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$\,_{3}F_{4}$ hypergeometric functions as a sum of a product of $\,_{2}F_{3}$ functions

Jack C. Straton

Abstract

This paper shows that certain $\,_{3}F_{4}$ hypergeometric functions may be expanded in sums of pair products of $\,_{2}F_{3}$ functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, $\,_{2}F_{1}$ functions, and $\,_{3}F_{2}$ functions into the realm of $\,_{P}F_{Q}$ functions where $P<Q$ for both the summand and terms in the series. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation.

$\,_{3}F_{4}$ hypergeometric functions as a sum of a product of $\,_{2}F_{3}$ functions

Abstract

This paper shows that certain hypergeometric functions may be expanded in sums of pair products of functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, functions, and functions into the realm of functions where for both the summand and terms in the series. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation.
Paper Structure (7 sections, 56 equations)

This paper contains 7 sections, 56 equations.

Table of Contents

  1. Introduction
  2. The transition amplitude
  3. Derivation of the Function $\Xi_{n}^{p}(k\alpha_{0},-\frac{z}{2})$
  4. The Fourier-Legendre series of a Bessel function of the first kind
  5. A more direct approach
  6. For p=1
  7. Discussion and Concluding Remarks