Multi-integral representations for associated Legendre and Ferrers functions

For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis, we obtain some new results for the associated Legendre function of the second kind including parameter values for which this function is identically zero.Comment: 22 page

Integral representations for a generalized Hermite linear functional

In this paper we find new integral representations for the {\it generalized Hermite linear functional} in the real line and the complex plane. As application, new integral representations for the Euler Gamma function are given.Comment: 4 figure

Identifying all abelian periods of a string in quadratic time and relevant problems

Abelian periodicity of strings has been studied extensively over the last years. In 2006 Constantinescu and Ilie defined the abelian period of a string and several algorithms for the computation of all abelian periods of a string were given. In contrast to the classical period of a word, its abelian version is more flexible, factors of the word are considered the same under any internal permutation of their letters. We show two O(|y|^2) algorithms for the computation of all abelian periods of a string y. The first one maps each letter to a suitable number such that each factor of the string can be identified by the unique sum of the numbers corresponding to its letters and hence abelian periods can be identified easily. The other one maps each letter to a prime number such that each factor of the string can be identified by the unique product of the numbers corresponding to its letters and so abelian periods can be identified easily. We also define weak abelian periods on strings and give an O(|y|log(|y|)) algorithm for their computation, together with some other algorithms for more basic problems.Comment: Accepted in the "International Journal of foundations of Computer Science