OFFSET
0,5
COMMENTS
These rational polynomials R(n;x) appear in the evaluation of an integral in thermal field theories in the Bose case. See the Haber and Weldon reference eq. (D1), p. 1857 and the W. Lang link.
REFERENCES
H. E. Haber and H. A. Weldon, On the relativistic Bose-Einstein integrals, J. Math. Phys. 23(10) (1982) 1852-1858.
LINKS
FORMULA
a(n, m)= numerator(R(n, x)[x^m]), m=0, ..., n, n=0, 1, ..., with the rational polynomials R(n, x) of degree n defined by R(n, x):=hypergeom([ -n, -n], [1/2], x/2) = 1 + sum(r(n, m)*x^m, m=1..n), n>=0.
The rational polynomials are R(n, x) = 1 + sum(((binomial(n, m)^2)/binomial(2*m, m))*(2*x)^m, m=1..n), n>=0.
a(n, m)=numerator(r(n, m)) with the rational triangle r(n, m) = (2^m)*(binomial(n, m)^2)/binomial(2*m, m)), m=1..n, n=1, 2, ... and r(n, 0)=1, n>=0, else 0.
EXAMPLE
The rows of the rational table are: [1/1]; [1/1,1/1]; [1/1,4/1,2/3]; [1/1, 9/1, 6/1, 2/5]; ...
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Nov 30 2004
STATUS
approved