Laguerre polynomials
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Polynomials
$n$ 
$L_n(x)$
0:
1
1:
-x + 1
2:
1/2*x^2 - 2*x + 1
3:
-1/6*x^3 + 3/2*x^2 - 3*x + 1
4:
1/24*x^4 - 2/3*x^3 + 3*x^2 - 4*x + 1
5:
-1/120*x^5 + 5/24*x^4 - 5/3*x^3 + 5*x^2 - 5*x + 1
6:
1/720*x^6 - 1/20*x^5 + 5/8*x^4 - 10/3*x^3 + 15/2*x^2 - 6*x + 1
7:
-1/5040*x^7 + 7/720*x^6 - 7/40*x^5 + 35/24*x^4 - 35/6*x^3 + 21/2*x^2 - 7*x + 1
8:
1/40320*x^8 - 1/630*x^7 + 7/180*x^6 - 7/15*x^5 + 35/12*x^4 - 28/3*x^3 + 14*x^2 - 8*x + 1
9:
-1/362880*x^9 + 1/4480*x^8 - 1/140*x^7 + 7/60*x^6 - 21/20*x^5 + 21/4*x^4 - 14*x^3 + 18*x^2 - 9*x + 1
10:
1/3628800*x^10 - 1/36288*x^9 + 1/896*x^8 - 1/42*x^7 + 7/24*x^6 - 21/10*x^5 + 35/4*x^4 - 20*x^3 + 45/2*x^2 - 10*x + 1
11:
-1/39916800*x^11 + 11/3628800*x^10 - 11/72576*x^9 + 11/2688*x^8 - 11/168*x^7 + 77/120*x^6 - 77/20*x^5 + 55/4*x^4 - 55/2*x^3 + 55/2*x^2 - 11*x + 1
12:
1/479001600*x^12 - 1/3326400*x^11 + 11/604800*x^10 - 11/18144*x^9 + 11/896*x^8 - 11/70*x^7 + 77/60*x^6 - 33/5*x^5 + 165/8*x^4 - 110/3*x^3 + 33*x^2 - 12*x + 1
13:
-1/6227020800*x^13 + 13/479001600*x^12 - 13/6652800*x^11 + 143/1814400*x^10 - 143/72576*x^9 + 143/4480*x^8 - 143/420*x^7 + 143/60*x^6 - 429/40*x^5 + 715/24*x^4 - 143/3*x^3 + 39*x^2 - 13*x + 1
14:
1/87178291200*x^14 - 1/444787200*x^13 + 13/68428800*x^12 - 13/1425600*x^11 + 143/518400*x^10 - 143/25920*x^9 + 143/1920*x^8 - 143/210*x^7 + 1001/240*x^6 - 1001/60*x^5 + 1001/24*x^4 - 182/3*x^3 + 91/2*x^2 - 14*x + 1
15:
-1/1307674368000*x^15 + 1/5811886080*x^14 - 1/59304960*x^13 + 13/13685760*x^12 - 13/380160*x^11 + 143/172800*x^10 - 143/10368*x^9 + 143/896*x^8 - 143/112*x^7 + 1001/144*x^6 - 1001/40*x^5 + 455/8*x^4 - 455/6*x^3 + 105/2*x^2 - 15*x + 1
16:
1/20922789888000*x^16 - 1/81729648000*x^15 + 1/726485760*x^14 - 1/11119680*x^13 + 13/3421440*x^12 - 13/118800*x^11 + 143/64800*x^10 - 143/4536*x^9 + 143/448*x^8 - 143/63*x^7 + 1001/90*x^6 - 182/5*x^5 + 455/6*x^4 - 280/3*x^3 + 60*x^2 - 16*x + 1
17:
-1/355687428096000*x^17 + 17/20922789888000*x^16 - 17/163459296000*x^15 + 17/2179457280*x^14 - 17/44478720*x^13 + 221/17107200*x^12 - 221/712800*x^11 + 2431/453600*x^10 - 2431/36288*x^9 + 2431/4032*x^8 - 2431/630*x^7 + 1547/90*x^6 - 1547/30*x^5 + 595/6*x^4 - 340/3*x^3 + 68*x^2 - 17*x + 1
18:
1/6402373705728000*x^18 - 1/19760412672000*x^17 + 17/2324754432000*x^16 - 17/27243216000*x^15 + 17/484323840*x^14 - 17/12355200*x^13 + 221/5702400*x^12 - 221/277200*x^11 + 2431/201600*x^10 - 2431/18144*x^9 + 2431/2240*x^8 - 221/35*x^7 + 1547/60*x^6 - 357/5*x^5 + 255/2*x^4 - 136*x^3 + 153/2*x^2 - 18*x + 1
19:
-1/121645100408832000*x^19 + 19/6402373705728000*x^18 - 19/39520825344000*x^17 + 323/6974263296000*x^16 - 323/108972864000*x^15 + 323/2421619200*x^14 - 323/74131200*x^13 + 4199/39916800*x^12 - 4199/2217600*x^11 + 46189/1814400*x^10 - 46189/181440*x^9 + 4199/2240*x^8 - 4199/420*x^7 + 2261/60*x^6 - 969/10*x^5 + 323/2*x^4 - 323/2*x^3 + 171/2*x^2 - 19*x + 1
