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Q-Pochhammer symbol — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Q-Pochhammer_symbol
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-07T17:23:12.441Z
SHA-256 of the archived text: 58179903ec69c7277f6605234de49f4ed99ae036fbb8fcab845b7ab152f0ba8c
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Q-Pochhammer symbol — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Q-Pochhammer_symbol (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_bd213628ce6b (Metraipolis archive dc_bd213628ce6b, sha256 58179903ec69c7277f6605234de49f4ed99ae036fbb8fcab845b7ab152f0ba8c)

BibTeX

@misc{dcbd213628ce6b,
  title = {Q-Pochhammer symbol — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Q-Pochhammer_symbol},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_bd213628ce6b; sha256 58179903ec69c7277f6605234de49f4ed99ae036fbb8fcab845b7ab152f0ba8c},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Q-Pochhammer_symbol. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In the mathematical field of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product ( a ; q ) n = ∏ k = 0 n − 1 ( 1 − a q k ) = ( 1 − a ) ( 1 − a q ) ( 1 − a q 2 ) ⋯ ( 1 − a q n − 1 ) , {\displaystyle (a;q)_{n}=\prod _{k=0}^{n-1}(1-aq^{k})=(1-a)(1-aq)(1-aq^{2})\cdots (1-aq^{n-1}),} with ( a ; q ) 0 = 1. {\displaystyle (a;q)_{0}=1.} It is a q-analog of the Pochhammer symbol x ( n ) = x ( x + 1 ) … ( x + n − 1 ) {\displaystyle x^{(n)}=x(x+1)\dots (x+n-1)} , in the sense that lim q → 1 ( q x ; q ) n ( 1 − q ) n = x ( n ) . {\displaystyle \lim _{q\to 1}{\frac {(q^{x};q)_{n}}{(1-q)^{n}}}=x^{(n)}.} The q-Pochhammer symbol is a major building block in the construction of q-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series. Unlike the ordinary Pochhammer symbol, the q-Pochhammer symbol can be extended to an infinite product: ( a ; q ) ∞ = ∏ k = 0 ∞ ( 1 − a q k ) . {\displaystyle (a;q)_{\infty }=\prod _{k=0}^{\infty }(1-aq^{k}).} This is an analytic function of q in the interior of the unit disk, and can also be considered as a formal power series in q. The special case ϕ ( q ) = ( q ; q ) ∞ = ∏ k = 1 ∞ ( 1 − q k ) {\displaystyle \phi (q)=(q;q)_{\infty }=\prod _{k=1}^{\infty }(1-q^{k})} is known as Euler's function, and is important in combinatorics, number theory, and the theory of modular forms.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; categoria «Number theory»).
Fonte: Wikipedia in inglese, voce «Q-Pochhammer symbol» — https://en.wikipedia.org/wiki/Q-Pochhammer_symbol
Wikidata Q518706 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q518706
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Q-Pochhammer_symbol?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1360135832, consultata il 2026-10-07T16:54:19.904Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 92a118327cfce919d270de15227b26e2f6406459cc9d94f3da4c4105d58388cc