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Bernoulli polynomials of the second kind $\psi_n$
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Numbers
$n$ 
0:
1
1:
x + 1/2
2:
x^2 - 1/6
3:
x^3 - 3/2*x^2 + 1/4
4:
x^4 - 4*x^3 + 4*x^2 - 19/30
5:
x^5 - 15/2*x^4 + 55/3*x^3 - 15*x^2 + 9/4
6:
x^6 - 12*x^5 + 105/2*x^4 - 100*x^3 + 72*x^2 - 863/84
7:
x^7 - 35/2*x^6 + 119*x^5 - 1575/4*x^4 + 1918/3*x^3 - 420*x^2 + 1375/24
8:
x^8 - 24*x^7 + 700/3*x^6 - 1176*x^5 + 3248*x^4 - 4704*x^3 + 2880*x^2 - 33953/90
9:
x^9 - 63/2*x^8 + 414*x^7 - 2940*x^6 + 60921/5*x^5 - 29547*x^4 + 39204*x^3 - 22680*x^2 + 57281/20
10:
x^10 - 40*x^9 + 1365/2*x^8 - 6480*x^7 + 37415*x^6 - 134568*x^5 + 295310*x^4 - 365280*x^3 + 201600*x^2 - 3250433/132
11:
x^11 - 99/2*x^10 + 3190/3*x^9 - 51975/4*x^8 + 99429*x^7 - 987525/2*x^6 + 1592096*x^5 - 3224925*x^4 + 3764112*x^3 - 1995840*x^2 + 1891755/8
12:
x^12 - 60*x^11 + 1584*x^10 - 24200*x^9 + 473319/2*x^8 - 1546380*x^7 + 6833860*x^6 - 20182800*x^5 + 38260728*x^4 - 42514560*x^3 + 21772800*x^2 - 13695779093/5460
13:
x^13 - 143/2*x^12 + 2275*x^11 - 42471*x^10 + 1548833/3*x^9 - 17144127/4*x^8 + 173413955/7*x^7 - 99657415*x^6 + 1368354988/5*x^5 - 490483422*x^4 + 522356640*x^3 - 259459200*x^2 + 24466579093/840
14:
x^14 - 84*x^13 + 19019/6*x^12 - 70980*x^11 + 5246241/5*x^10 - 10774764*x^9 + 314931617/4*x^8 - 412140300*x^7 + 4600447852/3*x^6 - 19796208432/5*x^5 + 6760458432*x^4 - 6936733440*x^3 + 3353011200*x^2 - 132282840127/360
15:
x^15 - 195/2*x^14 + 4305*x^13 - 455455/4*x^12 + 2010645*x^11 - 50008959/2*x^10 + 675182365/3*x^9 - 11864147295/8*x^8 + 7148825970*x^7 - 24894259390*x^6 + 60941259288*x^5 - 99737688960*x^4 + 99013795200*x^3 - 46702656000*x^2 + 240208245823/48
16:
x^16 - 112*x^15 + 5720*x^14 - 176400*x^13 + 10998988/3*x^12 - 54272400*x^11 + 589458584*x^10 - 14301086800/3*x^9 + 28818645856*x^8 - 129516266880*x^7 + 1277772845440/3*x^6 - 995165633280*x^5 + 1568627191296*x^4 - 1511816785920*x^3 + 697426329600*x^2 - 111956703448001/1530
17:
x^17 - 255/2*x^16 + 22372/3*x^15 - 265200*x^14 + 6407198*x^13 - 111291180*x^12 + 1434329780*x^11 - 13952967600*x^10 + 928729202401/9*x^9 - 579706822695*x^8 + 2452060831480*x^7 - 7669304479200*x^6 + 85968926965008/5*x^5 - 26204724862560*x^4 + 24588590342400*x^3 - 11115232128000*x^2 + 4573423873125/4
18:
x^18 - 144*x^17 + 19125/2*x^16 - 388416*x^15 + 10792314*x^14 - 217318752*x^13 + 3277547130*x^12 - 37729897920*x^11 + 1673578597977/5*x^10 - 2293802567056*x^9 + 12092677825410*x^8 - 48501172437120*x^7 + 145098027700272*x^6 - 1567399479766272/5*x^5 + 463465101767040*x^4 - 424405694361600*x^3 + 188305108992000*x^2 - 30342376302478019/1596
19:
x^19 - 323/2*x^18 + 12084*x^17 - 2223855/4*x^16 + 88011686/5*x^15 - 406668951*x^14 + 7093171732*x^13 - 95320932135*x^12 + 998233909257*x^11 - 81854000736651/10*x^10 + 157521686886928/3*x^9 - 261792607187475*x^8 + 7011240335453296/7*x^7 - 2879449701839688*x^6 + 30082965538822272/5*x^5 - 8652611611967040*x^4 + 7748235407001600*x^3 - 3379030566912000*x^2 + 56310194579604163/168
20:
x^20 - 180*x^19 + 45220/3*x^18 - 779760*x^17 + 55809555/2*x^16 - 733052376*x^15 + 102469372720/7*x^14 - 227016467280*x^13 + 2769288977455*x^12 - 26746824379860*x^11 + 204835481465316*x^10 - 1239825958993440*x^9 + 5882812601374960*x^8 - 151030551841260480/7*x^7 + 59835707603071680*x^6 - 121285016030877696*x^5 + 170061247969113600*x^4 - 149179920390144000*x^3 + 64023737057280000*x^2 - 12365722323469980029/1980
21:
x^21 - 399/2*x^20 + 18585*x^19 - 1073975*x^18 + 43132698*x^17 - 5113194975/4*x^16 + 28970107082*x^15 - 513378767850*x^14 + 7213058608077*x^13 - 161982267131685/2*x^12 + 8020363165546095/11*x^11 - 5258103386481855*x^10 + 90675458929607272/3*x^9 - 137184871326595890*x^8 + 484289209590356880*x^7 - 1299846755708298000*x^6 + 12812437590550327296/5*x^5 - 3510201084291054720*x^4 + 3020956027723468800*x^3 - 1277273554292736000*x^2 + 161867055619224199787/1320
22:
