Pochhammerin

Pochhammerin is not a standard term in mathematics. In most literature the related term Pochhammer or Pochhammer symbol is used to denote the rising factorial. The form “Pochhammerin” may appear as a misspelling, a possessive-style rendering in some languages, or an informal designation without a formal definition.

The Pochhammer symbol, often written as (a)_n, represents the rising factorial. For a complex number a and a nonnegative integer n, (a)_n is defined by (a)_0 = 1 and (a)_n = a(a+1)(a+2)…(a+n−1) for n ≥ 1. The expression (a)_n is also called the rising factorial, and it can be notated in other ways, such as (a)^{\overline{n}}. A key relation is (a)_n = Γ(a+n)/Γ(a), linking it to the gamma function.

Several properties follow from this definition. The sequence satisfies the recurrence (a)_{n+1} = (a+n)(a)_n. It extends to complex a by analytic continuation and is connected to binomial coefficients and to the theory of hypergeometric functions. The complementary concept is the falling factorial a^{\underline{n}} = a(a−1)…(a−n+1), which is distinct from the rising factorial but related in combinatorial contexts.

Applications of the rising factorial appear in series expansions, notably in hypergeometric functions such as 2F1, and in combinatorial identities. It also underpins generalized binomial theorems and appears in probability and statistics in certain distribution formulas.

Etymology: the symbol is named after Leopold Pochhammer, a 19th-century German mathematician. In German texts the term Pochhammerzeichen is commonly used, while “Pochhammerin” is rarely encountered and not a standard designation. See also Pochhammer symbol, rising factorial, gamma function, and hypergeometric function.