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{
"Title": "Hankel asymptotic coefficient polynomials $a_k(\\nu)$",
"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 $a_0(\\nu)=1$ CITE{DLMFHankel}.",
"Parameters": {
"k": {
"type": "Z",
"display": "$k$",
"title": "order of the asymptotic correction",
"constraints": "$k\\geq0$"
}
},
"Tags": [
"polynomial",
"special functions",
"recurrence"
],
"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 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",
"rigour details": "All coefficients were computed exactly in $\\mathbb{Q}[\\nu]$. Independent interpolation in $\\nu^2$ from the recurrence of HREF{Bessel_polynomials}[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 CITE{formula-half-integer}. 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 CITE{DLMFEquation}, 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."
},
"Display properties": {
"number-header": "$a_k(\\nu)$"
},
"Numbers": {
"0": "1",
"1": "1/2*v^2 - 1/8",
"2": "1/8*v^4 - 5/16*v^2 + 9/128",
"3": "1/48*v^6 - 35/192*v^4 + 259/768*v^2 - 75/1024",
"4": "1/384*v^8 - 7/128*v^6 + 329/1024*v^4 - 3229/6144*v^2 + 3675/32768",
"5": "1/3840*v^10 - 11/1024*v^8 + 1463/10240*v^6 - 17281/24576*v^4 + 352407/327680*v^2 - 59535/262144",
"6": "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",
"7": "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",
"8": "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",
"9": "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"
},
"Keywords": [
"Bessel asymptotic coefficients",
"large argument",
"Hankel expansion",
"fixed order"
],
"Comments": {
"comment-serialization": "The letter $v$ in the stored expressions denotes the Bessel order $\\nu$. It is a change of variable name only.",
"comment-conventions": "The indeterminate $\\nu$ denotes the Bessel order. In CITE{formula-K}, $z$ 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 CITE{DLMFHankel} CITE{DLMFModified}.",
"comment-accuracy": "Each coefficient polynomial is exact. The accuracy of a truncated expansion of HREF{Values_of_modified_Bessel_functions_of_the_second_kind}[$K_\\nu(z)$] depends on $z$, $\\nu$ and the truncation order."
},
"Formulas": {
"formula-recurrence": "$a_0(\\nu)=1$, $\\displaystyle a_{k+1}(\\nu)=\\frac{4\\nu^2-(2k+1)^2}{8(k+1)}a_k(\\nu)$ for $k\\geq0$.",
"formula-K": "$\\displaystyle K_\\nu(z)\\sim\\sqrt{\\frac{\\pi}{2z}}e^{-z}\\sum_{k=0}^{\\infty}\\frac{a_k(\\nu)}{z^k}$ as $z\\to+\\infty$ with $\\nu$ fixed, where $K_\\nu$ is the HREF{Values_of_modified_Bessel_functions_of_the_second_kind}[modified Bessel function of the second kind] CITE{DLMFModified}.",
"formula-structure": "$\\deg a_k=2k$, $a_k(-\\nu)=a_k(\\nu)$, and the leading coefficient is $1/(2^k k!)$. For $k\\geq1$, the roots are $\\nu=\\pm(j-\\tfrac12)$ with $1\\leq j\\leq k$.",
"formula-half-integer": "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 $k>n$ CITE{DLMFSpherical}.",
"formula-Bessel": "$\\displaystyle y_n(x)=\\sum_{k=0}^n a_k(n+\\tfrac12)x^k$ for $n\\geq0$, where $y_n$ is the HREF{Bessel_polynomials}[Bessel polynomial] normalized by $y_0(x)=1$ and $y_1(x)=x+1$ CITE{DLMFSpherical}."
},
"Similar tables": [
{
"table": "HREF{Bessel_polynomials}[Bessel polynomials $y_n$]",
"relation": "has coefficients obtained by specializing the Bessel order to half-integers"
},
{
"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 large-argument expansion uses these coefficient polynomials"
}
],
"Links": {
"DLMFHankel": {
"title": "DLMF 10.17.1, Hankel asymptotic coefficient polynomials",
"url": "https://dlmf.nist.gov/10.17#E1"
},
"DLMFModified": {
"title": "DLMF 10.40.2, large-argument expansion of the modified Bessel function K",
"url": "https://dlmf.nist.gov/10.40#E2"
},
"DLMFSpherical": {
"title": "DLMF 10.49, explicit formulas for spherical Bessel functions",
"url": "https://dlmf.nist.gov/10.49"
},
"DLMFEquation": {
"title": "DLMF 10.25.1, modified Bessel differential equation",
"url": "https://dlmf.nist.gov/10.25#E1"
}
},
"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 a(k):\n result = R.one()\n for j in range(1, k + 1):\n result *= (4*nu**2 - (2*j-1)**2) * (QQ(1)/QQ(8*j))\n return result\n[a(k) for k in range(21)]"
}
}
}