Debye derivative polynomials $V_k(p)$

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Polynomials

$k$

$V_k(p)$

7/24*p^3 - 3/8*p

-455/1152*p^6 + 33/64*p^4 - 15/128*p^2

95095/82944*p^9 - 6545/3072*p^7 + 5577/5120*p^5 - 105/1024*p^3

-40415375/7962624*p^12 + 2739737/221184*p^10 - 2448017/245760*p^8 + 114439/40960*p^6 - 4725/32768*p^4

5763232475/191102976*p^15 - 215656441/2359296*p^13 + 355886245/3538944*p^11 - 280397117/5898240*p^9 + 15602073/1835008*p^7 - 72765/262144*p^5

-6183948445675/27518828544*p^18 + 415138648925/509607936*p^16 - 4775249765/4194304*p^14 + 7176153985/9437184*p^12 - 75861726551/314572800*p^10 + 440748681/14680064*p^8 - 2837835/4194304*p^6

1329548915820125/660451885056*p^21 - 623575990562525/73383542784*p^19 + 117495020467825/8153726976*p^17 - 11287669528993/905969664*p^15 + 4806755210517/838860800*p^13 - 23169978705569/17616076800*p^11 + 28375388975/234881024*p^9 - 66891825/33554432*p^7

-2671063771882631125/126806761930752*p^24 + 59582343274078625/587068342272*p^22 - 237670164192005545/1174136684544*p^20 + 42077740772116621/195689447424*p^18 - 1257093219318079/9663676416*p^16 + 296916207863309/6710886400*p^14 - 29041565208893/3758096384*p^12 + 146540630595/268435456*p^10 - 14783093325/2147483648*p^8

6904699850316601458125/27390260577042432*p^27 - 461679745093525633675/338151365148672*p^25 + 29412227933816172445/9393093476352*p^23 - 37073088786771732695/9393093476352*p^21 + 6190356955971173843/2087354105856*p^19 - 519996976064854883/386547056640*p^17 + 3996930023590772587/11274289152000*p^15 - 1468251292588911/30064771072*p^13 + 517406211757245/188978561024*p^11 - 468131288625/17179869184*p^9

10:

-4464578923214714502823625/1314732507698036736*p^30 + 1491741571661309972478475/73040694872113152*p^28 - 286485860646564818213975/5410421842378752*p^26 + 17416724745836133903185/225434243432448*p^24 - 2318810066751674077595/33397665693696*p^22 + 16487466760916641832507/417470821171200*p^20 - 45614196450845087013377/3246995275776000*p^18 + 12736018679229356269/4294967296000*p^16 - 49042918914307396335/148159191842816*p^14 + 22818897912239625/1511828488192*p^12 - 33424574007825/274877906944*p^10

11:

1604407316678887857241980875/31553580184752881664*p^33 - 392956135061308232223935125/1168651117953810432*p^31 + 1136426675054854043018733575/1168651117953810432*p^29 - 209347382482809784715977475/129850124217090048*p^27 + 4057208263006803008962871/2404631929946112*p^25 - 7721779642093423553411219/6679533138739200*p^23 + 24317219180863821793468459/46756731971174400*p^21 - 185223434015774166216919/1236950581248000*p^19 + 109880525895631410224729/4233119766937600*p^17 - 258444671539364377513/107752139522048*p^15 + 28529686078127770725/314460325543936*p^13 - 1327867167401775/2199023255552*p^11

12:

-7612912717641322882613199251875/9087431093208829919232*p^36 + 508357655608598661513283761125/84142880492674351104*p^34 - 360533244162270085634519416625/18698417887260966912*p^32 + 334757877908934793781897774345/9349208943630483456*p^30 - 59254986053513352525941241997/1385067991648960512*p^28 + 3301248049948092476391432427/96185277197844480*p^26 - 6001391503930978947499296407/320617590659481600*p^24 + 285468063915881464182543121/41561539529932800*p^22 - 4541964002459331597708800737/2770769301995520000*p^20 + 46154973467474527585700329/193953851139686400*p^18 - 415019594369793972256701/22412445020585984*p^16 + 1485690169617868011075/2515682604351488*p^14 - 232376754295310625/70368744177664*p^12

Definition

The Debye derivative polynomials $V_k(p)\in\mathbb{Q}[p]$ are defined by $V_0(p)=1$ and the differential relation (1) [1].

