Bateman polynomials $F_n(x)$

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Polynomials

$n$

$F_n(x)$

-x

3/4*x^2 + 1/4

-5/12*x^3 - 7/12*x

35/192*x^4 + 65/96*x^2 + 9/64

-21/320*x^5 - 49/96*x^3 - 407/960*x

77/3840*x^6 + 217/768*x^4 + 2303/3840*x^2 + 25/256

-143/26880*x^7 - 473/3840*x^5 - 2051/3840*x^3 - 3023/8960*x

143/114688*x^8 + 2717/61440*x^6 + 8393/24576*x^4 + 231491/430080*x^2 + 1225/16384

-2431/9289728*x^9 - 10439/774144*x^7 - 186901/1105920*x^5 - 1242461/2322432*x^3 - 1456787/5160960*x

10:

46189/928972800*x^10 + 31603/8847360*x^8 + 1502501/22118400*x^6 + 35140391/92897280*x^4 + 7219663/14745600*x^2 + 3969/65536

11:

-4199/486604800*x^11 - 155363/185794560*x^9 - 3550807/154828800*x^7 - 100919/491520*x^5 - 489669869/928972800*x^3 - 11072177/45416448*x

12:

96577/70071091200*x^12 + 558467/3185049600*x^10 + 99058609/14863564800*x^8 + 143126893/1592524800*x^6 + 2561060957/6370099200*x^4 + 12272016991/27249868800*x^2 + 53361/1048576

13:

-7429/36436967424*x^13 - 1166353/35035545600*x^11 - 721259/424673280*x^9 - 366047179/11147673600*x^7 - 298657513/1274019840*x^5 - 2007748057/3892838400*x^3 - 225938249/1049624576*x

14:

7429/264505393152*x^14 + 453169/78479622144*x^12 + 9813709/25480396800*x^10 + 854817239/83235962880*x^8 + 19581791989/178362777600*x^6 + 23415419897/56056872960*x^4 + 8291566749497/19837904486400*x^2 + 184041/4194304

15:

-215441/59513713459200*x^15 - 1567519/1700391813120*x^13 - 7347281/93428121600*x^11 - 77774201/27745320960*x^9 - 2541919883/59454259200*x^7 - 964346539/3737124864*x^5 - 29944350920479/59513713459200*x^3 - 5213660383/26990346240*x

16:

6678671/15235510645555200*x^16 + 51921281/380887766138880*x^14 + 55546633/3805072588800*x^12 + 601919867/887850270720*x^10 + 756633004243/53271016243200*x^8 + 106932791825/837115969536*x^6 + 1630408326607681/3808877661388800*x^4 + 1503330129139/3847351173120*x^2 + 41409225/1073741824

17:

-392863/7848596393164800*x^17 - 392863/20927899238400*x^15 - 193047809/77732197171200*x^13 - 21545712967/146495294668800*x^11 - 10412994631/2536715059200*x^9 - 14257719757/271790899200*x^7 - 1056803269448257/3808877661388800*x^5 - 1465895066081/2989699891200*x^3 - 20616229578539/117461986836480*x

18:

392863/72655578039582720*x^18 + 120608941/49861671203635200*x^16 + 2138353309/5484783832399872*x^14 + 130679958659/4520426235494400*x^12 + 402018701057/383551316951040*x^10 + 35163877306583/1917756584755200*x^8 + 153765704797379/1068464382935040*x^6 + 19872306199487837/45706531936665600*x^4 + 25381947965913563/69067648259850240*x^2 + 147744025/4294967296

19:

-765049/1380455982752071680*x^19 - 1013173/3459789430456320*x^17 - 708125219/12465417800908800*x^15 - 12204206387/2350621642457088*x^13 - 103077892523/430516784332800*x^11 - 2133885882127/383551316951040*x^9 - 330325681439443/5342321914675200*x^7 - 4979084527103/16959752110080*x^5 - 164977336297110859/345338241299251200*x^3 - 159746458862789/991901222174720*x

20:

765049/14158522900021248000*x^20 + 97161223/2906223121583308800*x^18 + 628601477/81406810128384000*x^16 + 9507718817/11080371378585600*x^14 + 35754344825627/723268197679104000*x^12 + 466021021397/313103115878400*x^10 + 3463743849565733/153420526780416000*x^8 + 6774303011494589/42738575317401600*x^6 + 202498114146683983/461305257394176000*x^4 + 3646744382722889053/10498282535497236480*x^2 + 2133423721/68719476736

