In this paper, we prove that for $s\in(0,\infty)$, the harmonic mean of $E_k(s)$ and $E_k(1/s)$ is always less than or equal to $\Gamma(1-k,1)$. Where generalized exponential integral function, $\Gamma(u,s)$ upper incomplete gamma function $k\in \mathbb{N}$.

The multidimensional exponential Levy equations are used to describe many stochastic phenomena such as market fluctuations. Unfortunately in practice an exact solution does not exist for these equations. This motivates us to propose a numerical solution for n-dimensional exponential Levy equations by block pulse functions. We compute the jump integral of each block pulse function and present a ...

In this paper we use the Umbral Calculus to investigate the relation between natural exponential families and Sheeer polynomials. We give a new proof for the classiication of univariate natural exponential families with quadratic variance function. We also show how our methods apply to natural exponential families with variance function of any order and to multivariate natural exponential famil...