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16 Generalized Hypergeometric Functions & Meijer G-FunctionApplications

Β§16.24 Physical Applications

  1. Β§16.24(i) Random Walks
  2. Β§16.24(ii) Loop Integrals in Feynman Diagrams
  3. §16.24(iii) 3⁒j, 6⁒j, and 9⁒j Symbols

Β§16.24(i) Random Walks

Generalized hypergeometric functions and Appell functions appear in the evaluation of the so-called Watson integrals which characterize the simplest possible lattice walks. They are also potentially useful for the solution of more complicated restricted lattice walk problems, and the 3D Ising model; see Barber and Ninham (1970, pp.Β 147–148).

Β§16.24(ii) Loop Integrals in Feynman Diagrams

Appell functions are used for the evaluation of one-loop integrals in Feynman diagrams. See Cabral-Rosetti and Sanchis-Lozano (2000).

For an extension to two-loop integrals see Moch et al. (2002).

§16.24(iii) 3⁒j, 6⁒j, and 9⁒j Symbols

The 3⁒j symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. They can be expressed as F23 functions with unit argument. The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner 6⁒j symbols. These are balanced F34 functions with unit argument. Lastly, special cases of the 9⁒j symbols are F45 functions with unit argument. For further information see Chapter 34 and Varshalovich et al. (1988, §§8.2.5, 8.8, and 9.2.3).