A288455 Chebyshev coefficients of density of states of BCC lattice.

1, -48, 1728, -79872, 4058112, -216956928, 11977752576, -676117610496, 38792847949824, -2253773963526144, 132241430641901568, -7821943674035503104, 465750331610495975424, -27888626411947306254336, 1677958399935741979262976, -101375476324084742212288512, 6146869366762959307806867456

Offset: 0

Author

Yen-Lee Loh, Jun 16 2017

Keywords

• sign

Comments

This is the sequence of integers z^n g_n for n=0,2,4,6,... where g_n are the coefficients in the Chebyshev polynomial expansion of the density of states of the body-centered cubic lattice (z=8), g(w) = 1 / (Pi*sqrt(1-w^2)) * Sum_{n>=0} (2-delta_n) g_n T_n(w). Here |w| <= 1 and delta is the Kronecker delta.

The Chebyshev coefficients, g_n, are related to the number of walks on the lattice that return to the origin, W_n, as g_n = Sum_{k=0..n} a_{nk} z^{-k} W_k, where z is the coordination number of the lattice and a_{nk} are the coefficients of Chebyshev polynomials such that T_n(x) = Sum_{k=0..n} a_{nk} x^k.

For the bcc lattice (z=8), the even Chebyshev coefficients can be expressed in closed form in terms of the hypergeometric function pFq, as z^{2N} g_{2N} = (1 + delta_N) * 2^(2N-1) Binomial(2N,N)^3 * 4F3 (-N, -N, -N, -N; 1-2N, 1/2-N, 1/2-N; 1).

Links

• Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [math-ph], 2017.

Crossrefs

Related to numbers of walks returning to origin, W_n, on BCC lattice (A002897).

See also A288454, A288455, A288456, A288457, A288458, A288459, A288460, A288461.

Programs

Mathematica

zng[n_] :=
  If[OddQ[n], 0,
  (1 + KroneckerDelta[m]) 2^(2 m - 1) *
  Binomial[2m,m]^3 HypergeometricPFQ[{-m,-m,-m,-m}, {1-2m, 1/2-m, 1/2-m}, 1] /. m -> n/2];
Table[zng[n], {n,0,50}]
Wchain[n_] := If[OddQ[n], 0, Binomial[n, n/2]];
Wbcc[n_] := Wchain[n]^3;
ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
zng[n_] := Sum[ank[n, k]*8^(n-k)*Wbcc[k], {k, 0, n}];
Table[zng[n], {n,0,50}]