20:
1/2432902008176640000*x^20 - 1/6082255020441600*x^19 + 19/640237370572800*x^18 - 19/5928123801600*x^17 + 323/1394852659200*x^16 - 323/27243216000*x^15 + 323/726485760*x^14 - 323/25945920*x^13 + 4199/15966720*x^12 - 4199/997920*x^11 + 46189/907200*x^10 - 4199/9072*x^9 + 4199/1344*x^8 - 323/21*x^7 + 323/6*x^6 - 646/5*x^5 + 1615/8*x^4 - 190*x^3 + 95*x^2 - 20*x + 1
21:
-1/51090942171709440000*x^21 + 1/115852476579840000*x^20 - 1/579262382899200*x^19 + 19/91462481510400*x^18 - 19/1129166438400*x^17 + 323/332107776000*x^16 - 323/7783776000*x^15 + 323/242161920*x^14 - 323/9884160*x^13 + 4199/6842880*x^12 - 4199/475200*x^11 + 4199/43200*x^10 - 4199/5184*x^9 + 323/64*x^8 - 323/14*x^7 + 2261/30*x^6 - 6783/40*x^5 + 1995/8*x^4 - 665/3*x^3 + 105*x^2 - 21*x + 1
22:
1/1124000727777607680000*x^22 - 1/2322315553259520000*x^21 + 1/10532043325440000*x^20 - 1/78990324940800*x^19 + 19/16629542092800*x^18 - 19/256628736000*x^17 + 323/90574848000*x^16 - 323/2476656000*x^15 + 323/88058880*x^14 - 323/4043520*x^13 + 4199/3110400*x^12 - 4199/237600*x^11 + 46189/259200*x^10 - 3553/2592*x^9 + 3553/448*x^8 - 3553/105*x^7 + 24871/240*x^6 - 4389/20*x^5 + 7315/24*x^4 - 770/3*x^3 + 231/2*x^2 - 22*x + 1
23:
-1/25852016738884976640000*x^23 + 23/1124000727777607680000*x^22 - 23/4644631106519040000*x^21 + 23/31596129976320000*x^20 - 23/315961299763200*x^19 + 437/83147710464000*x^18 - 437/1539772416000*x^17 + 7429/634023936000*x^16 - 7429/19813248000*x^15 + 7429/792529920*x^14 - 7429/40435200*x^13 + 96577/34214400*x^12 - 96577/2851200*x^11 + 81719/259200*x^10 - 81719/36288*x^9 + 81719/6720*x^8 - 81719/1680*x^7 + 33649/240*x^6 - 33649/120*x^5 + 8855/24*x^4 - 1771/6*x^3 + 253/2*x^2 - 23*x + 1
24:
1/620448401733239439360000*x^24 - 1/1077167364120207360000*x^23 + 23/93666727314800640000*x^22 - 23/580578888314880000*x^21 + 23/5266021662720000*x^20 - 23/65825270784000*x^19 + 437/20786927616000*x^18 - 437/449100288000*x^17 + 7429/211341312000*x^16 - 7429/7429968000*x^15 + 7429/330220800*x^14 - 7429/18532800*x^13 + 96577/17107200*x^12 - 7429/118800*x^11 + 81719/151200*x^10 - 81719/22680*x^9 + 81719/4480*x^8 - 4807/70*x^7 + 33649/180*x^6 - 1771/5*x^5 + 1771/4*x^4 - 1012/3*x^3 + 138*x^2 - 24*x + 1
25:
-1/15511210043330985984000000*x^25 + 1/24817936069329577574400*x^24 - 1/86173389129616588800*x^23 + 23/11240007277776076800*x^22 - 23/92892622130380800*x^21 + 23/1053204332544000*x^20 - 23/15798064988160*x^19 + 437/5820339732480*x^18 - 437/143712092160*x^17 + 7429/76082872320*x^16 - 7429/2971987200*x^15 + 7429/145297152*x^14 - 7429/8895744*x^13 + 7429/684288*x^12 - 7429/66528*x^11 + 81719/90720*x^10 - 408595/72576*x^9 + 24035/896*x^8 - 24035/252*x^7 + 8855/36*x^6 - 1771/4*x^5 + 6325/12*x^4 - 1150/3*x^3 + 150*x^2 - 25*x + 1
