Computing Reciprocals of Bivariate Power Series

On Bivariate Generalized Exponential-Power Series Class of Distributions

In this paper, we introduce a new class of bivariate distributions by compounding the bivariate generalized exponential and power-series distributions. This new class contains the bivariate generalized exponential-Poisson, bivariate generalized exponential-logarithmic, bivariate generalized exponential-binomial and bivariate generalized exponential-negative binomial distributions as specia...

Integral Representations for Bivariate Complex Geometric Mean and Applications

In the paper, the authors survey integral representations (including the Lévy– Khintchine representations) and applications of some bivariate means (including the logarithmic mean, the identric mean, Stolarsky’s mean, the harmonic mean, the (weighted) geometric means and their reciprocals, and the Toader–Qi mean) and the multivariate (weighted) geometric means and their reciprocals, derive inte...

Variations on computing reciprocals of power series

Power series with restricted coefficients and a root on a given ray

We consider bounds on the smallest possible root with a specified argument φ of a power series f(z) = 1 + ∑∞ n=1aiz i with coefficients ai in the interval [−g, g]. We describe the form that the extremal power series must take and hence give an algorithm for computing the optimal root when φ/2π is rational. When g ≥ 2√2 + 3 we show that the smallest disc containing two roots has radius ( √ g + 1...

Congruences involving the reciprocals of central binomial coefficients

We present several congruences modulo a power of prime p concerning sums of the following type ∑p−1 k=1 m k ( 2k k ) −1 which reveal some interesting connections with the analogous infinite series.