New series expansions for the confluent hypergeometric function Mða; b; zÞ José L. López a,⇑ , Ester Pérez Sinusía b a Dpto. de Ingeniería Matemática e Informática, Universidad Pública de Navarra, Pamplona 31006, Spain b Dpto. de Matemática Aplicada, IUMA, Universidad de Zaragoza, Zaragoza 50009, Spain article info Keywords: Confluent hypergeometric function Series expansion Taylor expansion Multi-point Taylor expansion abstract Three new series expansions of the confluent hypergeometric function Mða; b; zÞ in terms of elementary functions are given. The starting point is an integral representation of Mða; b; zÞ. Then, different multi-point Taylor expansions of an appropriate function in the integrand are used. The new expansions are used to evaluate Mða; b; zÞ. They provide accurate evalu- ations of the confluent hypergeometric function, in particular improving the results in the region of small or moderate j z j where the power series definition is recommended for the evaluation of Mða; b; zÞ. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction The confluent hypergeometric function Mða; b; zÞ, also denoted by 1 F 1 ða; b; zÞ, is defined by means of the power series [11, eq. 13.2.2] Mða; b; zÞ :¼ X 1 n¼0 ðaÞ n ðbÞ n n! z n ; ð1Þ and is convergent for all a; b; z 2 C with b – 0; 1; 2; 3; ... This function is one of the standard solutions of Kummer’s equation z d 2 w dz 2 þðb  zÞ dw dz  aw ¼ 0: ð2Þ An extensive study of this function can be found in [14]. The numerical evaluation of the confluent hypergeometric function Mða; b; zÞ has been investigated by many authors in different papers (see [1,4,10,12] and references there in). As it is pointed out in these works, there is no optimal method for computing Mða; b; zÞ for all parameter and variable values, but a number of different methods must be considered depending on the parameter and variable region investigated. The algorithms are based on different computational approaches: Taylor series, expansions in terms of other special functions, asymptotic expansions, quadrature rules, numerical solutions of the confluent hypergeometric differential equation, recurrence relations and transformation formulae. http://dx.doi.org/10.1016/j.amc.2014.02.099 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail addresses: jl.lopez@unavarra.es (J.L. López), ester.perez@unizar.es (E. Pérez Sinusía). Applied Mathematics and Computation 235 (2014) 26–31 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc