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On the representation of Meijer’s G-function in the vicinity of singular unity. (English) Zbl 0581.33006
Complex analysis and applications, Proc. Int. Conf., Varna/Bulg. 1981, 383-398 (1984).

[For the entire collection see Zbl 0573.00007.]

Although the behaviors of Meijer’s G-function at the singular points 0 and $\infty$ have been explored in the case $p=q$, little information in regard to the behavior at the singular point $z={\left(-1\right)}^{m+n-p}$ has been available. This function is one solution of a certain p-th order linear differential equation. In the present paper a fundamental system of solutions in the neighbourhood of this critical point is found. The explicit, but very complicated, representations are obtained and the recurrences for the coefficients are included.

The complicated situation in regard to this G-function in a neighborhood of $z={\left(-1\right)}^{m+n-p}$ can now be better understood. The asymptotic behavior as $z\to 1$ is determined and a set of simple, explicit special cases is appended in order to display all of the various behaviors. Further, some new relations are included which connect G-functions with different parameters.

Reviewer: R.G.Buschman
##### MSC:
 33C60 Hypergeometric integrals and functions defined by them 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 34A25 Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)