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Bessel polynomials $y_n$
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Numbers
$n$ 
0:
1
1:
x + 1
2:
3*x^2 + 3*x + 1
3:
15*x^3 + 15*x^2 + 6*x + 1
4:
105*x^4 + 105*x^3 + 45*x^2 + 10*x + 1
5:
945*x^5 + 945*x^4 + 420*x^3 + 105*x^2 + 15*x + 1
6:
10395*x^6 + 10395*x^5 + 4725*x^4 + 1260*x^3 + 210*x^2 + 21*x + 1
7:
135135*x^7 + 135135*x^6 + 62370*x^5 + 17325*x^4 + 3150*x^3 + 378*x^2 + 28*x + 1
8:
2027025*x^8 + 2027025*x^7 + 945945*x^6 + 270270*x^5 + 51975*x^4 + 6930*x^3 + 630*x^2 + 36*x + 1
9:
34459425*x^9 + 34459425*x^8 + 16216200*x^7 + 4729725*x^6 + 945945*x^5 + 135135*x^4 + 13860*x^3 + 990*x^2 + 45*x + 1
10:
654729075*x^10 + 654729075*x^9 + 310134825*x^8 + 91891800*x^7 + 18918900*x^6 + 2837835*x^5 + 315315*x^4 + 25740*x^3 + 1485*x^2 + 55*x + 1
11:
13749310575*x^11 + 13749310575*x^10 + 6547290750*x^9 + 1964187225*x^8 + 413513100*x^7 + 64324260*x^6 + 7567560*x^5 + 675675*x^4 + 45045*x^3 + 2145*x^2 + 66*x + 1
12:
316234143225*x^12 + 316234143225*x^11 + 151242416325*x^10 + 45831035250*x^9 + 9820936125*x^8 + 1571349780*x^7 + 192972780*x^6 + 18378360*x^5 + 1351350*x^4 + 75075*x^3 + 3003*x^2 + 78*x + 1
13:
7905853580625*x^13 + 7905853580625*x^12 + 3794809718700*x^11 + 1159525191825*x^10 + 252070693875*x^9 + 41247931725*x^8 + 5237832600*x^7 + 523783260*x^6 + 41351310*x^5 + 2552550*x^4 + 120120*x^3 + 4095*x^2 + 91*x + 1
14:
213458046676875*x^14 + 213458046676875*x^13 + 102776096548125*x^12 + 31623414322500*x^11 + 6957151150950*x^10 + 1159525191825*x^9 + 151242416325*x^8 + 15713497800*x^7 + 1309458150*x^6 + 87297210*x^5 + 4594590*x^4 + 185640*x^3 + 5460*x^2 + 105*x + 1
15:
6190283353629375*x^15 + 6190283353629375*x^14 + 2988412653476250*x^13 + 924984868933125*x^12 + 205552193096250*x^11 + 34785755754750*x^10 + 4638100767300*x^9 + 496939367925*x^8 + 43212118950*x^7 + 3055402350*x^6 + 174594420*x^5 + 7936110*x^4 + 278460*x^3 + 7140*x^2 + 120*x + 1
16:
191898783962510625*x^16 + 191898783962510625*x^15 + 92854250304440625*x^14 + 28887988983603750*x^13 + 6474894082531875*x^12 + 1109981842719750*x^11 + 150738274937250*x^10 + 16564645597500*x^9 + 1490818103775*x^8 + 110430970650*x^7 + 6721885170*x^6 + 333316620*x^5 + 13226850*x^4 + 406980*x^3 + 9180*x^2 + 136*x + 1
17:
6332659870762850625*x^17 + 6332659870762850625*x^16 + 3070380543400170000*x^15 + 959493919812553125*x^14 + 216659917377028125*x^13 + 37554385678684875*x^12 + 5179915266025500*x^11 + 581419060472250*x^10 + 53835098191875*x^9 + 4141161399375*x^8 + 265034329560*x^7 + 14054850810*x^6 + 611080470*x^5 + 21366450*x^4 + 581400*x^3 + 11628*x^2 + 153*x + 1
18:
221643095476699771875*x^18 + 221643095476699771875*x^17 + 107655217802968460625*x^16 + 33774185977401870000*x^15 + 7675951358500425000*x^14 + 1343291487737574375*x^13 + 187771928393424375*x^12 + 21459648959248500*x^11 + 2034966711652875*x^10 + 161505294575625*x^9 + 10767019638375*x^8 + 602350749000*x^7 + 28109701620*x^6 + 1081142370*x^5 + 33575850*x^4 + 813960*x^3 + 14535*x^2 + 171*x + 1
19:
8200794532637891559375*x^19 + 8200794532637891559375*x^18 + 3989575718580595893750*x^17 + 1255977541034632040625*x^16 + 287080580807915895000*x^15 + 50661278966102805000*x^14 + 7164221267933730000*x^13 + 831561397170879375*x^12 + 80473683597181875*x^11 + 6557114959770375*x^10 + 452214824811750*x^9 + 26428139112375*x^8 + 1305093289500*x^7 + 54057118500*x^6 + 1853386920*x^5 + 51482970*x^4 + 1119195*x^3 + 17955*x^2 + 190*x + 1
