q-Pochhammer Symbol

The q-analog of the Pochhammer symbol defined by

(1)

(Koepf 1998, p. 25). -Pochhammer symbols are frequently called q-series and, for brevity, is often simply written . Note that this contention 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.

Min Max

Min Max Re Im

Letting gives the special case , sometimes known as "the" Euler function and defined by

			(2)
			(3)

This function is closely related to the pentagonal number theorem and other related and beautiful sum/product identities. As mentioned above, it is implemented in Mathematica as QPochhammer[q]. As can be seen in the plot above, along the real axis, reaches a maximum value (OEIS A143440) at value (OEIS A143441).

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

			(6)
			(7)

where is a Jacobi theta function (and in the latter case, care must be taken with the definition of the principal value the cube root).

Asymptotic results for -Pochhammer symbols include

			(8)
			(9)
			(10)

for (Watson 1936, Gordon and McIntosh 2000).

For ,

(11)

gives the normal Pochhammer symbol (Koekoek and Swarttouw 1998, p. 7). The -Pochhammer symbols are also called -shifted factorials (Koekoek and Swarttouw 1998, pp. 8-9).

The -Pochhammer symbol satisfies

(12)

(13)

(14)

(15)

(16)

(17)

(here, is a binomial coefficient so ), as well as 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 -bracket

(19)

and -binomial

(20)

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