26:
1/403291461126605635584000000*x^26 - 1/596585001666576384000000*x^25 + 1/1909072005333044428800*x^24 - 1/9943083361109606400*x^23 + 23/1729231888888627200*x^22 - 23/17863965794304000*x^21 + 23/243047153664000*x^20 - 23/4253325189120*x^19 + 437/1790873763840*x^18 - 437/49746493440*x^17 + 7429/29262643200*x^16 - 7429/1257379200*x^15 + 7429/67060224*x^14 - 7429/4447872*x^13 + 96577/4790016*x^12 - 96577/498960*x^11 + 1062347/725760*x^10 - 312455/36288*x^9 + 312455/8064*x^8 - 16445/126*x^7 + 23023/72*x^6 - 3289/6*x^5 + 7475/12*x^4 - 1300/3*x^3 + 325/2*x^2 - 26*x + 1
27:
-1/10888869450418352160768000000*x^27 + 1/14936720782466875392000000*x^26 - 1/44191481604931584000000*x^25 + 1/212119111703671603200*x^24 - 1/1473049386831052800*x^23 + 23/320228127571968000*x^22 - 23/3969770176512000*x^21 + 23/63012225024000*x^20 - 23/1260244500480*x^19 + 437/596957921280*x^18 - 437/18424627200*x^17 + 7429/11921817600*x^16 - 7429/558835200*x^15 + 7429/32288256*x^14 - 7429/2306304*x^13 + 96577/2661120*x^12 - 96577/295680*x^11 + 62491/26880*x^10 - 312455/24192*x^9 + 49335/896*x^8 - 9867/56*x^7 + 3289/8*x^6 - 2691/4*x^5 + 2925/4*x^4 - 975/2*x^3 + 351/2*x^2 - 27*x + 1
28:
1/304888344611713860501504000000*x^28 - 1/388888194657798291456000000*x^27 + 1/1066908627319062528000000*x^26 - 1/4734801600528384000000*x^25 + 1/30302730243381657600*x^24 - 1/263044533362688000*x^23 + 23/68620313051136000*x^22 - 23/992442544128000*x^21 + 23/18003492864000*x^20 - 23/405078589440*x^19 + 437/213199257600*x^18 - 437/7238246400*x^17 + 7429/5109350400*x^16 - 7429/259459200*x^15 + 7429/16144128*x^14 - 7429/1235520*x^13 + 96577/1520640*x^12 - 5681/10560*x^11 + 62491/17280*x^10 - 16445/864*x^9 + 9867/128*x^8 - 3289/14*x^7 + 2093/4*x^6 - 819*x^5 + 6825/8*x^4 - 546*x^3 + 189*x^2 - 28*x + 1
29:
-1/8841761993739701954543616000000*x^29 + 29/304888344611713860501504000000*x^28 - 29/777776389315596582912000000*x^27 + 29/3200725881957187584000000*x^26 - 29/18939206402113536000000*x^25 + 29/151513651216908288000*x^24 - 29/1578267200176128000*x^23 + 667/480342191357952000*x^22 - 667/7939540353024000*x^21 + 667/162031435776000*x^20 - 667/4050785894400*x^19 + 12673/2345191833600*x^18 - 12673/86858956800*x^17 + 215441/66421555200*x^16 - 215441/3632428800*x^15 + 215441/242161920*x^14 - 215441/19768320*x^13 + 164749/1520640*x^12 - 164749/190080*x^11 + 95381/17280*x^10 - 95381/3456*x^9 + 95381/896*x^8 - 8671/28*x^7 + 2639/4*x^6 - 7917/8*x^5 + 7917/8*x^4 - 609*x^3 + 203*x^2 - 29*x + 1
30:
