In 1757, V. Riccati first recorded the generalizations of the hyperbolic functions defined by F_(n, r)^α(x) congruent sum_(k = 0)^∞ α^k/((n k + r)!) x^(n k + r), for r = 0, ..., n - 1, where α is complex, with the value at x = 0 defined by F_(n, 0)^α(0) = 1. This is called the α-hyperbolic function of order n of the rth kind.

hyperbolic functions | Mittag-Leffler function