The identity is a generalization of the so-called Fibonacci identity (where n=1) which is actually found in Diophantus' Arithmetica (III, 19).
This system remained in use through the European Middle Ages, being presented in slightly modified form by Luca Pacioli in his seminal Summa de Arithmetica (1494).
The term "hypergeometric series" was first used by John Wallis in his 1655 book Arithmetica Infinitorum.