21:

-31367009/6243908598909370368000*x^21 - 322085629/89198694270133862400*x^19 - 28582995689/29062231215833088000*x^17 - 63454384003/483274659358310400*x^15 - 167029419962161/17950201633308672000*x^13 - 25715463653507/72326819767910400*x^11 - 69062176996073579/9665493187166208000*x^9 - 4889335546071413/69039237051187200*x^7 - 2974993444518794443/9687410405277696000*x^5 - 717829147207627871/1541705826891202560*x^3 - 1418473716888367/9522251732877312*x

22:

1348781387/3022051761872135258112000*x^22 + 14522925167/39247425478858899456000*x^20 + 11591257399/98364474884358144000*x^18 + 206781575749649/11057324206118141952000*x^16 + 1270433594146733/789808871865581568000*x^14 + 2438665039919047/31823800697880576000*x^12 + 846720909926623717/425281700235313152000*x^10 + 1632038900777073587/60754528605044736000*x^8 + 5629958371631519423/32788158294786048000*x^6 + 19459665047169300729497/44092786649088393216000*x^4 + 690769146779707001/2094895381233008640*x^2 + 7775536041/274877906944

23:

-58642669/1544604233845758020812800*x^23 - 9910611061/274731978352012296192000*x^21 - 104617157713/7849485095771779891200*x^19 - 12004620059563/4819859269333549056000*x^17 - 56826538238599/221146484122362839040*x^15 - 130598570544539659/8687897590521397248000*x^13 - 14024775459040543/28352113349020876800*x^11 - 3752061200478565441/425281700235313152000*x^9 - 32803315327103360761/413130794514304204800*x^7 - 9723003921965570488289/30525775372445810688000*x^5 - 8808931929092118964849/19400826125598893015040*x^3 - 11046045800444959/79640650856792064*x

24:

2756205443/889692038695156619988172800*x^24 + 4632770851/1381509376855833260851200*x^22 + 23281139593/16310618380824477696000*x^20 + 14411235430649/46270648985602070937600*x^18 + 550310955331/14427791579676672000*x^16 + 2147743793640191/794322065419099176960*x^14 + 16577780176893570157/149699158482830229504000*x^12 + 45040565495936257/17674044685103923200*x^10 + 84042246646815726337/2697997025399537664000*x^8 + 4893530382428081056451/26640676688679980236800*x^6 + 226348509022273627708133/511670139576234541056000*x^4 + 2179670630301086872583/6938293502643724615680*x^2 + 457028729521/17592186044416

25:

-2756205443/11348112738458630356992000000*x^25 - 1656479471243/5560575241844728874926080000*x^23 - 31902022434683/219785582681609836953600000*x^21 - 115233841510373/3139794038308711956480000*x^19 - 122287687837423567/23135324492801035468800000*x^17 - 167962769061635551/374247896207075573760000*x^15 - 156630833934970795159/6950318072417117798400000*x^13 - 22307801820308817221/34022536018825052160000*x^11 - 244609659944162174028991/23135324492801035468800000*x^9 - 91658629938373248089791/1046598012769570652160000*x^7 - 146951407131352194036882679/447711372129205223424000000*x^5 - 2027808057432009614301149/4576605137320764506112000*x^3 - 5515863820512167497/42475013790289100800*x

26:

2756205443/150418121788196747476992000000*x^26 + 85442368733/3372010642284850163220480000*x^24 + 1766471361856801/125717353293880826737459200000*x^22 + 1335067388355667/326538579984106043473920000*x^20 + 16860570096995941/24551772931135792742400000*x^18 + 10761724689372507341/155687124822143438684160000*x^16 + 1517972800165638852643/361416539765690125516800000*x^14 + 120619963414723547351/794322065419099176960000*x^12 + 223185091871109145400921/70766874919156108492800000*x^10 + 23123499450797225510973959/653077159968212086947840000*x^8 + 832091428315848028933379503/4276100452172817235968000000*x^6 + 1895218962852989893389179839/4283702408532235577720832000*x^4 + 2166532273918912439933473/7215825242749473600307200*x^2 + 1690195005625/70368744177664

27:

