q-Pochhammer Symbol

The -Pochhammer symbol, also called a -shifted factorial (Koekoek and Swarttouw 1998, pp. 8-9), is defined by

(1)

(Koepf 1998, p. 25), where and are complex numbers and is an integer or . The infinite product is used for ; for negative , and the denominator factors must be nonzero. For brevity, is often simply written . This convention has the slightly curious side-effect that the argument is not taken literally, so for example means , not (cf. Andrews 1986b).

The -Pochhammer symbol is implemented in the Wolfram Language as QPochhammer[a, q, n], with the special cases and represented as QPochhammer[a, q] and QPochhammer[q], respectively.

The special case , sometimes known as "the" Euler function , is defined by

			(2)
			(3)

This function is closely related to the pentagonal number theorem and other related and beautiful sum/product identities. As can be seen in the plot above, on the real interval , reaches a maximum value (OEIS A143440) at value (OEIS A143441).

For a nonnegative integer , the finite -Pochhammer symbol is given by the sum

(4)

where is a q-binomial coefficient (Koekoek and Swarttouw 1998, p. 11).

It is closely related to the Dedekind eta function,

(5)

where is the half-period ratio with , is the square of the nome, and means (Berndt 1994, p. 139). Other representations for in terms of special functions include

			(6)
			(7)

where is a Jacobi theta function and denotes its derivative with respect to the first argument, evaluated at . For complex , care must be taken with the definition of the principal value of the cube root.

Asymptotic results for -Pochhammer symbols include

			(8)
			(9)
			(10)

for as (Watson 1936, Gordon and McIntosh 2000).

For a fixed nonnegative integer ,

(11)

gives the ordinary Pochhammer symbol (Koekoek and Swarttouw 1998, p. 7). This normalized limit gives the sense in which the -Pochhammer symbol is a q-analog of the ordinary Pochhammer symbol. With held fixed instead, the unscaled limit is .

For a nonnegative integer , the -Pochhammer symbol satisfies the following identities wherever the denominators are nonzero, with for the infinite products and in formulas containing their reciprocals:

(12)

(13)

(14)

(15)

(16)

(17)

where is a binomial coefficient so . It also satisfies many other identities, some of which are given by Koekoek and Swarttouw (1998, p. 9).

A generalized -Pochhammer symbol can be defined using the concise notation

(18)

(Gordon and McIntosh 2000).

The q-bracket

(19)

and q-factorial

(20)

symbols are sometimes also used when discussing q-series, where is a nonnegative integer and is a q-binomial coefficient.