# §16.24 Physical Applications

## §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. 147–148).

## §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) $\mathit{3j}$, $\mathit{6j}$, and $\mathit{9j}$ Symbols

The $\mathit{3j}$ 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 $\mathop{{{}_{3}F_{2}}\/}\nolimits$ functions with unit argument. The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner $\mathit{6j}$ symbols. These are balanced $\mathop{{{}_{4}F_{3}}\/}\nolimits$ functions with unit argument. Lastly, special cases of the $\mathit{9j}$ symbols are $\mathop{{{}_{5}F_{4}}\/}\nolimits$ functions with unit argument. For further information see Chapter 34 and Varshalovich et al. (1988, §§8.2.5, 8.8, and 9.2.3).