Two-Variable Expansion of the Scattering Amplitude: An Application of Appell's Generalized Hypergeometric Functions

Appell's polynomials in two variables orthogonal in a triangle are described and some of their properties and those of related generalized hypergeometric functions are given. An application to the expansion of the scattering amplitude is suggested, the equal-mass case being discussed in some detail. A simple crossing matrix is derived. Difficulties introduced by inequality of the particle masses are explained. A Neumann formula is presented which permits an analytic continuation in the parameters to be made of the expansion coefficients for parts of the amplitude: This is in analogy with the Froissart-Gribov continuation. A conjectured analog of the Sommerfeld-Watson transformation then suggests the existence of fixed cuts in the partial-wave scattering amplitude.