Hankel asymptotic coefficient polynomials $a_k(\nu)$

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Polynomials

$k$

$a_k(\nu)$

1/2*v^2 - 1/8

1/8*v^4 - 5/16*v^2 + 9/128

1/48*v^6 - 35/192*v^4 + 259/768*v^2 - 75/1024

1/384*v^8 - 7/128*v^6 + 329/1024*v^4 - 3229/6144*v^2 + 3675/32768

1/3840*v^10 - 11/1024*v^8 + 1463/10240*v^6 - 17281/24576*v^4 + 352407/327680*v^2 - 59535/262144

1/46080*v^12 - 143/92160*v^10 + 9581/245760*v^8 - 308737/737280*v^6 + 21967231/11796480*v^4 - 7156487/2621440*v^2 + 2401245/4194304

1/645120*v^14 - 13/73728*v^12 + 11011/1474560*v^10 - 1218503/8257536*v^8 + 32953349/23592960*v^6 - 109750823/18874368*v^4 + 2430898831/293601280*v^2 - 57972915/33554432

1/10321920*v^16 - 17/1032192*v^14 + 6409/5898240*v^12 - 585871/16515072*v^10 + 800744659/1321205760*v^8 - 199081441/37748736*v^6 + 24615717239/1174405120*v^4 - 13724532225/469762048*v^2 + 13043905875/2147483648

1/185794560*v^18 - 323/247726080*v^16 + 31331/247726080*v^14 - 18824117/2972712960*v^12 + 1395693013/7927234560*v^10 - 86428869397/31708938240*v^8 + 4248959006581/190253629440*v^6 - 7251187604321/84557168640*v^4 + 3535825093925/30064771072*v^2 - 418854310875/17179869184

10:

1/3715891200*v^20 - 19/212336640*v^18 + 34561/2831155200*v^16 - 5275559/5945425920*v^14 + 2539727359/67947724800*v^12 - 8432486387/9059696640*v^10 + 102100383570113/7610145177600*v^8 - 22844784140209/217432719360*v^6 + 1518546718552307/3865470566400*v^4 - 127936483908305/240518168576*v^2 + 30241281245175/274877906944

11:

1/81749606400*v^22 - 23/4246732800*v^20 + 5681/5662310400*v^18 - 48281071/475634073600*v^16 + 835175609/135895449600*v^14 - 457664785643/1993133260800*v^12 + 80275722533653/15220290355200*v^10 - 626292713429209/8697308774400*v^8 + 1402858291920059/2576980377600*v^6 - 47932577150805141/24051816857600*v^4 + 16150238253691605/6047313952768*v^2 - 1212400457192925/2199023255552

12:

1/1961990553600*v^24 - 23/78479622144*v^22 + 2921/40768634880*v^20 - 1113913/114152177664*v^18 + 7446629017/9132174213120*v^16 - 62328812257/1435055947776*v^14 + 596723105157827/401815665377280*v^12 - 468499062143069/14611478740992*v^10 + 140098972937988943/333976656936960*v^8 - 53385819864323939/17317308137472*v^6 + 46957659689155191059/4233119766937600*v^4 - 713067703435998435/48378511622144*v^2 + 213786613951685775/70368744177664

13:

1/51011754393600*v^26 - 1/69759664128*v^24 + 3151/697596641280*v^22 - 122659/152202903552*v^20 + 182654201/2029372047360*v^18 - 117346396451/17858474016768*v^16 + 1107869547133859/3482402433269760*v^14 - 725618525742529/71433896067072*v^12 + 108488627043954253/519519244124160*v^10 - 9873251528160562349/3740538557693952*v^8 + 1444153424274134673137/76196155804876800*v^6 - 91071110775152596505/1354598325420032*v^4 + 1785020911343464631025/20125460834811904*v^2 - 10278202593831046875/562949953421312

14:

1/1428329123020800*v^28 - 29/45343781683200*v^26 + 203/797253304320*v^24 - 6818741/117196235735040*v^22 + 68673653/8117488189440*v^20 - 20933020613/25512105738240*v^18 + 21112855176027341/390029072526213120*v^16 - 22598393151913723/9286406488719360*v^14 + 240263122506317747/3265549534494720*v^12 - 380580069838093332839/261837699038576640*v^10 + 10885314206448634969657/609569246439014400*v^8 - 1216869440159017586089/9675702324428800*v^6 + 1738656980011646566572945/3944590323623133184*v^4 - 93053573381450253980025/161003686678495232*v^2 + 1070401384414690453125/9007199254740992

15:

1/42849873690624000*v^30 - 899/34279898952499200*v^28 + 141143/10882507603968000*v^26 - 10474249/2812709657640960*v^24 + 38801858567/56254193152819200*v^22 - 185672450497/2143016882012160*v^20 + 88397905198434467/11700872175786393600*v^18 - 4310488250512099849/9360697740629114880*v^16 + 173905933274150443321/8914950229170585600*v^14 - 19496357734940737129513/34562576273092116480*v^12 + 37264786645740806230201063/3456257627309211648000*v^10 - 1892240069308674888235193/14629661914536345600*v^8 + 2651057007994964860831686877/2958442742717349888000*v^6 - 98088651358745125825192613/31556722588985065472*v^4 + 10444611855038041861946595/2576058986855923712*v^2 - 60013837619516978071875/72057594037927936