20:
319830986772877770815625*x^20 + 319830986772877770815625*x^19 + 155815096120119939628125*x^18 + 49204767195827349356250*x^17 + 11303797869311688365625*x^16 + 2009564065655411265000*x^15 + 287080580807915895000*x^14 + 33774185977401870000*x^13 + 3326245588683517500*x^12 + 277187132390293125*x^11 + 19671344879311125*x^10 + 1192202719958250*x^9 + 61665657928875*x^8 + 2710578370500*x^7 + 100391791500*x^6 + 3088978200*x^5 + 77224455*x^4 + 1514205*x^3 + 21945*x^2 + 210*x + 1
21:
13113070457687988603440625*x^21 + 13113070457687988603440625*x^20 + 6396619735457555416312500*x^19 + 2025596249561559215165625*x^18 + 467445288360359818884375*x^17 + 83648104232906493905625*x^16 + 12057384393932467590000*x^15 + 1435402904039579475000*x^14 + 143540290403957947500*x^13 + 12196233825172897500*x^12 + 886998823648938000*x^11 + 55437426478058625*x^10 + 2980506799895625*x^9 + 137561852302875*x^8 + 5421156741000*x^7 + 180705224700*x^6 + 5019589575*x^5 + 113565375*x^4 + 2018940*x^3 + 26565*x^2 + 231*x + 1
22:
563862029680583509947946875*x^22 + 563862029680583509947946875*x^21 + 275374479611447760672253125*x^20 + 87420469717919924022937500*x^19 + 20255962495615592151656250*x^18 + 3646073249210806587298125*x^17 + 529771326808407794735625*x^16 + 63731888939357328690000*x^15 + 6459313068178107637500*x^14 + 558212240459836462500*x^13 + 41467195005587851500*x^12 + 2660996470946814000*x^11 + 147833137274823000*x^10 + 7107362368981875*x^9 + 294775397791875*x^8 + 10480903032600*x^7 + 316234143225*x^6 + 7972289325*x^5 + 164038875*x^4 + 2656500*x^3 + 31878*x^2 + 253*x + 1
23:
25373791335626257947657609375*x^23 + 25373791335626257947657609375*x^22 + 12404964652972837218854831250*x^21 + 3947034207764084569635628125*x^20 + 917914932038159202240843750*x^19 + 166098892464047855643581250*x^18 + 24307154994738710581987500*x^17 + 2951583106503986284955625*x^16 + 302726472461947311277500*x^15 + 26554953724732220287500*x^14 + 2009564065655411265000*x^13 + 131941075017779527500*x^12 + 7539490001015973000*x^11 + 375268733082243000*x^10 + 16245399700530000*x^9 + 609202488769875*x^8 + 19651693186125*x^7 + 539458244325*x^6 + 12401338950*x^5 + 233107875*x^4 + 3453450*x^3 + 37950*x^2 + 276*x + 1
24:
1192568192774434123539907640625*x^24 + 1192568192774434123539907640625*x^23 + 583597200719403932796125015625*x^22 + 186074469794592558282822468750*x^21 + 43417376285404930265991909375*x^20 + 7894068415528169139271256250*x^19 + 1162692247248334989505068750*x^18 + 142370479254898161980212500*x^17 + 14757915532519931424778125*x^16 + 1311814714001771682202500*x^15 + 100908824153982437092500*x^14 + 6759442766295474255000*x^13 + 395823225053338582500*x^12 + 20298626925812235000*x^11 + 911366923199733000*x^10 + 35739879341166000*x^9 + 1218404977539750*x^8 + 35835440515875*x^7 + 899097073875*x^6 + 18928359450*x^5 + 326351025*x^4 + 4440150*x^3 + 44850*x^2 + 300*x + 1
25:
