A remarkable sequence derived from Euler products

An infinite product of the form a(0)=Pi(m=1)(infinity)[1/(1-alpha(m))], 0 < alpha < 1 was introduced by Euler in a famous theorem of number theory. A g eneralized form of this infinite product is used to define a sequence a(n)= Pi [GRAPHICS] 1/(1-alpha(m-n))], which is shown to have many interesting properties. The a(n) coefficients are in fact the partial-fraction expansion coefficients a ssociated with the characteristic function of a first-order Markov process driven by an uncorrelated sequence of random variates with exponential dens ity. The a(n) coefficients are recursively calculated from a(0), and monoto nically converge to zero O(alpha(n2)). The sum of the sequence is equal to 1, and the alternating sum is equal to a(0)(2)(alpha)/a(0)(alpha(2)). A mor e remarkable property is that the a(n) sequence is orthogonal to all expone ntially increasing sequences of the form alpha(-kn), where k is a positive integer. Various other expressions are also derived for the moments of a(n) alpha(-kn), k greater than or equal to 0. The z-transform of the a(n) seque nce is shown to be characterized by an infinite set of zeros alpha(m) on th e real axis and an essential singularity at the origin. [S0022-2488(00)0301 0-3].