-146078888479/109654810783595428910727168000000*x^27 - 101979601391/49327400262526058889216000000*x^25 - 26853709631149/20691883486747944183398400000*x^23 - 399774433113748789/925736874254940633248563200000*x^21 - 7402376242562111927/88165416595708631737958400000*x^19 - 72386088100601202623/7291876560547330444492800000*x^17 - 147857494712061298874257/204923178047146301168025600000*x^15 - 2179393289880381179953/68239486529186247475200000*x^13 - 32649556955909755422791/38993992302392141414400000*x^11 - 4044430353765008615896694803/326730661501743752911257600000*x^9 - 4056134907761623095522547901/42535946603192760926208000000*x^7 - 99293613370348285346747403533/295050931199924389281792000000*x^5 - 39429953640909693416639196869/91126033054231193198788608000*x^3 - 43108777127399077963/352869345334709452800*x

28:

146078888479/1563079484624342113927456358400000*x^28 + 110581718578603/682296600431260446555635712000000*x^26 + 9820359993409/85688741027473839441838080000*x^24 + 3532366321225512853/81464844934434775725873561600000*x^22 + 143912353897091749/14848912268750927450603520000*x^20 + 77797380771717551/58276735748630013542400000*x^18 + 12054553324658400463829/104920667160138906198029107200*x^16 + 68541088703454034281199/11152281798484152444518400000*x^14 + 274433619412920300738707/1372588529044203377786880000*x^12 + 9935086626338893241844084791/2613845292013950023290060800000*x^10 + 1095677702254915558844366801/27659738539830158976614400000*x^8 + 37139858454540523859850101707/181569803815338085711872000000*x^6 + 7084753958163071397362313186151/16038181817544690002986795008000*x^4 + 42839111941531244071081529/148873868166199665858969600*x^2 + 25145962430625/1125899906842624

29:

-5037203051/795250965861507391296425164800000*x^29 - 1365082026821/111648534616024436709104025600000*x^27 - 1093073062067/112748343457202420318208000000*x^25 - 3387061996901797/816423282567320192459735040000*x^23 - 4483528843445815841/4231939996594014323422003200000*x^21 - 485805822338397557/2878870745982322668994560000*x^19 - 115134110286607247337569/6756255082281671990024601600000*x^17 - 1457611676168108733317/1338273815818098293342208000*x^15 - 297329825984745042791279/6862942645221016888934400000*x^13 - 9521392567604374122608947/9171386989522631660666880000*x^11 - 2124581968541548272235043423/149362588115082858473717760000*x^9 - 11170917611068524764466689/108710523994547788185600000*x^7 - 2066888717680880307668215360367953/6014318181579258751120048128000000*x^5 - 484436979986968686007887454439/1145584415538906428784771072000*x^3 - 18894620352267602119043/163731376235305186099200*x

30:

297194980009/715725869275356652166782648320000000*x^30 + 337492604417/379695421366236950751608832000000*x^28 + 5857798688429257/7443235641068295780606935040000000*x^26 + 2005386037239631439/5290422871036234847139083059200000*x^24 + 9728974914140864737/88468609883549077711355904000000*x^22 + 340566984512149582123/16927759986376057293688012800000*x^20 + 2875639112811678008777353/1216125914810700958204428288000000*x^18 + 196739019992085297749549/1100566438742715799280025600000*x^16 + 45946428088475879364321191/5353095263272393173368832000000*x^14 + 9257234523982315126114108699/36318692478509621376240844800000*x^12 + 28472439231498401639604743977/6349421761977206534307840000000*x^10 + 1670043611763682604206964787293/38170439184965619387727872000000*x^8 + 412321575412159857691638212813723/1929727758795484091268464640000000*x^6 + 32607297560432713742688309881639/74022377619437030783015976960000*x^4 + 2721302074944277755236479380481/9843540163149121906595069952000*x^2 + 93990019574025/4503599627370496

Definition

For an integer $n\geq 0$, the Bateman polynomial $F_n(x)$ is the polynomial in $\mathbb{Q}[x]$ defined by the terminating hypergeometric series $F_n(x)={}_3F_2\left(-n,n+1,\frac{x+1}{2};1,1;1\right)$ [4].

Parameters

$n$

— index ($n\geq 0$)

Formulas

(1)

$F_n(x)=\sum_{j=0}^{n} \frac{(-n)_j(n+1)_j\left(\frac{x+1}{2}\right)_j}{(j!)^3}$, where $(a)_j$ is the rising factorial.