58435841445947272053455474390625*x^25 + 58435841445947272053455474390625*x^24 + 28621636626586418964957783375000*x^23 + 9143022811270661613805958578125*x^22 + 2139856402637814420252458390625*x^21 + 390756386568644372393927184375*x^20 + 57889835047206573687989212500*x^19 + 7142252375954057792673993750*x^18 + 747445016088215350396115625*x^17 + 67230504092590798712878125*x^16 + 5247258856007086728810000*x^15 + 357767649273210458782500*x^14 + 21404902093269001807500*x^13 + 1126573794382579042500*x^12 + 52196469237802890000*x^11 + 2126522820799377000*x^10 + 75947243599977750*x^9 + 2365139074047750*x^8 + 63707449806000*x^7 + 1466947857375*x^6 + 28392539175*x^5 + 450675225*x^4 + 5651100*x^3 + 52650*x^2 + 325*x + 1
26:
2980227913743310874726229193921875*x^26 + 2980227913743310874726229193921875*x^25 + 1460896036148681801336386859765625*x^24 + 467486731567578176427643795125000*x^23 + 109716273735247939365671502937500*x^22 + 20114650184795455550373108871875*x^21 + 2995798963692940188353441746875*x^20 + 372148939589185116565644937500*x^19 + 39282388067747317859706965625*x^18 + 3571126187977028896336996875*x^17 + 282368117188881354594088125*x^16 + 19557964826935505080110000*x^15 + 1192558830910701529275000*x^14 + 64214706279807005422500*x^13 + 3057843156181285972500*x^12 + 128751290786580462000*x^11 + 4784676346798598250*x^10 + 156361972117601250*x^9 + 4467484917645750*x^8 + 110649781242000*x^7 + 2347116571800*x^6 + 41912795925*x^5 + 614557125*x^4 + 7125300*x^3 + 61425*x^2 + 351*x + 1
27:
157952079428395476360490147277859375*x^27 + 157952079428395476360490147277859375*x^26 + 77485925757326082742881959041968750*x^25 + 24835232614527590622718576616015625*x^24 + 5843584144594727205345547439062500*x^23 + 1075219482605429805783580728787500*x^22 + 160917201478363644402984870975000*x^21 + 20114650184795455550373108871875*x^20 + 2139856402637814420252458390625*x^19 + 196411940338736589298534828125*x^18 + 15712955227098927143882786250*x^17 + 1103802639920172567958708125*x^16 + 68452876894274267780385000*x^15 + 3761147082102981746175000*x^14 + 183470589370877158350000*x^13 + 7950392206071343528500*x^12 + 305784315618128597250*x^11 + 10413707343032243250*x^10 + 312723944235202500*x^9 + 8229577479873750*x^8 + 188104628111400*x^7 + 3688326041400*x^6 + 60964066800*x^5 + 828316125*x^4 + 8906625*x^3 + 71253*x^2 + 378*x + 1
28:
8687364368561751199826958100282265625*x^28 + 8687364368561751199826958100282265625*x^27 + 4264706144566677861733233976502203125*x^26 + 1368918021712760795124247943074781250*x^25 + 322858023988858678095341496008203125*x^24 + 59604558274866217494524583878437500*x^23 + 8960162355045248381529839406562500*x^22 + 1126420410348545510820894096825000*x^21 + 120687901108772733302238653231250*x^20 + 11174805658219697527985060484375*x^19 + 903494925558188310773260209375*x^18 + 64280271383586520134065943750*x^17 + 4047276346373966082515263125*x^16 + 226421054342599501119735000*x^15 + 11283441246308945238525000*x^14 + 501486277613730899490000*x^13 + 19875980515178358821250*x^12 + 701505194653353840750*x^11 + 21984493279734735750*x^10 + 608988733510657500*x^9 + 14813239463772750*x^8 + 313507713519000*x^7 + 5700140245800*x^6 + 87470182800*x^5 + 1104421500*x^4 + 11044215*x^3 + 82215*x^2 + 406*x + 1
29:
