Numerical calculation of the extension of k-beta function and some new extensions by using two parameter k-Mittag-Leffler function

A numerical method for efficient calculation of recently defined extension of k -beta functions, based on weighted quadrature formulas of Gaussian type, is proposed. The modified moments of an even exponential weight function on (-1, 1), with essential singularities at +/- 1, are calculated in symbolic form in terms of the Meijer G -function. A similar problem with respect the two -parameter Mittag-Leffler function Es1,s2(z) is also considered. The MATHEMATICA package OrthogonalPolynomials by Cvetkovic<acute accent> and Milovanovic<acute accent> (2004) [4] is applied. Also, a new extension of k -gamma and k -beta functions by using two parameter k-Mittag-Leffler function is presented, as well as their basic properties, including some identities, a functional relation, summation and derivative formulas, integral representations and Mellin transform.