RECIPROCALS OF BINARY POWER SERIES

Reciprocals of Binary Power Series

If A is a set of nonnegative integers containing 0, then there is a unique nonempty set B of nonnegative integers such that every positive integer can be written in the form a + b, where a ∈ A and b ∈ B, in an even number of ways. We compute the natural density of B for several specific sets A, including the Prouhet-Thue-Morse sequence, {0} ∪ {2 : n ∈ N}, and random sets, and we also study the ...

Variations on computing reciprocals of power series

Computing Reciprocals of Bivariate Power Series

We consider the multiplicative complexity of the inversion and division of bivariate power series modulo the \triangular" ideal generated by all monomials of total degree n + 1. For inversion, we obtain a lower bound of 7 8 n 2 ? O(n) opposed to an upper bound of 7 3 n 2 + O(n). The former bound holds for all elds with characteristic distinct from two while the latter is valid over elds of char...

The Series of Reciprocals of Non-central Binomial Coefficients

Utilizing Gamma-Beta function, we can build one series involving reciprocal of non-central binomial coefficients, then We can structure several new series of reciprocals of non-central binomial coefficients by item splitting, these new created denominator of series contain 1 to 4 odd factors of binomial coefficients. As the result of splitting items, some identities of series of numbers values ...

On Certain Series Involving Reciprocals of Binomial Coefficients

We evaluate the following family of series involving reciprocals of binomial coefficients in terms of elementary functions for m = 3,4. ∞ ∑ k=0 xk (mk+1) (mk k ) .