ascending factorial

The Pochhammer symbol (x)_n | congruent | (Γ(x + n))/(Γ(x)) | = | x(x + 1)...(x + n - 1) (Abramowitz and Stegun 1972, p. 256; Spanier 1987; Koepf 1998, p. 5) for n>=0 is an unfortunate notation used in the theory of special functions for the rising factorial, also known as the rising factorial power or ascending Factorial. The Pochhammer symbol is implemented in the Wolfram Language as Pochhammer[x, n]. In combinatorics, the notation x^(n), 〈x〉_n, or x^(n^_) is used for the rising factorial, while (x)_n or x^(n__) denotes the falling factorial . Extreme caution is therefore needed in interpreting the notations (x)_n and x^(n).

factorial | falling factorial | generalized hypergeometric function | Hankel's symbol | harmonic logarithm | hypergeometric function | Kramp's symbol | rising factorial