Factorialx

Factorialx is a generalized factorial function parameterized by a base value x. For a nonnegative integer n, factorialx(n) is the product x(x+1)(x+2)…(x+n−1). Equivalently, factorialx(n) = Γ(n+x)/Γ(x), where Γ denotes the gamma function. This construction is commonly known in mathematics as the rising factorial (x)^{(n)} or the Pochhammer symbol (x)_{n}, and factorialx is a name used in some contexts for that quantity when x is fixed and n varies.

For an integer n ≥ 0, factorialx(0) = 1 and factorialx(n+1) = (x+n)·factorialx(n). The gamma-function expression provides a natural extension to non-integer n and to complex values, making factorialx a flexible tool in analysis and special functions. The rising factorial is closely related to binomial coefficients and hypergeometric functions. In particular, binomial(n+x−1, n) = factorialx(n)/n!, and many identities involving (x)_{n} appear in series expansions and combinatorial identities.

Examples illustrate the concept: factorialx(5) with x = 3 equals 3·4·5·6·7 = 2520; factorialx(3) with x = 1.5 equals 1.5·2.5·3.5 = 13.125. While not universally adopted as a distinct notation, factorialx aligns with the rising factorial and the Pochhammer symbol, and it appears in contexts ranging from polynomial expansions to probability distributions and analytic combinatorics.