Specialization of Appellʹs functions to univariate hypergeometric functions

Univariate specializations of Appellʹs hypergeometric functions F 1 , F 2 , F 3 , F 4 satisfy ordinary Fuchsian equations of order at most 4. In special cases, these differential equations are of order 2 and could be simple (pullback) transformations of Eulerʹs differential equation for the Gauss hypergeometric function. The paper classifies these cases, and presents corresponding relations between univariate specializations of Appellʹs functions and univariate hypergeometric functions. The computational aspect and interesting identities are discussed.