Debye polynomials $U_k(p)$

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Polynomials

$k$

$U_k(p)$

-5/24*p^3 + 1/8*p

385/1152*p^6 - 77/192*p^4 + 9/128*p^2

-85085/82944*p^9 + 17017/9216*p^7 - 4563/5120*p^5 + 75/1024*p^3

37182145/7962624*p^12 - 7436429/663552*p^10 + 144001/16384*p^8 - 96833/40960*p^6 + 3675/32768*p^4

-5391411025/191102976*p^15 + 5391411025/63700992*p^13 - 108313205/1179648*p^11 + 250881631/5898240*p^9 - 67608983/9175040*p^7 + 59535/262144*p^5

5849680962125/27518828544*p^18 - 1169936192425/1528823808*p^16 + 4445922195/4194304*p^14 - 33010308331/47185920*p^12 + 1441372804469/6606028800*p^10 - 388895895/14680064*p^8 + 2401245/4194304*p^6

-1267709431363375/660451885056*p^21 + 1774793203908725/220150628352*p^19 - 36927006432745/2717908992*p^17 + 10559432785187/905969664*p^15 - 1602251736839/301989888*p^13 + 1007390378503/838860800*p^11 - 25388505925/234881024*p^9 + 57972915/33554432*p^7

2562040760785380875/126806761930752*p^24 - 512408152157076175/5283615080448*p^22 + 75358832548684685/391378894848*p^20 - 39803268297948155/195689447424*p^18 + 3542717254441859/28991029248*p^16 - 276439228010667/6710886400*p^14 + 667955999804539/93952409600*p^12 - 928090660435/1879048192*p^10 + 13043905875/2147483648*p^8

-6653619855759634132375/27390260577042432*p^27 + 1330723971151926826475/1014454095446016*p^25 - 3128960418491082175/1043677052928*p^23 + 35348759075759093965/9393093476352*p^21 - 17618708259302571707/6262062317568*p^19 + 817138105244771959/644245094400*p^17 - 3739063570455884033/11274289152000*p^15 + 1359491937582325/30064771072*p^13 - 472414367256615/188978561024*p^11 + 418854310875/17179869184*p^9

10:

4318199286388002551911375/1314732507698036736*p^30 - 4318199286388002551911375/219122084616339456*p^28 + 91891691150784941691275/1803473947459584*p^26 - 16705838021516291703055/225434243432448*p^24 + 19941766574064397067317/300578991243264*p^22 - 5227733363217471800551/139156940390400*p^20 + 1232816120293110459821/92771293593600*p^18 - 35892416277828185849/12884901888000*p^16 + 45660648644355162105/148159191842816*p^14 - 20993386079260455/1511828488192*p^12 + 30241281245175/274877906944*p^10

11:

-1556514560957130010757145625/31553580184752881664*p^33 + 3424332034105686023665720375/10517860061584293888*p^31 - 365967912305800454531456575/389550372651270144*p^29 + 201734750392525792544487385/129850124217090048*p^27 - 11694306169843138084657687/7213895789838336*p^25 + 164293183874328160710877/148434069749760*p^23 - 23186185730591085896097833/46756731971174400*p^21 + 527174389121818780771231/3710851743744000*p^19 - 329641577686894230674187/13469017440256000*p^17 + 241770821762631191867/107752139522048*p^15 - 26416375998266454375/314460325543936*p^13 + 1212400457192925/2199023255552*p^11

12:

7404339766473067461171741738125/9087431093208829919232*p^36 - 1480867953294613492234348347625/252428641478023053312*p^34 + 38826657063629086145255937175/2077601987473440768*p^32 - 323782209780772997264458503055/9349208943630483456*p^30 + 171527591207538652048777279465/4155203974946881536*p^28 - 1058890883945614567899138703/32061759065948160*p^26 + 40295057240679430076066704447/2244323134616371200*p^24 - 12275126748382902959849354203/1870269278846976000*p^22 + 1440134927609056360249131941/923589767331840000*p^20 - 1247431715337149394208117/5541538603991040*p^18 + 4288535808487871046652577/246536895226445824*p^16 - 1383228778609739182725/2515682604351488*p^14 + 213786613951685775/70368744177664*p^12

Definition

The Debye polynomials $U_k(p)\in\mathbb{Q}[p]$ are defined by $U_0(p)=1$ and the differential-integral recurrence (1) [1].

Parameters

$k$

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

Formulas

(1)

$U_0(p)=1$, $\displaystyle U_{k+1}(p)=\frac12p^2(1-p^2)U_k^{\prime}(p)+\frac18\int_0^p(1-5t^2)U_k(t)\,dt$ for $k\geq0$, where the prime denotes differentiation with respect to $p$ [1].

(2)

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

(3)

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

(4)

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

(5)

$\displaystyle \sum_{k=0}^{\infty}U_k(1)h^k=\exp\!\left(-\sum_{m=1}^{\infty}\frac{B_{2m}}{2m(2m-1)}h^{2m-1}\right)$ as a formal power series in $h$, where $B_{2m}$ are the Bernoulli numbers [2].

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. Each stored object is a polynomial in the indeterminate $p$, with its rational coefficients unchanged.

(7)

The endpoint values $U_k(1)$ are the coefficients of the reciprocal scaled-Gamma expansion, whose logarithm is given by the Bernoulli series in (5) [2].

(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(k):
    U = R(1)
    for _ in range(k):
        f = (1 - 5*p**2)*U
        integral = R([QQ(0)] + [c/QQ(j+1) for j, c in enumerate(f.list())])
        U = p**2*(1-p**2)*U.derivative()/QQ(2) + integral/QQ(8)
    return U

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

Links

[1]

DLMF 10.41(ii), Uniform expansions for real variable

[2]

DLMF 5.11(i), logarithmic Gamma and scaled-Gamma expansions

[3]

DLMF 10.32.9, integral representation of the modified Bessel function K

Similar tables

Values of modified Bessel functions of the first kind $I_\nu(x)$ — holds values of the function whose uniform large-order expansion uses these coefficients

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

Bernoulli numbers — determines the logarithm of the formal generating series of the endpoint values at $p=1$

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 corrections; the finite cutoff is an editorial choice that keeps the longest expanded entry at 706 characters and leaves the derivative and fixed-order coefficient families to separate tables)

How they were obtained:

All coefficients were computed exactly in $\mathbb{Q}[p]$ from (1). An independent expansion of the Bessel integral [3], using exact Gaussian moments, reproduces every coefficient of all thirteen polynomials. The first three nonconstant polynomials agree with DLMF 10.41.10.

The reciprocal scaled-Gamma expansion [2] independently reproduces all thirteen values at $p=1$. The recurrence, degree, parity and lowest exponent were checked over the stored range. The attached checks include known Gaussian moments and a deliberate coefficient corruption as controls.