New analytic representations of the Lerch transcendent

López, José L. ; Pérez Sinusía, Ester (Universidad de Zaragoza)

Resumen: We consider an integral representation of the Lerch transcendent function (z, s, a) of the form (z, s, a) = 1 0h(t, z)g(t, s, a)dt, and two different analytical methods for the approximation of this integral transform to obtain new convergent expansions of the Lerch transcendent in the variable z. The first method uses multi-point Taylor expansions of h(t, z) at certain appropriately selected base points that provides convergent expansions of the Lerch transcendent in terms of elementary functions of z uniformly valid in compact sets of the complex z−plane. The second method expands g(t, s, a) in a Taylor series at a selected point in [0, 1] giving a uniform convergent expansion of (z, s, a) in terms of elementary functions of z valid in a large unbounded region of the complex plane. We provide explicit and/or recursive algorithms for the computation of the coefficients of the expansions. Numerical experiments illustrate the accuracy of the new approximations.

Idioma: Inglés

DOI: 10.1007/s11075-025-02113-w

Año: 2025

Publicado en: NUMERICAL ALGORITHMS (2025), [22 pp.]
ISSN: 1017-1398

Financiación: info:eu-repo/grantAgreement/ES/MCINN/PID2022-136441NB-I00

Tipo y forma: Artículo (Versión definitiva)

Área (Departamento): Área Matemática Aplicada (Dpto. Matemática Aplicada)