Appell’s Function F₄ as a Double Average

A quadratic transformation of a double hypergeometric series of order two (Appell’s F₄ with equal denominator parameters) into a series of order three revives longstanding doubts about the accepted classification by order. Both series can be represented as double Dirichlet averages of xᵗ. Appell’s F₄ with unrestricted parameters can be represented as such an average with two rows and three columns. There are six cases in which a restriction on the parameters reduces the number of columns to two.