Hankel derivative asymptotic coefficient polynomials $b_k(\nu)$

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Polynomials

$k$

$b_k(\nu)$

1/2*v^2 + 3/8

1/8*v^4 + 7/16*v^2 - 15/128

1/48*v^6 + 25/192*v^4 - 341/768*v^2 + 105/1024

1/384*v^8 + 7/384*v^6 - 973/3072*v^4 + 1341/2048*v^2 - 4725/32768

1/3840*v^10 + 1/1024*v^8 - 1057/10240*v^6 + 18251/24576*v^4 - 422553/327680*v^2 + 72765/262144

1/46080*v^12 - 11/92160*v^10 - 4939/245760*v^8 + 270611/737280*v^6 - 23654609/11796480*v^4 + 8349421/2621440*v^2 - 2837835/4194304

1/645120*v^14 - 13/368640*v^12 - 429/163840*v^10 + 4369937/41287680*v^8 - 31263947/23592960*v^6 + 197847299/31457280*v^4 - 555804741/58720256*v^2 + 66891825/33554432

1/10321920*v^16 - 5/1032192*v^14 - 1391/5898240*v^12 + 339053/16515072*v^10 - 661458941/1321205760*v^8 + 196358747/37748736*v^6 - 79804000483/3523215360*v^4 + 15446253747/469762048*v^2 - 14783093325/2147483648

1/185794560*v^18 - 17/35389440*v^16 - 3349/247726080*v^14 + 8632039/2972712960*v^12 - 142094381/1132462080*v^10 + 76923041039/31708938240*v^8 - 4279689925859/190253629440*v^6 + 1116233049421/12079595520*v^4 - 3930320436475/30064771072*v^2 + 468131288625/17179869184

10:

1/3715891200*v^20 - 19/495452160*v^18 - 3553/19818086400*v^16 + 1867909/5945425920*v^14 - 1203840703/52848230400*v^12 + 47045264279/63417876480*v^10 - 94957438655047/7610145177600*v^8 + 54335798506231/507343011840*v^6 - 3804594429089897/9019431321600*v^4 + 20112317604285/34359738368*v^2 - 33424574007825/274877906944

11:

1/81749606400*v^22 - 11/4246732800*v^20 + 361/5662310400*v^18 + 12684533/475634073600*v^16 - 430958551/135895449600*v^14 + 108190413643/664377753600*v^12 - 68473337333027/15220290355200*v^10 + 598911889412147/8697308774400*v^8 - 12960433609753549/23192823398400*v^6 + 153837425383836749/72155450572800*v^4 - 17624993498100915/6047313952768*v^2 + 1327867167401775/2199023255552

12:

1/1961990553600*v^24 - 299/1961990553600*v^22 + 1909/203843174400*v^20 + 5079251/2853804441600*v^18 - 16069157299/45660871065600*v^16 + 977372842499/35876398694400*v^14 - 2321634669384521/2009078326886400*v^12 + 10443622865711503/365286968524800*v^10 - 682359446561748757/1669883284684800*v^8 + 1375676723381455513/432932703436800*v^6 - 2002315058562976573/169324790677504*v^4 + 772754215903629225/48378511622144*v^2 - 232376754295310625/70368744177664

13:

1/51011754393600*v^26 - 5/627836977152*v^24 + 5359/6278369771520*v^22 + 40963/456608710656*v^20 - 583938191/18264348426240*v^18 + 582140815681/160726266150912*v^16 - 2348313273985423/10447207299809280*v^14 + 5403895371473779/642905064603648*v^12 - 897598605176687723/4675673197117440*v^10 + 120254378804418391/46179488366592*v^8 - 165785185362631330207/8466239533875200*v^6 + 96759527981468167731/1354598325420032*v^4 - 1922931146523727230975/20125460834811904*v^2 + 11100458801337530625/562949953421312

14:

1/1428329123020800*v^28 - 17/45343781683200*v^26 + 31/507343011840*v^24 + 327727/117196235735040*v^22 - 19640827/8117488189440*v^20 + 70462046497/178584740167680*v^18 - 13485556353585499/390029072526213120*v^16 + 1571355504082291/844218771701760*v^14 - 1452830173663501051/22858846741463040*v^12 + 357576548568971404573/261837699038576640*v^10 - 10835839155504602198143/609569246439014400*v^8 + 8811755010176492975021/67729916271001600*v^6 - 167411866071013846111965/358599120329375744*v^4 + 99728685043643926170675/161003686678495232*v^2 - 1149690375852815671875/9007199254740992

15:

1/42849873690624000*v^30 - 551/34279898952499200*v^28 + 40223/10882507603968000*v^26 - 89581/2812709657640960*v^24 - 8656578793/56254193152819200*v^22 + 77210293187/2143016882012160*v^20 - 50812543123787173/11700872175786393600*v^18 + 3036785350745414819/9360697740629114880*v^16 - 140663699400488580839/8914950229170585600*v^14 + 17376343149858834867083/34562576273092116480*v^12 - 35578238721270257675183537/3456257627309211648000*v^10 + 1895849274535450081205443/14629661914536345600*v^8 - 2743957992338842787445616403/2958442742717349888000*v^6 + 103595558322605875897269007/31556722588985065472*v^4 - 11143817169458417061419205/2576058986855923712*v^2 + 64152722972587114490625/72057594037927936

16:

1/1371195958099968000*v^32 - 31/48971284217856000*v^30 + 178901/914130638733312000*v^28 - 23247241/1828261277466624000*v^26 - 527640181/64290506460364800*v^24 + 420806125247/150011181740851200*v^22 - 21359272039165481/46803488703145574400*v^20 + 614987640762838933/13372425343755878400*v^18 - 1022006822367346189883/332824808555701862400*v^16 + 621300015050910875511049/4493134915501975142400*v^14 - 8238474329024438318876677/1975004358462406656000*v^12 + 756952289022836395919029793/9216687006157897728000*v^10 - 71409334232400639425409077321/71002625825216397312000*v^8 + 7962047842021784614423715611/1127025806749466624000*v^6 - 12479113044887424958492082493/504907561423761047552*v^4 + 60660545164342760777881185/1873497444986126336*v^2 - 61394155884765868567528125/9223372036854775808

17:

1/46620662575398912000*v^34 - 23/997233424072704000*v^32 + 4619/498616712036352000*v^30 - 259811/221607427571712000*v^28 - 381869129/1063715652344217600*v^26 + 678848213291/3600268361780428800*v^24 - 348033513671273/8509725218753740800*v^22 + 185098235826332593/34038900875014963200*v^20 - 4483947539489450289217/9258581038004069990400*v^18 + 194184317794553170115783/6535468968002872934400*v^16 - 452562849863940620732917321/359450793240158011392000*v^14 + 365269515699151514474593447/10054567643081342976000*v^12 - 241343414162073070785809547541/348558344960153223168000*v^10 + 427778431250690998677450850843/51638273327430107136000*v^8 - 131439305300916381728565262989/2295034370108004761600*v^6 + 114467284645199304417831913305/577037213055726911488*v^4 - 1054064914131090035898418437225/4076730440289810907136*v^2 + 3918391713821821611515765625/73786976294838206464

18:

1/1678343852714360832000*v^36 - 1/1278738173496655872*v^34 + 2719/6838172050784256000*v^32 - 557597/7180080653323468800*v^30 - 108553351/9117562734379008000*v^28 + 442588179643/40117276031267635200*v^26 - 21384873046359793/6739702373252962713600*v^24 + 3192394705912463/5835240150002565120*v^22 - 1008049814409623116993/15871853208006977126400*v^20 + 2062951320530515684921999/399970700841775823585280*v^18 - 104234974610806137076143827/352115062765869072384000*v^16 + 588533394991209673233042319/49296108787221670133760*v^14 - 12497619978350282086590007202381/37644301255696548102144000*v^12 + 1471505897704350361871649548753/239011436544105067315200*v^10 - 25569828617849568437183750510197/354091017102377877504000*v^8 + 895065916653175173883971453123017/1817667221125539771187200*v^6 - 2123141080376998966129663734612565/1255632975609261759397888*v^4 + 71560175416909727031846223455825/32613843522318487257088*v^2 - 531595142508493798628972203125/1180591620717411303424

19:

1/63777066403145711616000*v^38 - 37/1491861202412765184000*v^36 + 39997/2557476346993311744000*v^34 - 4897357/1148812904531755008000*v^32 - 76811149/382937634843918336000*v^30 + 452499573733/802345520625352704000*v^28 - 2906795402958637/13479404746505925427200*v^26 + 855260974878735553/17972539662007900569600*v^24 - 224367000708062353987/31743706416013954252800*v^22 + 112828507560391318454785633/151988866319874812962406400*v^20 - 3655062086268692066730426971/65180410507548652732416000*v^18 + 60117426803763984005913256187/19718443514888668053504000*v^16 - 97731272215015353855935588384809/828174627625324058247168000*v^14 + 319299969907780189811427233986111/100384803348524128272384000*v^12 - 367249942077612492545170566919537/6373638307842801795072000*v^10 + 48303357962693712983755832469961769/72706688845021590847488000*v^8 - 1180388338865098524787440911827609951/263682924877944969473556480*v^6 + 21891430334994529946106771435277815/1435009114982013439311872*v^4 - 15038683754060730766737826053876375/762662186983447702011904*v^2 + 38190914185478633427818266171875/9444732965739290427392

20:

1/2551082656125828464640000*v^40 - 29/39247425478858899456000*v^38 + 1739/3060228107513364480000*v^36 - 6750983/33050463561144336384000*v^34 + 128007457/14139235748083138560000*v^32 + 65131033511/2592193220481908736000*v^30 - 17467351688473483/1347940474650592542720000*v^28 + 6014712215470507/1659003661108421591040*v^26 - 5103643338461314120421/7520816597024844546048000*v^24 + 2114727416570419370215571/23382902510749971224985600*v^22 - 451936039438989861992715255503/51442385523649936694968320000*v^20 + 75305252311840002614471340527/120333065552397512736768000*v^18 - 2155038154703172753954365397379997/66253970210025924659773440000*v^16 + 619731424701170928054920715414281/509645924692507112767488000*v^14 - 19713376724076152042383792464119173/617752635990917712445440000*v^12 + 171085486449053832966624892798117151/302012399817781992751104000*v^10 - 130516352832473654514077205658023447151/20283301913688074574888960000*v^8 + 34810408118831680921647628397264880671/811332076547522982995558400*v^6 - 63358054314286635140076600286101265137/436242770954532085550809088*v^4 + 1141430885103354525287323382661011325/6101297495867581616095232*v^2 - 11587123363874217382000061956546875/302231454903657293676544

