Alternative to Euler's formula for  with  and for even indexed Bernoulli numbers
E. Salinas-Hernández, Abelardo Santaella-Quintas, Martha P. Ramírez-Torres, Gonzalo Ares de Parga

 

Abstract:

This paper proposes an alternative mechanism to get an original result for the expression  the first related result was obtained by Leonhard Euler in 1732; later, we will be able to reproduce even indexed Bernoulli numbers from both results.

 

 

Computations of hitting time densities for the generalized Cox-Ingersoll-Ross diffusion
Jonathan Gutierrez-Pavón

Abstract:
We give explicit formulae for the density function of first hitting time of the so-called generalized Cox-Ingersoll-Ross process. In fact, we treat the several cases of the diffusion depending on the values of the parameters. To find the density function we use the eigenvalues and eigenfunctions associated with the infinitesimal operator. It turns out that a very important tool in this analysis is the so-called Kummer equation, where we use the known solutions; this allows us to compute the eigenfunctions interms of the confluent hypergeometric functions.