16:

1/1371195958099968000*v^32 - 31/31163544502272000*v^30 + 150133/249308356018176000*v^28 - 35525783/166205570678784000*v^26 + 2025696023/40912140474777600*v^24 - 3549012086567/450033545222553600*v^22 + 3772233263916269/4254862609376870400*v^20 - 605129705387818307/8509725218753740800*v^18 + 1107450938293990867463/272311207000119705600*v^16 - 67023012335159307973873/408466810500179558400*v^14 + 63208097653461501970843861/13825030509236846592000*v^12 - 213634243363940980435258403/2513641910770335744000*v^10 + 19346612376761802772790102207/19364352497786290176000*v^8 - 14684369482628924167293740447/2151594721976254464000*v^6 + 1076988408371033295105062321/45900687402160095232*v^4 - 627865873216946822324684745/20608471894847389696*v^2 + 57673297952355815927071875/9223372036854775808

17:

1/46620662575398912000*v^34 - 1/28492383544934400*v^32 + 1829/71230958862336000*v^30 - 1477057/132964456543027200*v^28 + 2406895397/759796894531584000*v^26 - 64641783203/102864810336583680*v^24 + 759258257337631/8509725218753740800*v^22 - 8938187286017569/972540025000427520*v^20 + 911334418804666279721/1322654434000581427200*v^18 - 48873250753973442679913/1307093793600574586880*v^16 + 74373041320367504435388377/51350113320022573056000*v^14 - 11235075889782471690729511/287273361230895513600*v^12 + 82483911551541297483353892841/116186114986717741056000*v^10 - 4033071555493603079288701393/491793079308858163200*v^8 + 90718283995251916868431834913/1639310264362860544000*v^6 - 762516176438345213567727096957/4039260491390088380416*v^4 + 995289296049024392169352570455/4076730440289810907136*v^2 - 3694483615889146090857721875/73786976294838206464

18:

1/1678343852714360832000*v^36 - 37/31968454337416396800*v^34 + 6919/6838172050784256000*v^32 - 3783953/7180080653323468800*v^30 + 1663915049/9117562734379008000*v^28 - 356276633437/8023455206253527040*v^26 + 52733395574200007/6739702373252962713600*v^24 - 5918704382139109/5835240150002565120*v^22 + 1544696474476994589407/15871853208006977126400*v^20 - 2759830423783778267361533/399970700841775823585280*v^18 + 883172452605480648904122211/2464805439361083506688000*v^16 - 3304668495954822006407412073/246480543936108350668800*v^14 + 13266656052098881994590907422819/37644301255696548102144000*v^12 - 1497914918151136347529090593523/239011436544105067315200*v^10 + 25246872981369830361853887041603/354091017102377877504000*v^8 - 173046333198138604206215974305767/363533444225107954237440*v^6 + 289278131349656999382681667547645/179376139372751679913984*v^4 - 67780326029953687871863136407875/32613843522318487257088*v^2 + 502860269940467106811189921875/1180591620717411303424

19:

1/63777066403145711616000*v^38 - 37/1032826986285760512000*v^36 + 7289/196728949768716288000*v^34 - 2030893/88370223425519616000*v^32 + 843851341/88370223425519616000*v^30 - 57855234001/20572962067316736000*v^28 + 8166282364263323/13479404746505925427200*v^26 - 402981509257338029/4147509152771053977600*v^24 + 28560877562339813521/2441823570462611865600*v^22 - 12371057005542702694323143/11691451255374985612492800*v^20 + 9689501551288066550885173691/135374698746447201828864000*v^18 - 5430161244757473233222833157/1516803347299128311808000*v^16 + 107686921493536382062349146072871/828174627625324058247168000*v^14 - 77351168152868766597320968970683/23165723849659414216704000*v^12 + 85781865586988024362118644435033/1470839609502185029632000*v^10 - 10984868508277649834801492255094037/16778466656543444041728000*v^8 + 263460628663767082812546732782317423/60849905741064223724666880*v^6 - 1609364084981131227886931950905665/110385316537077956870144*v^4 + 185693664789670040623766540457496125/9914608430784820126154752*v^2 - 36232405765710498380237842265625/9444732965739290427392

20:

