New series expansions for the confluent hypergeometric function M(a,b,z)M(a,b,z)

Three new series expansions of the confluent hypergeometric function M(a,b,z)M(a,b,z) in terms of elementary functions are given. The starting point is an integral representation of M(a,b,z)M(a,b,z). Then, different multi-point Taylor expansions of an appropriate function in the integrand are used. The new expansions are used to evaluate M(a,b,z)M(a,b,z). They provide accurate evaluations of the confluent hypergeometric function, in particular improving the results in the region of small or moderate ∣z∣∣z∣ where the power series definition is recommended for the evaluation of M(a,b,z)M(a,b,z).