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{
"Title": "Hankel derivative asymptotic coefficient polynomials $b_k(\\nu)$",
"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 CITE{DLMFHankel}.",
"Parameters": {
"k": {
"type": "Z",
"display": "$k$",
"title": "order of the asymptotic correction",
"constraints": "$k\\geq0$"
}
},
"Tags": [
"polynomial",
"special functions",
"recurrence"
],
"Links": {
"DLMFHankel": {
"title": "DLMF 10.17.8, Hankel derivative coefficient polynomials",
"url": "https://dlmf.nist.gov/10.17#E8"
},
"DLMFModified": {
"title": "DLMF 10.40.4, large-argument expansion of the derivative of K",
"url": "https://dlmf.nist.gov/10.40#E4"
},
"DLMFRecurrence": {
"title": "DLMF 10.29.1, modified Bessel recurrences and derivatives",
"url": "https://dlmf.nist.gov/10.29#E1"
},
"DLMFSpherical": {
"title": "DLMF 10.49(ii), finite half-integer expansions",
"url": "https://dlmf.nist.gov/10.49#ii"
}
},
"Data properties": {
"type": "Q[]",
"rigour": "exact",
"complete": "no",
"complete-note": "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",
"rigour details": "All coefficients were computed exactly in $\\mathbb{Q}[\\nu]$ from the product in CITE{DLMFHankel}. 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 HREF{Bessel_polynomials}[the stored Bessel family] through degree 20. Both relations with HREF{T569}[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."
},
"Display properties": {
"number-header": "$b_k(\\nu)$"
},
"Numbers": {
"0": "1",
"1": "1/2*v^2 + 3/8",
"2": "1/8*v^4 + 7/16*v^2 - 15/128",
"3": "1/48*v^6 + 25/192*v^4 - 341/768*v^2 + 105/1024",
"4": "1/384*v^8 + 7/384*v^6 - 973/3072*v^4 + 1341/2048*v^2 - 4725/32768",
"5": "1/3840*v^10 + 1/1024*v^8 - 1057/10240*v^6 + 18251/24576*v^4 - 422553/327680*v^2 + 72765/262144",
"6": "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",
"7": "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",
"8": "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",
"9": "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"
},
"Keywords": [
"Bessel asymptotic coefficients",
"large argument",
"Hankel expansion",
"fixed order",
"Bessel derivative coefficients"
],
"Comments": {
"comment-conventions": "The indeterminate $\\nu$ denotes the Bessel order. In CITE{formula-K}, $z$ 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 $b_k(\\nu)$ CITE{DLMFModified} CITE{DLMFHankel}.",
"comment-serialization": "The letter $v$ in the stored expressions denotes $\\nu$. This is a change of variable name only; the coefficients are not rescaled.",
"comment-accuracy": "The coefficient polynomials are exact. The accuracy of a truncated expansion of the derivative of HREF{Values_of_modified_Bessel_functions_of_the_second_kind}[$K_\\nu(z)$] depends on $z$, $\\nu$ and the number of terms retained."
},
"Formulas": {
"formula-K": "$\\displaystyle K_\\nu^{\\prime}(z)\\sim-\\sqrt{\\frac{\\pi}{2z}}e^{-z}\\sum_{k=0}^{\\infty}\\frac{b_k(\\nu)}{z^k}$ as $z\\to+\\infty$ with $\\nu$ fixed, where $K_\\nu$ is the HREF{Values_of_modified_Bessel_functions_of_the_second_kind}[modified Bessel function of the second kind] CITE{DLMFModified}.",
"formula-a": "$b_0(\\nu)=1$ and $\\displaystyle b_k(\\nu)=a_k(\\nu)+(k-\\tfrac12)a_{k-1}(\\nu)$ for $k\\geq1$, where $a_k(\\nu)$ are the HREF{T569}[Hankel asymptotic coefficient polynomials].",
"formula-shift": "$\\displaystyle b_k(\\nu)=\\tfrac12\\bigl(a_k(\\nu-1)+a_k(\\nu+1)\\bigr)$ for $k\\geq0$, from $K_\\nu^{\\prime}(z)=-\\tfrac12(K_{\\nu-1}(z)+K_{\\nu+1}(z))$ CITE{DLMFRecurrence}.",
"formula-structure": "$\\deg b_k=2k$, $b_k(-\\nu)=b_k(\\nu)$, and the leading coefficient is $1/(2^k k!)$. For $k\\geq2$, $b_k(\\pm(j-\\tfrac12))=0$ for $1\\leq j<k$.",
"formula-half-integer": "$\\displaystyle \\sum_{k=0}^{n+1}b_k(n+\\tfrac12)x^k=(1+\\tfrac12x)y_n(x)+x^2y_n^{\\prime}(x)$ for integers $n\\geq0$, where $y_n$ is the HREF{Bessel_polynomials}[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 $K_{n+1/2}(z)=\\sqrt{\\pi/(2z)}e^{-z}y_n(1/z)$ CITE{DLMFSpherical}.",
"formula-half": "$b_0(\\tfrac12)=1$, $b_1(\\tfrac12)=\\tfrac12$, and $b_k(\\tfrac12)=0$ for $k\\geq2$."
},
"Similar tables": [
{
"table": "HREF{T569}[Hankel asymptotic coefficient polynomials $a_k(\\nu)$]",
"relation": "holds the coefficients for the functions themselves, which determine the derivative coefficients"
},
{
"table": "HREF{Bessel_polynomials}[Bessel polynomials $y_n$]",
"relation": "gives the elementary half-integer specialization used to express these derivative coefficients"
},
{
"table": "HREF{Values_of_modified_Bessel_functions_of_the_second_kind}[Values of modified Bessel functions of the second kind $K_\\nu(x)$]",
"relation": "holds values of the function whose derivative has this fixed-order large-argument expansion"
},
{
"table": "HREF{T568}[Debye derivative polynomials $V_k(p)$]",
"relation": "holds derivative coefficients for the complementary large-order asymptotic regime"
}
],
"Programs": {
"program-sage": {
"language": "Sage",
"code": "import numberdb.sage as numberdb\nfrom sage.rings.rational_field import QQ\nfrom sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing\nR = PolynomialRing(QQ, 'v')\nnu = R.gen()\ndef b(k):\n if k == 0:\n return R.one()\n product = R.one()\n for j in range(1, k):\n product *= (4*nu**2-(2*j-1)**2) * (QQ(1)/QQ(8*j))\n return product * (4*nu**2+4*k**2-1) * (QQ(1)/QQ(8*k))\nvalues = [b(k) for k in range(21)]\nvalues"
}
}
}