On some solutions of the extended confluent hypergeometric differential equation

The extended confluent hypergeometric equation is defined (Section 1) as a linear second-order differential equation with (Section 2) a regular singularity at the origin and an (Section 3) irregular singularity of arbitrary degree M+1 at infinity; the original confluent hypergeometric equation is the particular case M=0, whereas the case M=1 is reducible (Section 3.2) to the former. Six types of solutions of the extended confluent hypergeometric equation of degree M, are obtained viz.: (i) functions of the first kind, i.e., regular ascending power series expansions, with infinite radius of convergence about the origin, for all values of the coefficients and degree M (Section 2.1); (ii) functions of the second kind, i.e. power series expansions with a logarithmic singularity, at the origin, for some values of the coefficients and all M (Section 2.2): (iii) only one asymptotic power series expansion exists, (Section 2.3) for M=0; (iv) concerning normal integrals (Section 3.2), valid as asymptotic expansions in the neighbourhood of the point-at-infinity, one exists for degree zero M=0 and two for degree unity M=1; (v) for degree greater than one M>1, two Laurent series expansions (Section 3.3) valid in the neighbourhood of infinity are obtained; (vi) an integral representation (Section 4) using the complex Laplace transform (Section 4.1) is obtained for (Sections 4.2–4.3) degree unity or zero M⩽1, using paths in a complex cut-plane (Fig. 1).