1/265252859812191058636308480000000*x^30 - 1/294725399791323398484787200000*x^29 + 29/20325889640780924033433600000*x^28 - 29/77777638931559658291200000*x^27 + 29/426763450927625011200000*x^26 - 29/3156534400352256000000*x^25 + 29/30302730243381657600*x^24 - 29/368262346707763200*x^23 + 667/128091251028787200*x^22 - 667/2381862105907200*x^21 + 667/54010478592000*x^20 - 667/1485288161280*x^19 + 12673/938076733440*x^18 - 12673/37638881280*x^17 + 215441/30996725760*x^16 - 215441/1816214400*x^15 + 215441/129153024*x^14 - 12673/658944*x^13 + 164749/912384*x^12 - 8671/6336*x^11 + 95381/11520*x^10 - 476905/12096*x^9 + 130065/896*x^8 - 5655/14*x^7 + 13195/16*x^6 - 23751/20*x^5 + 9135/8*x^4 - 2030/3*x^3 + 435/2*x^2 - 30*x + 1
31:
-1/8222838654177922817725562880000000*x^31 + 31/265252859812191058636308480000000*x^30 - 31/589450799582646796969574400000*x^29 + 899/60977668922342772100300800000*x^28 - 899/311110555726238633164800000*x^27 + 899/2133817254638125056000000*x^26 - 899/18939206402113536000000*x^25 + 899/212119111703671603200*x^24 - 899/2946098773662105600*x^23 + 20677/1152821259259084800*x^22 - 20677/23818621059072000*x^21 + 20677/594115264512000*x^20 - 20677/17823457935360*x^19 + 392863/12194997534720*x^18 - 392863/526944337920*x^17 + 6678671/464950886400*x^16 - 6678671/29059430400*x^15 + 392863/129153024*x^14 - 392863/11860992*x^13 + 268801/912384*x^12 - 268801/126720*x^11 + 2956811/241920*x^10 - 1344005/24192*x^9 + 175305/896*x^8 - 58435/112*x^7 + 81809/80*x^6 - 56637/40*x^5 + 31465/24*x^4 - 4495/6*x^3 + 465/2*x^2 - 31*x + 1
32:
1/263130836933693530167218012160000000*x^32 - 1/256963707943060088053923840000000*x^31 + 31/16578303738261941164769280000000*x^30 - 31/55261012460873137215897600000*x^29 + 899/7622208615292846512537600000*x^28 - 899/48611024332224786432000000*x^27 + 899/400090735244648448000000*x^26 - 899/4142951400462336000000*x^25 + 899/53029777925917900800*x^24 - 899/828590280092467200*x^23 + 20677/360256643518464000*x^22 - 20677/8187650989056000*x^21 + 20677/222793224192000*x^20 - 20677/7240779786240*x^19 + 392863/5335311421440*x^18 - 392863/247005158400*x^17 + 6678671/232475443200*x^16 - 392863/908107200*x^15 + 392863/72648576*x^14 - 20677/370656*x^13 + 268801/570240*x^12 - 268801/83160*x^11 + 268801/15120*x^10 - 58435/756*x^9 + 58435/224*x^8 - 23374/35*x^7 + 6293/5*x^6 - 25172/15*x^5 + 4495/3*x^4 - 2480/3*x^3 + 248*x^2 - 32*x + 1
33:
-1/8683317618811886495518194401280000000*x^33 + 1/7973661725263440308097515520000000*x^32 - 1/15573558057155156851752960000000*x^31 + 31/1507118521660176469524480000000*x^30 - 31/6698304540711895420108800000*x^29 + 899/1154880093226188865536000000*x^28 - 899/8838368060404506624000000*x^27 + 899/84867731718561792000000*x^26 - 899/1004351854657536000000*x^25 + 899/14462666707068518400*x^24 - 899/251087963664384000*x^23 + 