Simultaneous use of general and special linear transformations allows 36 transformations for the Appel function F 4 to be obtained. This result is important because F 4 is widely used in physics and mathematics. The transformations express F 4 in terms of the Appel functions F 1, F 2, and F 3; the Horn function H 2; and non-Hornian series G, K, and Γ. Until now, it was not possible to obtain “linear multiplets” having such a large dimensionality. A program based on Maple V4 was developed for formula generation. Details of program operation and possible applications of the results obtained are discussed.