(2)

$F_0(x)=1$, $F_1(x)=-x$, and $(n+1)^2F_{n+1}(x)=-(2n+1)xF_n(x)+n^2F_{n-1}(x)$ for $n\geq1$ [4].

(3)

$\sum_{n=0}^{\infty}F_n(x)t^n=(1-t)^x\, {}_2F_1\left(\frac{x+1}{2},\frac{x+1}{2};1;t^2\right)$ [4].

(4)

$F_n\!\left(\frac{d}{dy}\right)\operatorname{sech} y =\operatorname{sech} y\,P_n(\tanh y)$, where $P_n$ is the Legendre polynomial [4].

(5)

If $B_n(x)=i^nF_n(ix)$, then $\int_{-\infty}^{\infty}B_m(x)B_n(x)\operatorname{sech}^2(\pi x/2)\,\mathrm{d}x =\frac{4}{\pi(2n+1)}\delta_{m,n}$ [2].

(6)

$F_n(-1)=1$, $F_n(1)=(-1)^n$, $F_{2k+1}(0)=0$, and $F_{2k}(0)=\left(\binom{2k}{k}/4^k\right)^2$.

(7)

Touchard's polynomials $Q_n$ satisfy $Q_n(x)=(-1)^n2^n n!\binom{2n}{n}^{-1}F_n(2x+1)$ [1].

Comments

(8)

This table holds Bateman's $F_n(x)$, not Bateman's polynomials $Z_n(x)={}_2F_2(-n,n+1;1,1;x)$. OEIS A073768 [6] is a table of the coefficients of $n!Z_n(-x)$.

(9)

Pasternack's polynomials $F_n^m(x)$ add a second parameter $m$ and specialise to this table at $m=0$ [4].

(10)

The polynomials $B_n(x)=i^nF_n(ix)$ are real polynomials orthogonal on the real line for the weight $\operatorname{sech}^2(\pi x/2)$ [2].

(11)

Carlitz relates $F_n$ to Touchard's polynomials $Q_n$ in (7). These $Q_n$ are not the Touchard polynomials $T_n$ whose coefficients are Stirling numbers of the second kind.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.rational_field import QQ

R = PolynomialRing(QQ, 'x')
x = R.gen()

def bateman_polynomial(n):
    values = [R.one()]
    if n >= 1:
        values.append(-x)
    for k in range(1, n):
        numerator = -QQ(2*k + 1)*x*values[k] + QQ(k*k)*values[k - 1]
        values.append(numerator / QQ((k + 1)*(k + 1)))
    return values[n]

bateman_polynomial(31)      # the next one after this table

References

[1]

Leonard Carlitz, Some polynomials of Touchard connected with the Bernoulli numbers, Canadian Journal of Mathematics 9 (1957), 188-190. (doi) (MR)

[2]

Erik Koelink, On Jacobi and continuous Hahn polynomials, Proceedings of the American Mathematical Society 124 (1996), no. 3, 887-898. (arXiv) (doi) (MR)

[3]

Simon Pasternack, A generalization of the polynomial $F_n(x)$, London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 28 (1939), no. 187, 209-226. (doi) (MR)

Links

[4]

Wikipedia: Bateman polynomials

[5]

J-STAGE: Bateman, Some Properties of a certain Set of Polynomials

[6]

OEIS A073768: triangle of coefficients of $n!Z_n(-x)$

Similar tables

Legendre polynomials — appear in Bateman's defining differential identity (4)

Touchard polynomials — are the Bell-polynomial family $T_n$, not the $Q_n$ family related to $F_n$ in (7)

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (it holds every index $n$ with $0\leq n\leq30$)

How they were obtained:

The generator computes $F_n(x)$ by the three-term recurrence (2), with every division made in Sage's rational field.

Before the draft was created, the entries were checked against the terminating hypergeometric formula (1), against the ordinary generating function (3), against the six polynomials printed by [4], and against the special values in (6) for every entry. The orthogonality formula (5) was checked in ball arithmetic on the pairs $(0,0)$, $(1,1)$, $(2,2)$, $(3,3)$, $(5,5)$, $(3,4)$ and $(2,5)$ on the cutoff interval $[-120,120]$, after the control integral for $\operatorname{sech}^2(\pi x/2)$ returned the known value $4/\pi$ and a wrong target was rejected.