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...

{\displaystyle n=0} . These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation ( x...

and introduced the Pochhammer symbol, now generally used for expressing hypergeometric functions in a compact notation. Pochhammer was born in Stendal...

{x}{2}}} if | q | > 1 {\displaystyle |q|>1} . Here ( ⋅ ; ⋅ ) ∞ {\displaystyle (\cdot ;\cdot )_{\infty }} is the infinite q-Pochhammer symbol. The q {\displaystyle...

where ( ⋅ ⋅ ) ∞ {\displaystyle (\cdot \cdot )_{\infty }} is the q-Pochhammer symbol. elliptic hypergeometric series Jacobi theta function Ramanujan theta...

at x = 0. Here, ( q ; q ) n {\displaystyle (q;q)_{n}} is the q-Pochhammer symbol, and [ n ] ! q {\displaystyle [n]!_{q}} is the q-factorial. If f ( x...

computers Pentax Q series – cameras In mathematics q-Pochhammer symbol q-series Hypergeometric q-series P series (disambiguation) R series (disambiguation)...

expansions of these J-fractions are defined in terms of the q-Pochhammer symbol, Pochhammer symbol, and the binomial coefficients. The radii of convergence...

q-derivative q-difference polynomial Quantum calculus LLT polynomial q-binomial coefficient q-Pochhammer symbol q-Vandermonde identity q-Bessel polynomials q-Charlier...

c , d ∣ q ) {\displaystyle p_{n}(\cos {\theta })=p_{n}(\cos {\theta };a,b,c,d\mid q)} and using the definition of the q-Pochhammer symbol p n ( cos...

(1-q)} is the q-Pochhammer symbol. That this is the q-analog of the exponential follows from the property ( d d z ) q e q ( z ) = e q ( z ) {\displaystyle...

taking the limit as q → 1 {\displaystyle q\rightarrow 1} . The q-factorial also has a concise definition in terms of the q-Pochhammer symbol, a basic building-block...

q ) n = ( 1 − q n ) ( 1 − q n − 1 ) ⋯ ( 1 − q ) {\displaystyle (q;q)_{n}=(1-q^{n})(1-q^{n-1})\cdots (1-q)} is the q-Pochhammer symbol. The function A...

of q-Pochhammer symbols: ∑ n = − ∞ ∞ q n ( n + 1 ) 2 z n = ( q ; q ) ∞ ( − 1 z ; q ) ∞ ( − z q ; q ) ∞ , {\displaystyle \sum _{n=-\infty }^{\infty }q^{\frac...

formulation of Euler's generating function is a special case of a q {\displaystyle q} -Pochhammer symbol and is similar to the product formulation of many modular...

; q ) n {\displaystyle (a;q)_{n}} denotes the q-Pochhammer symbol. Identities that follow from this include φ ( q ) = f ( q , q ) = ∑ n = − ∞ ∞ q n 2...

and the q-Pochhammer symbol by p n ( x ; a | q ) = 2 ϕ 1 ( q − n , 0 ; a q ; q , q x ) = 1 ( a − 1 q − n ; q ) n 2 ϕ 0 ( q − n , x − 1 ; ; q , x / a )...

q-Pochhammer symbol: Γ ( z ; 0 , q ) = 1 ( z ; q ) ∞ . {\displaystyle \Gamma (z;0,q)={\frac {1}{(z;q)_{\infty }}}.} Define Γ ~ ( z ; p , q ) := ( q ;...

the q-Pochhammer symbol by P n ( x ; a , b ; q ) = 1 ( b − 1 q − n ; q ) n 2 ϕ 1 ( q − n , a q x − 1 ; a q ; q , x b ) {\displaystyle P_{n}(x;a,b;q)={\frac...

functions and the q-Pochhammer symbol by p n ( x ; a , b , c ∣ q ) = ( a b , a c ; q ) n a n 3 ϕ 2 ( q − n , a e i θ , a e − i θ ; a b , a c ∣ q ; q ) {\displaystyle...

The q-Laguerre polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by L n ( α ) ( x ; q ) = ( q α + 1 ; q ) n...

q-Bessel function is the same as the Hahn–Exton q-Bessel function. The three Jackson q-Bessel functions are given in terms of the q-Pochhammer symbol...

q-Pochhammer symbol by ： P n ( x ; a ∣ q ) = a − n e i n ϕ ( a 2 ; q ) n ( q ; q ) n 3 ϕ 2 ( q − n , a e i ( θ + 2 ϕ ) , a e − i θ ; a 2 , 0 ∣ q ; q )...

}\\&=\left(q^{2};q^{2}\right)_{\infty }\,\theta \left(-w^{2}q;q^{2}\right)\end{aligned}}} where (  ;  )∞ is the q-Pochhammer symbol and θ(  ;  ) is the q-theta...

h_{n}(x;q)=\sum _{k=0}^{n}{\frac {(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}}}x^{k}} where (q;q)n is the descending q-Pochhammer symbol. Furthermore, the h n ( x ; q )...

and the q-Pochhammer symbol by P n ( α , β ) ( x ; q ) = ( q n + 1 ; q ) n ( q ; q ) n 4 ϕ 3 [ q − n , q n + α + β + 1 , q 1 2 α + 1 4 e i θ , q 1 2 α +...

q-Pochhammer symbol by 。 P n ( α ) ( x | q ) = ( q α + 1 ; q ) n ( q ; q ) n {\displaystyle P_{n}^{(\alpha )}(x|q)={\frac {(q^{\alpha +1};q)_{n}}{(q;q)_{n}}}}...

Euler function may be expressed as a q-Pochhammer symbol: ϕ ( q ) = ( q ; q ) ∞ . {\displaystyle \phi (q)=(q;q)_{\infty }.} The logarithm of the Euler...

q-Pochhammer symbol and the basic hypergeometric series by C n ( x ; β | q ) = ( β ; q ) n ( q ; q ) n e i n θ 2 ϕ 1 ( q − n , β ; β − 1 q 1 − n ; q ...

integral. A q-analogue of Dixon's formula for the basic hypergeometric series in terms of the q-Pochhammer symbol is given by 4 φ 3 [ a − q a 1 / 2 b c...