Asymptotic expansion for inverse moments of binomial and Poisson distributions

An asymptotic expansion for inverse moments of positive binomial and Poisson distributions is derived. The expansion coefficients of the asymptotic series are given by the positive central moments of the distribution. Compared to previous results, a single expansion formula covers all (also non-integer) inverse moments. In addition, the approach can be generalized to other positive distributions.Comment: 8 page

Chebyshev Series Expansion of Inverse Polynomials

An inverse polynomial has a Chebyshev series expansion 1/\sum(j=0..k)b_j*T_j(x)=\sum'(n=0..oo) a_n*T_n(x) if the polynomial has no roots in [-1,1]. If the inverse polynomial is decomposed into partial fractions, the a_n are linear combinations of simple functions of the polynomial roots. If the first k of the coefficients a_n are known, the others become linear combinations of these with expansion coefficients derived recursively from the b_j's. On a closely related theme, finding a polynomial with minimum relative error towards a given f(x) is approximately equivalent to finding the b_j in f(x)/sum_(j=0..k)b_j*T_j(x)=1+sum_(n=k+1..oo) a_n*T_n(x), and may be handled with a Newton method providing the Chebyshev expansion of f(x) is known.Comment: LaTeX2e, 24 pages, 1 PostScript figure. More references. Corrected typos in (1.1), (3.4), (4.2), (A.5), (E.8) and (E.11