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

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Abstract
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.