1/2551082656125828464640000*v^40 - 41/39247425478858899456000*v^38 + 34891/27542052967620280320000*v^36 - 30629747/33050463561144336384000*v^34 + 6464393617/14139235748083138560000*v^32 - 139183977847/864064406827302912000*v^30 + 56450809480564157/1347940474650592542720000*v^28 - 67921827293807341/8295018305542107955200*v^26 + 9145782828878158585019/7520816597024844546048000*v^24 - 3218502372400418688187801/23382902510749971224985600*v^22 + 609500651636574029180210413897/51442385523649936694968320000*v^20 - 277938323731459452802615010351/360999196657192538210304000*v^18 + 2470155987131452647175514972426323/66253970210025924659773440000*v^16 - 672511633221265656693269037460171/509645924692507112767488000*v^14 + 6836410238471868862741037133545329/205917545330305904148480000*v^12 - 57461701120415405526432719839918327/100670799939260664250368000*v^10 + 385300842407974432774929915306179955187/60849905741064223724666880000*v^8 - 11229785111249253536538174042045883103/270444025515840994331852800*v^6 + 60665979530699261805801923578492902423/436242770954532085550809088*v^4 - 1086893092372685962197875102828942175/6101297495867581616095232*v^2 + 11021897833929133607268351617203125/302231454903657293676544

Definition

The Hankel asymptotic coefficient polynomials $a_k(\nu)\in\mathbb{Q}[\nu]$ are $a_k(\nu)=\frac{1}{8^k k!}\prod_{j=1}^k(4\nu^2-(2j-1)^2)$ for $k\geq0$, with the empty product giving $a_0(\nu)=1$ [1].

Parameters

$k$

— order of the asymptotic correction ($k\geq0$)

Formulas

(1)

$a_0(\nu)=1$, $\displaystyle a_{k+1}(\nu)=\frac{4\nu^2-(2k+1)^2}{8(k+1)}a_k(\nu)$ for $k\geq0$.

(2)

$\displaystyle K_\nu(z)\sim\sqrt{\frac{\pi}{2z}}e^{-z}\sum_{k=0}^{\infty}\frac{a_k(\nu)}{z^k}$ as $z\to+\infty$ with $\nu$ fixed, where $K_\nu$ is the modified Bessel function of the second kind [2].

(3)

$\deg a_k=2k$, $a_k(-\nu)=a_k(\nu)$, and the leading coefficient is $1/(2^k k!)$. For $k\geq1$, the roots are $\nu=\pm(j-\tfrac12)$ with $1\leq j\leq k$.

(4)

For integers $n,k\geq0$, $\displaystyle a_k(n+\tfrac12)=\frac{(n+k)!}{2^k k!(n-k)!}$ when $k\leq n$, and $a_k(n+\tfrac12)=0$ when $k>n$ [3].

(5)

$\displaystyle y_n(x)=\sum_{k=0}^n a_k(n+\tfrac12)x^k$ for $n\geq0$, where $y_n$ is the Bessel polynomial normalized by $y_0(x)=1$ and $y_1(x)=x+1$ [3].

Comments

(6)

The letter $v$ in the stored expressions denotes the Bessel order $\nu$. It is a change of variable name only.

(7)

The indeterminate $\nu$ denotes the Bessel order. In (2), $z$ tends to positive infinity with $\nu$ fixed, and the square root is positive. The coefficients are stored in $\nu$, without clearing denominators or absorbing the external signs or phase factors that occur in other Bessel expansions [1] [2].

(8)

Each coefficient polynomial is exact. The accuracy of a truncated expansion of $K_\nu(z)$ depends on $z$, $\nu$ and the truncation order.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
R = PolynomialRing(QQ, 'v')
nu = R.gen()
def a(k):
    result = R.one()
    for j in range(1, k + 1):
        result *= (4*nu**2 - (2*j-1)**2) * (QQ(1)/QQ(8*j))
    return result
[a(k) for k in range(21)]

Links

[1]

DLMF 10.17.1, Hankel asymptotic coefficient polynomials

[2]

DLMF 10.40.2, large-argument expansion of the modified Bessel function K

[3]

DLMF 10.49, explicit formulas for spherical Bessel functions

[4]

DLMF 10.25.1, modified Bessel differential equation

Similar tables

Bessel polynomials $y_n$ — has coefficients obtained by specializing the Bessel order to half-integers

Values of modified Bessel functions of the second kind $K_\nu(x)$ — holds values of the function whose large-argument expansion uses these coefficient polynomials

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (it holds every correction order $0\leq k\leq20$, retaining each full polynomial in $\nu$ through degree 40. This initial prefix covers successive fixed-order, large-argument corrections. The finite cutoff is an editorial choice that keeps the longest expanded entry at 1,213 characters and leaves the derivative and large-order coefficient families to separate tables)

How they were obtained:

All coefficients were computed exactly in $\mathbb{Q}[\nu]$. Independent interpolation in $\nu^2$ from the recurrence of Bessel polynomials reproduces all 21 polynomials.

The 21 half-integer nodes determine each even polynomial of degree at most 40; all 441 specializations also agree with the factorial formula in (4). The Bessel polynomials used in this reconstruction agree with the stored family through degree 20. The recurrence, degree, leading coefficient, evenness and all half-integer roots were checked throughout the range. Substitution into the modified Bessel differential equation [4], after removing $e^{-z}z^{-1/2}$, gives the expected truncated-series residual. The checks include known Bessel polynomials, interpolation of $1,t,t^2$ and deliberate coefficient corruption as controls.