x^22 - 220*x^21 + 45353/2*x^20 - 1455300*x^19 + 195535802/3*x^18 - 2164128120*x^17 + 220944743965/4*x^16 - 1108963070600*x^15 + 17773292421313*x^14 - 229451847983820*x^13 + 4794295038648115/2*x^12 - 20284599731022900*x^11 + 693338933092243856/5*x^10 - 2283113383183531360/3*x^9 + 3318281460396026740*x^8 - 11314221342121051200*x^7 + 29471976683031856512*x^6 - 56632097478664350720*x^5 + 75920678645023872000*x^4 - 64188285602918400000*x^3 + 26761922089943040000*x^2 - 20953816286242674495191/8280
23:
x^23 - 483/2*x^22 + 82225/3*x^21 - 7772919/4*x^20 + 96505332*x^19 - 3567769821*x^18 + 101862493150*x^17 - 18396761470395/8*x^16 + 625338073007263/15*x^15 - 1225903285065459/2*x^14 + 7350842198684775*x^13 - 288270768999258045/4*x^12 + 577130864847801178*x^11 - 18802278018335709744/5*x^10 + 178127049133900266400/9*x^9 - 83201302267339412310*x^8 + 274808539869932924064*x^7 - 696382421496005458368*x^6 + 1306829253148332756480*x^5 - 1717133441463494208000*x^4 + 1427876882647971840000*x^3 - 587545834974658560000*x^2 + 4380881778942163832799/80
24:
x^24 - 264*x^23 + 32844*x^22 - 2560360*x^21 + 701379756/5*x^20 - 5742375408*x^19 + 546869431768/3*x^18 - 4596834498960*x^17 + 187147249265823/2*x^16 - 7770003687519304/5*x^15 + 21193792880719212*x^14 - 238166141203073160*x^13 + 2206461762371899472*x^12 - 16816147857376060512*x^11 + 524570755795132953984/5*x^10 - 1594575825768529098880/3*x^9 + 2160924649322773961088*x^8 - 6931499577979483685376*x^7 + 17122891461428588571648*x^6 - 31433687292974729748480*x^5 + 40536880042625584128000*x^4 - 33187814234683637760000*x^3 + 13488008733331292160000*x^2 - 101543126947618093900697699/81900
25:
x^25 - 575/2*x^24 + 39050*x^23 - 3332700*x^22 + 601513825/3*x^21 - 9043336995*x^20 + 317469004600*x^19 - 26669126400400/3*x^18 + 201871340883575*x^17 - 15037787741528475/4*x^16 + 173509032243521030/3*x^15 - 738136989823775100*x^14 + 101759601425189238025/13*x^13 - 68920232017862927075*x^12 + 502237397680531175900*x^11 - 3011873150404344081240*x^10 + 132617842888136145324400/9*x^9 - 58089926109537343051800*x^8 + 1269488313357571254974400/7*x^7 - 437522128149656054102400*x^6 + 786879491429707553664000*x^5 - 997124064225417878400000*x^4 + 804491388770775552000000*x^3 - 323150209236062208000000*x^2 + 192060902780872132330221667/6552
Definition
The Bernoulli polynomials of the second kind, also called the Fontana-Bessel polynomials, are defined by the generating function $\frac{t}{\log(1+t)}\,(1+t)^{x} = \sum_{n \geq 0} \psi_n(x)\, \frac{t^{n}}{n!}$. They are monic of degree $n$ with rational coefficients.
Parameters
$n$
—   integer ($0 \leq n \leq 25$)
Formulas
(1)
$\frac{t}{\log(1+t)}\,(1+t)^{x} = \sum_{n \geq 0} \psi_n(x)\, \frac{t^{n}}{n!}$.
(2)
$\psi_n(0) = b_n$, the Cauchy numbers of the first kind: $1, \tfrac{1}{2}, -\tfrac{1}{6}, \tfrac{1}{4}, -\tfrac{19}{30}, \tfrac{9}{4}, \dots$
(3)
$\psi_n$ is monic of degree $n$.
(4)
$\psi_n(x) = n! \int_{0}^{1} \binom{x+u}{n}\, du$, which is the form in which they arise in the Gregory-Newton interpolation of an integral.
Comments
(5)
The convention here puts the $n!$ in the generating function, so that $\psi_1(x) = x + \tfrac{1}{2}$ and $\psi_n$ is monic. Some authors write $\sum b_n(x) t^{n}$ without the factorial, giving $b_n = \psi_n / n!$; every statement here is about the monic form.
(6)
Not the Bernoulli polynomials Bernoulli_polynomials, which come from $\frac{t e^{xt}}{e^{t}-1}$ and whose constant terms are the Bernoulli numbers. These come from $\frac{t}{\log(1+t)}$ and their constant terms are the Cauchy numbers; the two families are related by the change from $e^{t}-1$ to $\log(1+t)$, which is the inverse substitution.
(7)
Twenty-five is where the longest entry reaches 636 characters.
Programs
(P1)
Sage
R. = QQ[]
P. = PowerSeriesRing(R, default_prec=30)
series = t/log(1+t) * sum(binomial(x, k)*t^k for k in range(30))
psi = [series[n] * factorial(n) for n in range(26)]

series[26] * factorial(26)       # the next one after this table
Links
Data properties
Entries are of type: rational polynomial
Table is complete: false
How well the digits are known: exact


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