Definition

The Hankel derivative asymptotic coefficient polynomials $b_k(\nu)\in\mathbb{Q}[\nu]$ are $b_0(\nu)=1$ and $b_k(\nu)=\frac{4\nu^2+4k^2-1}{8^k k!}\prod_{j=1}^{k-1}(4\nu^2-(2j-1)^2)$ for $k\geq1$, with the empty product equal to 1 [1].

Parameters

$k$

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

Formulas

(1)

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

(2)

$b_0(\nu)=1$ and $\displaystyle b_k(\nu)=a_k(\nu)+(k-\tfrac12)a_{k-1}(\nu)$ for $k\geq1$, where $a_k(\nu)$ are the Hankel asymptotic coefficient polynomials.

(3)

$\displaystyle b_k(\nu)=\tfrac12\bigl(a_k(\nu-1)+a_k(\nu+1)\bigr)$ for $k\geq0$, from $K_\nu^{\prime}(z)=-\tfrac12(K_{\nu-1}(z)+K_{\nu+1}(z))$ [3].

(4)

$\deg b_k=2k$, $b_k(-\nu)=b_k(\nu)$, and the leading coefficient is $1/(2^k k!)$. For $k\geq2$, $b_k(\pm(j-\tfrac12))=0$ for $1\leq j<k$.

(5)

$\displaystyle \sum_{k=0}^{n+1}b_k(n+\tfrac12)x^k=(1+\tfrac12x)y_n(x)+x^2y_n^{\prime}(x)$ for integers $n\geq0$, where $y_n$ is the Bessel polynomial normalized by $y_0(x)=1$ and $y_1(x)=1+x$, and the prime denotes differentiation with respect to $x$. This follows by differentiating $K_{n+1/2}(z)=\sqrt{\pi/(2z)}e^{-z}y_n(1/z)$ [4].

(6)

$b_0(\tfrac12)=1$, $b_1(\tfrac12)=\tfrac12$, and $b_k(\tfrac12)=0$ for $k\geq2$.

Comments

(7)

The indeterminate $\nu$ denotes the Bessel order. In (1), $z$ tends to positive infinity with $\nu$ fixed, the square root is positive, and the prime differentiates with respect to $z$. The overall minus sign and the phase factors in other Bessel expansions are external to $b_k(\nu)$ [2] [1].

(8)

The letter $v$ in the stored expressions denotes $\nu$. This is a change of variable name only; the coefficients are not rescaled.

(9)

The coefficient polynomials are exact. The accuracy of a truncated expansion of the derivative of $K_\nu(z)$ depends on $z$, $\nu$ and the number of terms retained.

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 b(k):
    if k == 0:
        return R.one()
    product = R.one()
    for j in range(1, k):
        product *= (4*nu**2-(2*j-1)**2) * (QQ(1)/QQ(8*j))
    return product * (4*nu**2+4*k**2-1) * (QQ(1)/QQ(8*k))
values = [b(k) for k in range(21)]
values

Links

[1]

DLMF 10.17.8, Hankel derivative coefficient polynomials

[2]

DLMF 10.40.4, large-argument expansion of the derivative of K

[3]

DLMF 10.29.1, modified Bessel recurrences and derivatives

[4]

DLMF 10.49(ii), finite half-integer expansions

Similar tables

Hankel asymptotic coefficient polynomials $a_k(\nu)$ — holds the coefficients for the functions themselves, which determine the derivative coefficients

Bessel polynomials $y_n$ — gives the elementary half-integer specialization used to express these derivative coefficients

Values of modified Bessel functions of the second kind $K_\nu(x)$ — holds values of the function whose derivative has this fixed-order large-argument expansion

Debye derivative polynomials $V_k(p)$ — holds derivative coefficients for the complementary large-order asymptotic regime

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 derivative corrections. The finite cutoff is an editorial choice that keeps the longest expanded entry at 1,209 characters and leaves the function and large-order coefficient families to separate tables)

How they were obtained:

All coefficients were computed exactly in $\mathbb{Q}[\nu]$ from the product in [1]. Independent interpolation in $\nu^2$ from the elementary half-integer Bessel recurrence reconstructs all 21 polynomials, with all 441 specializations agreeing.

The Bessel polynomials used in this reconstruction agree with the stored Bessel family through degree 20. Both relations with the stored $a_k$ polynomials, degree, leading coefficient, evenness and the stated half-integer roots were checked throughout the range. The attached verifier includes known nonzero low-order polynomials, interpolation of $1,t,t^2$ and deliberate coefficient corruption as controls.