(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).

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

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.

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G. E. "The Fifth and Seventh Order Mock Theta Functions." Trans.
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G. E.; Askey, R.; and Roy, R. Special
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-Series."
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R. W. "Experiments and Discoveries in q-Trigonometry." In Symbolic
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The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its -Analogue. Delft, Netherlands: Technische Universiteit
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