Feb 14, 2026
Fasten your seatbelt.
By Taylor’s theorem, every sufficiently well-behaved function on the real numbers has a power series representation f(x) = \sum_{n=0}^\infty f^{n}(0) x^n/n! around x = 0, where f^{n}(x) denotes the n-th derivative of f. Perhaps the most well-known is the exponential function with power series representation \exp(x) = \sum_{n=0}^\infty x^n/n!.
The generalized hypergeometic series is {}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q,z) = \sum_{n=0}^\infty \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n}\frac{x^n}{n!}, where (a)_n = a(a + 1)\cdots(a + n - 1) is the Pochhammer symbol.