A Remarkable Sequence Derived from Euler Products.

An infinite product of the form a sub 0 = prod from m=1 to inf (1 over (1 -alpha sup m)), 0 < alpha < 1 was introduced by Euler in a famous theorem of number theory. A generalized form of this infinite product is used to define a sequence a sub n, which is shown to have many interesting properties. The a sub n coefficients are in fact the partial-fraction expansion coefficients of a sub 0, as a function of alpha, and furthermore emerge as the parameters that define the probability distribution of a first-order Markov process driven by white exponential noise. 

The alpha sub n coefficients are recursively calculated from a sub 0, and monotonically converge to zero O(alpha sup n sup 2). The sum of the sequence is equal to 1, and the alternating sum is equal to a sub 0 sup 2 (alpha)/a sub 0 (alpha sup 2). A more remarkable property is that the a sub n sequence is orthogonal to all exponentially- increasing sequences of the form alpha sup (-kn), where k is a positive integer. 

Various other expressions are also derived for the moments of a sub n alpha sup (-kn), k >= 0. The z-transform of the a sub n sequence is shown to be characterized by an infinite set of zeros alpha sup m on the real axis and an essential singularity at the origin. The generating function of the alpha sub n sequence is developed separately in an appendix, with analysis of the zeros, extrema, growth rate, and derivatives. Several calculator programs used for numerical computations are also included in an appendix.