20677/120085547839488000*x^22 - 20677/2977327632384000*x^21 + 20677/87767027712000*x^20 - 20677/3071845969920*x^19 + 392863/2425141555200*x^18 - 392863/119760076800*x^17 + 392863/7044710400*x^16 - 392863/495331200*x^15 + 20677/2201472*x^14 - 20677/224640*x^13 + 268801/362880*x^12 - 268801/55440*x^11 + 128557/5040*x^10 - 642785/6048*x^9 + 385671/1120*x^8 - 29667/35*x^7 + 69223/45*x^6 - 9889/5*x^5 + 1705*x^4 - 2728/3*x^3 + 264*x^2 - 33*x + 1
34:
1/295232799039604140847618609643520000000*x^34 - 1/255391694670937838103476305920000000*x^33 + 1/469038925015496488711618560000000*x^32 - 1/1374137475631337369272320000000*x^31 + 31/177308061371785467002880000000*x^30 - 31/985044785398808150016000000*x^29 + 899/203802369392856858624000000*x^28 - 899/1819664012436221952000000*x^27 + 899/19968878051426304000000*x^26 - 899/265857843879936000000*x^25 + 899/4253725502078976000*x^24 - 899/81234341185536000*x^23 + 20677/42383134531584000*x^22 - 20677/1138389977088000*x^21 + 20677/36139364352000*x^20 - 20677/1355226163200*x^19 + 392863/1141243084800*x^18 - 392863/59880038400*x^17 + 6678671/63402393600*x^16 - 351509/247665600*x^15 + 351509/22014720*x^14 - 351509/2358720*x^13 + 4569617/3991680*x^12 - 198679/27720*x^11 + 2185469/60480*x^10 - 2185469/15120*x^9 + 504339/1120*x^8 - 336226/315*x^7 + 168113/90*x^6 - 11594/5*x^5 + 5797/3*x^4 - 2992/3*x^3 + 561/2*x^2 - 34*x + 1
35:
-1/10333147966386144929666651337523200000000*x^35 + 1/8435222829702975452789103132672000000*x^34 - 1/14593811124053590748770074624000000*x^33 + 1/40203336429899699032424448000000*x^32 - 1/157044282929295699345408000000*x^31 + 31/25329723053112209571840000000*x^30 - 31/168864820354081397145600000*x^29 + 899/40760473878571371724800000*x^28 - 899/415923202842565017600000*x^27 + 899/5134854356081049600000*x^26 - 899/75959383965696000000*x^25 + 899/1336885157796249600*x^24 - 899/27851774120755200*x^23 + 20677/15742307111731200*x^22 - 20677/455355990835200*x^21 + 20677/15488299008000*x^20 - 20677/619531960320*x^19 + 392863/554318069760*x^18 - 392863/30795448320*x^17 + 351509/1811496960*x^16 - 351509/141523200*x^15 + 351509/13208832*x^14 - 351509/1482624*x^13 + 198679/114048*x^12 - 198679/19008*x^11 + 2185469/43200*x^10 - 168113/864*x^9 + 168113/288*x^8 - 168113/126*x^7 + 40579/18*x^6 - 40579/15*x^5 + 6545/3*x^4 - 6545/6*x^3 + 595/2*x^2 - 35*x + 1
36:
