q-Pochhammer symbol(qポッホハマー記号;q-類似の数式に頻出する乗積を略記する記号, , )
←,< https://shinichiwanko2000.livedoor.blog/archives/28316953.html >2025年04月20日 /wiki/Ramanujan_congruences{exceptional(be treated as a special case? ) reseach!? 別格とされている研究！・？・ }
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 ) ,
with ( a ; q ) _0 = 1. It is a *q-analog< https://en.wikipedia.org/wiki/Q-analog > of the Pochhammer symbol ( x ) _n = x ( x + 1 ) … ( x + n − 1 ) , in the sense that
lim [q → 1] ( 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役割をはたす(演じる? )? 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 ) .
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 )
is known as Euler's function, and is important in combinatorics, number theory, and the theory of modular forms.
< https://en.wikipedia.org/wiki/Q-Pochhammer_symbol >
{TVﾃﾚﾋﾞのage, period, epoch時代?!に、used well使われていたコトmay beかな？・！？・ ・ }
↓
https://shinichiwanko2000.livedoor.blog/archives/28562584.html
2025年05月21日 /wiki/q-Pochhammer_symbol
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 ) ,
with ( a ; q ) _0 = 1. It is a *q-analog< https://en.wikipedia.org/wiki/Q-analog > of the Pochhammer symbol ( x ) _n = x ( x + 1 ) … ( x + n − 1 ) , in the sense that
lim [q → 1] ( 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役割をはたす(演じる? )? 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 ) .
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 )
is known as Euler's function, and is important in combinatorics, number theory, and the theory of modular forms.
< https://en.wikipedia.org/wiki/Q-Pochhammer_symbol >
{TVﾃﾚﾋﾞのage, period, epoch時代?!に、used well使われていたコトmay beかな？・！？・ ・ }
↓
https://shinichiwanko2000.livedoor.blog/archives/28562584.html
2025年05月21日 /wiki/q-Pochhammer_symbol
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