495179769008019818390136611716089140625*x^29 + 495179769008019818390136611716089140625*x^28 + 243246202319729033595154826807903437500*x^27 + 78186279317055760798442622902540390625*x^26 + 18480393293122270734177347231509546875*x^25 + 3422295054281901987810619857686953125*x^24 + 516572838382173884952546393613125000*x^23 + 65281182872472523922574544247812500*x^22 + 7040127564678409442630588105156250*x^21 + 657078572703318214645521556481250*x^20 + 53639067159454548134328290325000*x^19 + 3860387409203168236940293621875*x^18 + 246407706970414993847252784375*x^17 + 14009802737448344131783603125*x^16 + 711609027933884146376310000*x^15 + 32345864906085643017105000*x^14 + 1316401478736043611161250*x^13 + 47936188301312512451250*x^12 + 1558900432563008535000*x^11 + 45126065153139720750*x^10 + 1157078593670249250*x^9 + 26099517150456750*x^8 + 513012622122000*x^7 + 8674126461000*x^6 + 123916092300*x^5 + 1457836380*x^4 + 13592880*x^3 + 94395*x^2 + 435*x + 1
30:
29215606371473169285018060091249259296875*x^30 + 29215606371473169285018060091249259296875*x^29 + 14360213301232574733313961739766585078125*x^28 + 4621677844074851638307941709350165312500*x^27 + 1094607910438780651178196720635565468750*x^26 + 203284326224344978075950819546605015625*x^25 + 30800655488537117890295578719182578125*x^24 + 3911194347750745128926422694499375000*x^23 + 424327688671071405496734537610781250*x^22 + 39894056199844320174906665929218750*x^21 + 3285392863516591073227607782406250*x^20 + 238937662801206623507462384175000*x^19 + 15441549636812672947761174487500*x^18 + 890858632893038823909298528125*x^17 + 46032208994473130718717553125*x^16 + 2134827083801652439128930000*x^15 + 88951128491735518297038750*x^14 + 3329721387391169134113750*x^13 + 111851106036395862386250*x^12 + 3363943038688597365000*x^11 + 90252130306279441500*x^10 + 2148860245387605750*x^9 + 45080984168970750*x^8 + 825281174718000*x^7 + 13011189691500*x^6 + 173482529220*x^5 + 1906401420*x^4 + 16613520*x^3 + 107880*x^2 + 465*x + 1
Definition
The Bessel polynomials are $y_0 = 1$, $y_1 = x + 1$, and $y_n(x) = (2n-1)\,x\,y_{n-1}(x) + y_{n-2}(x)$ for $n \geq 2$. They are the polynomial solutions of $x^{2} y'' + 2(x+1) y' = n(n+1) y$.
Parameters
$n$
—   integer ($0 \leq n \leq 30$)
Formulas
(1)
$y_n(x) = (2n-1)\,x\,y_{n-1}(x) + y_{n-2}(x)$, with $y_0 = 1$ and $y_1 = x+1$.
(2)
$y_n(x) = \sum_{k=0}^{n} \frac{(n+k)!}{(n-k)!\,k!\,2^{k}}\, x^{k}$; the coefficients are integers despite the division.
(3)
$x^{2} y_n''(x) + 2(x+1) y_n'(x) = n(n+1)\, y_n(x)$.
(4)
$y_n$ has degree $n$, constant term $1$, and leading coefficient $\frac{(2n)!}{n!\,2^{n}} = (2n-1)!!$.
Comments
(5)
Not the Bessel functions, and not their zeros: those are transcendental and are listed separately Zeros_of_Bessel_J_functions. The connection is that $y_n$ appears in the closed form of a modified Bessel function of half-integer order.
(6)
Some authors write the reverse polynomials $\theta_n(x) = x^{n} y_n(1/x)$, which satisfy the same recurrence with the roles of the two ends exchanged. The table holds $y_n$.
Programs
(P1)
Sage
R. = ZZ[]
def bessel_polynomial(n):
    a, b = R(1), x + 1
    if n == 0: return R(1)
    for i in range(2, n + 1):
        a, b = b, (2*i - 1)*x*b + a
    return b

bessel_polynomial(31)            # the next one after this table
Links
Data properties
Entries are of type: integral polynomial
Table is complete: false
How well the digits are known: exact


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