1/371993326789901217467999448150835200000000*x^36 - 1/287031887955170692490740314931200000000*x^35 + 1/468623490539054191821616840704000000*x^34 - 1/1216150927004465895730839552000000*x^33 + 1/4467037381099966559158272000000*x^32 - 1/21811705962402180464640000000*x^31 + 31/4221620508852034928640000000*x^30 - 31/32834826179960271667200000*x^29 + 899/9057883084126971494400000*x^28 - 899/103980800710641254400000*x^27 + 899/1426348432244736000000*x^26 - 899/23209811767296000000*x^25 + 899/445628385932083200*x^24 - 899/10057585099161600*x^23 + 20677/6122008321228800*x^22 - 20677/189731662848000*x^21 + 20677/6883688448000*x^20 - 20677/292556759040*x^19 + 392863/277159034880*x^18 - 20677/855429120*x^17 + 351509/1006387200*x^16 - 351509/82555200*x^15 + 351509/8072064*x^14 - 15283/41184*x^13 + 198679/76032*x^12 - 198679/13200*x^11 + 168113/2400*x^10 - 168113/648*x^9 + 168113/224*x^8 - 11594/7*x^7 + 40579/15*x^6 - 15708/5*x^5 + 19635/8*x^4 - 1190*x^3 + 315*x^2 - 36*x + 1
37:
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48:
1/12413915592536072670862289047373375038521486354677760000000000*x^48 - 1/5387984198149337096728424065700249582691617341440000000000*x^47 + 47/229275923325503706243762726200010620540068823040000000000*x^46 - 47/325060382786630963483595547542548138761912320000000000*x^45 + 47/642094583282234001942904785269230891381555200000000*x^44 - 47/1658302126245439054604609466087889698816000000000*x^43 + 2021/231390994359828705293666437128542748672000000000*x^42 - 2021/918218231586621846403438242573582336000000000*x^41 + 82861/179164532992511579786036730258259968000000000*x^40 - 82861/1007800498082877636296456607702712320000000*x^39 + 82861/6625907285226019962501358367539200000000*x^38 - 82861/50474362976098490019054669004800000000*x^37 + 3065857/16370063667923834600774487244800000000*x^36 - 3065857/164205885557877970532460134400000000*x^35 + 3065857/1876638692090033948942401536000000*x^34 - 3065857/24350848080753035669667840000000*x^33 + 3065857/357771872628143774760960000000*x^32 - 3065857/5939572104178168135680000000*x^31 + 95041567/3448783802426033111040000000*x^30 - 95041567/72807658051216254566400000*x^29 + 2756205443/50212177966356037632000000*x^28 - 2756205443/1344969052670251008000000*x^27 + 2756205443/40588915169746944000000*x^26 - 2756205443/1380983800154112000000*x^25 + 2756205443/53029777925917900800*x^24 - 2756205443/2301639666923520000*x^23 + 63392725189/2601853536522240000*x^22 - 63392725189/145144722078720000*x^21 + 63392725189/9215537909760000*x^20 - 2185956041/23038844774400*x^19 + 41533164779/36377123328000*x^18 - 1339779509/112275072000*x^17 + 22776251653/211341312000*x^16 - 22776251653/27243216000*x^15 + 1339779509/242161920*x^14 - 1339779509/43243200*x^13 + 17417133617/119750400*x^12 - 470733341/831600*x^11 + 272529829/151200*x^10 - 20963833/4536*x^9 + 20963833/2240*x^8 - 511313/35*x^7 + 511313/30*x^6 - 71346/5*x^5 + 16215/2*x^4 - 8648/3*x^3 + 564*x^2 - 48*x + 1
49:
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50:
