Stokes Phenomenon Arising in the Confluence of the Gauss Hypergeometric Equation

It is shown that by the application of Borel’s method of summation to the later terms of an asymptotic expansion, the ‘sum’ of such terms can normally be replaced by an easily calculable series involving ‘basic converging factors’. As particular consequences, [i] the remainder in a truncated asymptotic expansion can be written down once the general term in the expansion is known; [ii] the converging factor for a given asymptotic expansion can conveniently be calculated from the basic converging factors; and [iii] the Stokes phenomenon is simply expressed in terms of discontinuities in these basic quantities. Formulae and tables are given for the basic converging factors.

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Error bounds, duality, and the Stokes phenomenon. I

St Petersburg Mathematical Journal ◽

10.1090/s1061-0022-2010-01125-7 ◽

2010 ◽

Vol 21 (6) ◽

pp. 903-903 ◽

Cited By ~ 3

Author(s):

V. P. Gurariĭ

Keyword(s):

Error Bounds ◽

Stokes Phenomenon

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THE GENERALIZED MATRIX VALUED HYPERGEOMETRIC EQUATION

International Journal of Mathematics ◽

10.1142/s0129167x10005970 ◽

2010 ◽

Vol 21 (02) ◽

pp. 145-155 ◽

Cited By ~ 1

Author(s):

P. ROMÁN ◽

S. SIMONDI

Keyword(s):

Differential Equation ◽

Orthogonal Polynomials ◽

Unit Circle ◽

Hypergeometric Functions ◽

Spherical Functions ◽

Hypergeometric Equation ◽

Necessary Condition ◽

Hypergeometric Differential Equation ◽

The Matrix ◽

Generalized Matrix

The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in [4]. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao's equation in order to get a generalized matrix valued hypergeometric equation. We take advantage of the tools and strategies developed in [4] to identify the corresponding matrix hypergeometric functions nFm. We prove that, if n = m + 1, these functions are analytic for |z| < 1 and we give a necessary condition for the convergence on the unit circle |z| = 1.

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Hyperasymptotics

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1990.0111 ◽

1990 ◽

Vol 430 (1880) ◽

pp. 653-668 ◽

Cited By ~ 99

Keyword(s):

Asymptotic Expansion ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Transition Point ◽

Asymptotic Series ◽

Stokes Phenomenon ◽

Numerical Computations ◽

Borel Summation ◽

Nested Sequence ◽

Exponentially Small

We develop a technique for systematically reducing the exponentially small (‘superasymptotic’) remainder of an asymptotic expansion truncated near its least term, for solutions of ordinary differential equations of Schrödinger type where one transition point dominates. This is achieved by repeatedly applying Borel summation to a resurgence formula discovered by Dingle, relating the late to the early terms of the original expansion. The improvements form a nested sequence of asymptotic series truncated at their least terms. Each such ‘hyperseries’ involves the terms of the original asymptotic series for the particular function being approximated, together with terminating integrals that are universal in form, and is half the length of its predecessor. The hyperasymptotic sequence is therefore finite, and leads to an ultimate approximation whose error is less than the square of the original superasymptotic remainder. The Stokes phenomenon is automatically and exactly incorporated into the scheme. Numerical computations confirm the efficacy of the technique.

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