Special moments

In this article, we show that a linear combination X˜ of n independent, unbiased Bernoulli random variables {Xk} can match the first 2n moments of a random variable Y which is uniform on an interval. More generally, for each p⩾2, each Xk can be uniform on an arithmetic progression of length p. All values of X˜ lie in the range of Y, and their ordering as real numbers coincides with dictionary order on the vector (X1,…,Xn). The construction involves the roots of truncated q-exponential series. It applies to a construction in numerical cubature using error-correcting codes [G. Kuperberg, Numerical cubature using error-correcting codes, arXiv:math.NA/0402047]. For example, when n=2 and p=2, the values of X˜ are the 4-point Chebyshev quadrature formula.