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

• Published in: Applied mathematics and computation Vol. 479; p. 128857
• Main Authors: Laxmi, Parik, Jain, Shilpi, Agarwal, Praveen, Milovanović, Gradimir V.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.10.2024
• Subjects: Gaussian quadrature rule; k-beta function; k-gamma function; Mellin transform; Modified moments; Orthogonal polynomial; Gaussian quadrature rule; k-beta function; k-gamma function; Orthogonal polynomial; Modified moments; Mellin transform
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2024.128857

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 Cvetković and Milovanović (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. •Very efficient algorithm for numerical calculating the values of the extension of the k-beta function Bk(v1,v2;s) and the function Bs1,s2(v1,v2;s), defined by the two-parameter Mittag-Leffler function Es1,s2(z).•Methods are based on quadrature formulas of Gaussian type.•New extensions and their properties.