RADIUS OF CONVERGENCE FOR SOME (MULTIVARIABLE) HYPERGEOMETRIC FUNCTIONS
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Date
2024-07-15
Authors
Kashou, Layth
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Publisher
An-Najah National University
Abstract
Background: Hypergeometric functions are a class of special functions in mathematics that play a crucial role in various branches of science and engineering. Its importance lies in their versatility and their ability to represent a vast array of mathematical and physical phenomena. The key aspects that underscore their significance and applications include solutions to differential equations, solving Schrödinger's equation for various physical systems, studying complex integrals and contour integrals, solving problems involving electromagnetic fields and wave propagation in different media and they have applications in celestial mechanics for predicting the positions and orbits of celestial bodies.
Aims: We have two main objectives. The first one is deriving new transformation formulas for the Kampé de Fériet function taking into account the radius of convergence of each transformation. While the other is developing alternative methods for determining the radius of convergence for well-known multivariable (double and triple) hypergeometric series.
Methods: In this thesis, we will use Miller-Paris transformation formulas for generalized hypergeometric functions ${}_{r+1}F_{r+1}(z)$, ${}_{r+2}F_{r+1}(z)$ and its radii of convergence to derive new transformation formulas for the Kampé de Fériet function. Also, we will use Mathematica to develop an alternative method to calculate the radii of convergence for some well-known multivariable hypergeometric series.
Results: While testing the Kampé de Fériet transformations we derived using various parameter values and variables within the radius of convergence on Mathematica, we found that the left-hand side and the right-hand side are equal for all tested cases. Also, when we try to calculate the radii of convergence for the selected well-known multivariable hypergeometric series by plotting them on Mathematica the results indicate that our findings are identical with the ones presented in Srivastava’s book.