General Overview of Multivariable Special Functions

The theory of one-variable (ordinary) hypergeometric and basic hypergeometric series goes back to work of Euler, Gauss and Jacobi. The theory of elliptic hypergeometric series is of a much more recent vintage [20]. The three theories deal with the study of series ∑ k≥0 ck with f (k) := ck+1/ck a rational function in k (hypergeometric theory), a rational function in qk (basic hypergeometric theory), or a doubly periodic meromorphic function in k (elliptic hypergeometric theory, see [21, Ch. 11] for an overview). Examples of elementary functions admitting hypergeometric and basic hypergeometric series representations are