1/30414093201713378043612608166064768844377641568960512000000000000*x^50 - 1/12165637280685351217445043266425907537751056627584204800000000*x^49 + 1/10133808646968222588459011467243571460017539881369600000000*x^48 - 1/13195063342406539828722671181306733671897838387200000000*x^47 + 47/1122984114247365091806184781387807121012581990400000000*x^46 - 47/2653554145196987457008943245245290928668672000000000*x^45 + 47/7862382652435518391137609615541602751610880000000*x^44 - 47/28428036449921812364650447990078109122560000000*x^43 + 2021/5288937013938941835283804277223834255360000000*x^42 - 2021/26984372520096642016754103455223644160000000*x^41 + 82861/6581554273194302930915634989078937600000000*x^40 - 82861/45248185628210832650044990549917696000000*x^39 + 82861/356987657816259034714358899802112000000*x^38 - 82861/3213877805825046711417358516224000000*x^37 + 3065857/1216061872474341998914676195328000000*x^36 - 3065857/14074790190675254617068011520000000*x^35 + 3065857/183833994327186999080071987200000*x^34 - 3065857/2703441093046867633530470400000*x^33 + 3065857/44684976744576324521164800000*x^32 - 3065857/829115779440381021388800000*x^31 + 95041567/534913406090568400896000000*x^30 - 95041567/12481312808779929354240000*x^29 + 2756205443/9468582130798567096320000*x^28 - 2756205443/277777281898427351040000*x^27 + 2756205443/9144931091306250240000*x^26 - 2756205443/338200114323456000000*x^25 + 2756205443/14069124755855769600*x^24 - 2756205443/659490222930739200*x^23 + 63392725189/802857662698291200*x^22 - 2185956041/1658796823756800*x^21 + 2185956041/112843321344000*x^20 - 70514711/282108303360*x^19 + 1339779509/475129774080*x^18 - 1339779509/48392847360*x^17 + 1339779509/5693276160*x^16 - 1339779509/778377600*x^15 + 1339779509/124540416*x^14 - 253471799/4447872*x^13 + 173428073/684288*x^12 - 13340621/14256*x^11 + 146746831/51840*x^10 - 17895955/2592*x^9 + 2556565/192*x^8 - 59455/3*x^7 + 264845/12*x^6 - 52969/3*x^5 + 57575/6*x^4 - 9800/3*x^3 + 1225/2*x^2 - 50*x + 1
Definition
The Laguerre polynomials $L_n$, $n\geq 0$, can be defined as the system of orthogonal polynomials with respect to the inner product (5) such that $\deg L_n=n$ and $L_n(0)=1$.
Parameters
$n$
—   integer ($n \geq 0$)
Formulas
(1)
$L_0(x) = 1$, $L_1(x) = 1-x$, and $(n+1)L_{n+1}(x) = (2n+1-x) L_n(x) - n L_{n-1}(x)$ for $n\geq 1$ (recurrence formula).
(2)
$L_n(x) = \sum_{k=0}^n \binom{n}{k} \frac{(-1)^k}{k!} x^k$ (closed form).
(3)
$\sum_{n=0}^\infty L_n(x)t^n = \frac{1}{1-t} e^{-tx/(1-t)}$ (generating function)..
(4)
$L_n$ is a solution of Laguerre's equation $xy'' + (1-x)y' + ny = 0$.
Comments
(5)
The $L_n$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{0}^{\infty} f(x)g(x) e^{-x} dx$.
Programs
(P1)
Sage
polynomials = {n: laguerre(n,x) for n in [0..100]}
Links
Data properties
Entries are of type: rational polynomial
Table is complete: no