Parameters

$k$

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

Formulas

(1)

$\displaystyle V_{k+1}(p)=U_{k+1}(p)-\frac12p(1-p^2)U_k(p)-p^2(1-p^2)U_k^{\prime}(p)$ for $k\geq0$, where $U_k$ are the Debye polynomials and the prime denotes differentiation with respect to $p$ [1].

(2)

$\displaystyle I_\nu^{\prime}(\nu z)\sim\frac{(1+z^2)^{1/4}e^{\nu\eta}}{z\sqrt{2\pi\nu}}\sum_{k=0}^{\infty}\frac{V_k(p)}{\nu^k}$, where $I_\nu$ is the modified Bessel function of the first kind [1].

(3)

$\displaystyle K_\nu^{\prime}(\nu z)\sim-\sqrt{\frac{\pi}{2\nu}}\frac{(1+z^2)^{1/4}e^{-\nu\eta}}{z}\sum_{k=0}^{\infty}(-1)^k\frac{V_k(p)}{\nu^k}$, where $K_\nu$ is the modified Bessel function of the second kind [1].

(4)

$\deg V_k=3k$, $V_k(-p)=(-1)^kV_k(p)$, and the lowest nonzero power of $p$ in $V_k$ is $p^k$. In particular, $V_k(0)=0$ for $k>0$.

(5)

$V_k(1)=U_k(1)$ for $k\geq0$.

Comments

(6)

In the asymptotic expansions, $z>0$, $p=(1+z^2)^{-1/2}$ and $\eta=\sqrt{1+z^2}+\log\!\left(z/(1+\sqrt{1+z^2})\right)$, with positive square roots and the real logarithm. The order $\nu$ tends to positive infinity. A prime on a Bessel function denotes differentiation with respect to its full argument, so $I_\nu^{\prime}(\nu z)$ differs from $dI_\nu(\nu z)/dz$ by a factor of $1/\nu$.

(7)

Each entry retains the full polynomial in the indeterminate $p$ with reduced rational coefficients. The factor $(-1)^k$ and the overall minus sign in (3) are external to $V_k(p)$.

(8)

The polynomials are exact. The accuracy of a truncated Bessel expansion 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, 'p')
p = R.gen()
def debye_derivative(k):
    U = R(1)
    V = R(1)
    for j in range(k):
        f = (1-5*p**2)*U
        integral = R([QQ(0)] + [c/QQ(m+1) for m, c in enumerate(f.list())])
        next_U = p**2*(1-p**2)*U.derivative()/QQ(2) + integral/QQ(8)
        V = next_U - p*(1-p**2)*U/QQ(2) - p**2*(1-p**2)*U.derivative()
        U = next_U
    return V

values = [debye_derivative(k) for k in range(13)]
values

Links

[1]

DLMF 10.41(ii), uniform expansions for real variable

[2]

DLMF 10.32.9, integral representation of the modified Bessel function K

Similar tables

Debye polynomials $U_k(p)$ — holds the coefficient polynomials for the functions themselves, from which the derivative coefficients are obtained

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

Values of modified Bessel functions of the second kind $K_\nu(x)$ — holds values of the function whose derivative uses these coefficients with alternating external signs

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (it holds every correction order $0\leq k\leq12$, retaining each full polynomial in $p$ through degree 36. This initial prefix covers successive large-order derivative corrections; the finite cutoff is an editorial choice that keeps the longest expanded entry at 702 characters and leaves the function and fixed-order coefficient families to separate tables)

How they were obtained:

All coefficients were computed exactly in $\mathbb{Q}[p]$ from (1) and the recurrence for $U_k$. An independent saddle expansion of the differentiated Bessel integral [2], using exact Gaussian moments, reproduces every coefficient of all thirteen polynomials.

The first three nonconstant polynomials agree with DLMF 10.41.11. Degree, parity, lowest exponent, and the relations with the stored $U_k$ polynomials were checked throughout the stored range. The attached checks include known Gaussian moments and deliberate coefficient corruption as controls.