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On the extended hypergeometric equation and functions of arbitrary degree

The extended hypergeometric equation is a generalization of the Gaussian hypergeometric equation (Section 1), which is distinct from the generalized hypergeometric equation, because it remains of the second-order, but has an irregular singularity at infinity, of degree intermediate between those of Mathieu and Hill equations (Section 2). The solutions in the neighbourhood of the regular singularities (Section 3) at the origin (Subsections 3.1, 3.2) and point unity (Subsection 3.3), can be obtained by the Frobenius- Puchs method, in terms of power series with power (Subsection 3.1) or logarithmic (Subsection 3.2) singularities, with the difference that the recurrence formulas for the coefficients are multiple. The point at infinity is an irregular singularity (Section 4) of the extended hypergeometric equation. Because the corresponding solutions for this equation involve an essential singularity, the Frobenius-Fuchs method (Subsection 4.1) fails to give the solution in the neighbourhood of this point. The method of normal integrals (Subsection 4.2) yields the asymptotic expansions for extended hypergeometric functions only in the cases of lower degree, e.g. in the case of the first degree they generalize the asymptotic expansions of Mathieu functions. The solution of the extended hypergeometric equation in the neighbourhood of the irregular singularity at infinity, for any degree, is formally expressible as a Laurent series multiplied by a complex power, whose exponent on index is the root of an infinite determinant, and whose coefficients are determined from infinite linear systems of equations (Subsection 4.3). The method is similar to that used to solve Hill's equation, into which the extended hypergeometric type can be transformed (Section 5). The conclusion (Section 6) indicates some physical problems leading to differential equations with irregular singularities, and the cases in which the corresponding solutions are specified by extended hypergeometric functions.