝑎^61𝑎^22𝑎^43𝑎4+11628𝑎^71𝑎^53𝑎4−1279080𝑎^41𝑎^72𝑎^24+2441880𝑎^51𝑎^52𝑎3𝑎^24−1162800𝑎^61𝑎^32𝑎^23𝑎^24+116280𝑎^71𝑎2𝑎^33𝑎^24−193800𝑎^61𝑎^42𝑎^34+116280𝑎^71𝑎^22𝑎3𝑎^34+−6120𝑎^81𝑎^23𝑎^34−3060𝑎^81𝑎2𝑎^44+817190𝑎^31𝑎^92𝑎5−2558160𝑎^41𝑎^72𝑎3𝑎5+2441880𝑎^51𝑎^52𝑎^23𝑎5+−775200𝑎^61𝑎^32𝑎^33𝑎5+58140𝑎^71𝑎2𝑎^43𝑎5+813960𝑎^51𝑎^62𝑎4𝑎5−1162800𝑎^61𝑎^42𝑎3𝑎4𝑎5+348840𝑎^71𝑎^22𝑎^23𝑎4𝑎5−12240𝑎^81𝑎^33𝑎4𝑎5+116280𝑎^71𝑎^32𝑎^24𝑎5−36720𝑎^81𝑎2𝑎3𝑎^24𝑎5+680𝑎^91𝑎^34𝑎5−116280𝑎^61𝑎^52𝑎^25+116280𝑎^71𝑎^32𝑎3𝑎^25−18360𝑎^81𝑎2𝑎^23𝑎^25−18360𝑎^81𝑎^22𝑎4𝑎^25+2040𝑎^91𝑎3𝑎4𝑎^25+680𝑎^91𝑎2𝑎^35−319770𝑎^41𝑎^82𝑎6+813960𝑎^51𝑎^62𝑎3𝑎6−581400𝑎^61𝑎^42𝑎^23𝑎6+116280𝑎^71𝑎^22𝑎^33𝑎6−3060𝑎^81𝑎^43𝑎6−232560𝑎^61𝑎^52𝑎4𝑎6+232560𝑎^71𝑎^32𝑎3𝑎4𝑎6+−36720𝑎^81𝑎2𝑎^23𝑎4𝑎6−18360𝑎^81𝑎^22𝑎^24𝑎6+2040𝑎^91𝑎3𝑎^24𝑎6+58140𝑎^71𝑎^42𝑎5𝑎6−36720𝑎^81𝑎^22𝑎3𝑎5𝑎6+2040𝑎^91𝑎^23𝑎5𝑎6+4080𝑎^91𝑎2𝑎4𝑎5𝑎6−120𝑎1^10𝑎^25𝑎6−6120𝑎^81𝑎^32𝑎^26+2040𝑎^91𝑎2𝑎3𝑎^26−120𝑎1^10𝑎4𝑎^26+116280𝑎^51𝑎^72𝑎7−232560𝑎^61𝑎^52𝑎3𝑎7+116280𝑎^71𝑎^32𝑎^23𝑎7−12240𝑎^81𝑎2𝑎^33𝑎7+58140𝑎^71𝑎^42𝑎4𝑎7+−36720𝑎^81𝑎^22𝑎3𝑎4𝑎7+2040𝑎^91𝑎^23𝑎4𝑎7+2040𝑎^91𝑎2𝑎^24𝑎7−12240𝑎^81𝑎^32𝑎5𝑎7+4080𝑎^91𝑎2𝑎3𝑎5𝑎7+−240𝑎1^10𝑎4𝑎5𝑎7+2040𝑎^91𝑎^22𝑎6𝑎7−240𝑎1^10𝑎3𝑎6𝑎7−120𝑎1^10𝑎2𝑎^27−38760𝑎^61𝑎^62𝑎8+58140𝑎^71𝑎^42𝑎3𝑎8−18360𝑎^81𝑎^22𝑎^23𝑎8+680𝑎^91𝑎^33𝑎8−12240𝑎^81𝑎^32𝑎4𝑎8+4080𝑎^91𝑎2𝑎3𝑎4𝑎8+−120𝑎1^10𝑎^24𝑎8+2040𝑎^91𝑎^22𝑎5𝑎8−240𝑎1^10𝑎3𝑎5𝑎8−240𝑎1^10𝑎2𝑎6𝑎8+15𝑎1^11𝑎7𝑎8+11628𝑎^71𝑎^52𝑎9+−12240𝑎^81𝑎^32𝑎3𝑎9+2040𝑎^91𝑎2𝑎^23𝑎9+2040𝑎^91𝑎^22𝑎4𝑎9−240𝑎1^10𝑎3𝑎4𝑎9−240𝑎1^10𝑎2𝑎5𝑎9+15𝑎1^11𝑎6𝑎9−3060𝑎^81𝑎^42𝑎10+2040𝑎^91𝑎^22𝑎3𝑎10−120𝑎1^10𝑎^23𝑎10−240𝑎1^10𝑎2𝑎4𝑎10+15𝑎1^11𝑎5𝑎10+680𝑎^91𝑎^32𝑎11−240𝑎1^10𝑎2𝑎3𝑎11+15𝑎1^11𝑎4𝑎11−120𝑎1^10𝑎^22𝑎12+15𝑎1^11𝑎3𝑎12+15𝑎1^11𝑎2𝑎13−𝑎1^12𝑎14)