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 (...

functions, the Pochhammer k-symbol and the k-gamma function, introduced by Rafael Díaz and Eddy Pariguan are generalizations of the Pochhammer symbol and gamma...

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

In mathematics, the generalized Pochhammer symbol of parameter α > 0 {\displaystyle \alpha >0} and partition κ = ( κ 1 , κ 2 , … , κ m ) {\displaystyle...

be encountered as representing the rising factorial, also called "Pochhammer symbol". Another notation for the same is x ( n ) {\displaystyle x^{(n)}}...

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

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

in expansions of falling and rising factorials (also known as the Pochhammer symbol) as polynomials. That is, the falling factorial, defined as   ( x...

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 forms, and...

{k+1}}}}} where s k + 1 ¯ {\displaystyle s^{\overline {k+1}}} is the Pochhammer symbol. An alternative expansion is γ ( s , z ) = ∑ k = 0 ∞ ( − 1 ) k k ...

{(-x)^{k}}{k!}};|x|<\Re (a),} where ( s ) k {\displaystyle (s)_{k}} is the Pochhammer symbol. Series at a = −n is given by Φ ( z , s , a ) = ∑ k = 0 n z k ( a...

&b_{q}\end{matrix}};z\right].} Using the rising factorial or Pochhammer symbol ( a ) 0 = 1 , ( a ) n = a ( a + 1 ) ( a + 2 ) ⋯ ( a + n − 1 ) , n...

{(r)_{k}}{k!}},} where ( ⋅ ) k {\displaystyle (\cdot )_{k}} is the Pochhammer symbol, here standing for a falling factorial. This agrees with the usual...

where ( a ; q ) ∞ {\displaystyle (a;q)_{\infty }} is the infinite q-Pochhammer symbol. It enjoys a particularly elegant form when expressed in terms of...

Moment (mathematics) Cumulant Factorial moment generating function The Pochhammer symbol (x)r is used especially in the theory of special functions, to denote...

{1-z}{2}}\right),} with ( λ ) n {\displaystyle (\lambda )_{n}} the (rising) Pochhammer symbol. The nonpolynomial solution for the special case of integer degree...

polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by P n ( x ; a , b ; q ) = 1 ( b − 1 q − n ; q ) n 2 ϕ 1 ( q − n ...

is a basic hypergeometric function, x = cos θ, and (,,,)n is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral values...

polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by p n ( x ; a | q ) = 2 ϕ 1 ( q − n , 0 ; a q ; q , q x ) = 1 ( a...

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

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

x^{(n)}=x(x+1)(x+2)\cdots (x+n-1)} is the rising factorial, also called the "Pochhammer symbol". The value of the characteristic function for t = 0, is one: φ X...

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 ; q ) n 1 ϕ 1 ( q −...

factorial, and the ( x ) n {\displaystyle (x)_{n}} notation is a Pochhammer symbol. Falling factorials count the number of different sequences of n {\displaystyle...

where ( q ; q ) ∞ {\displaystyle (q;q)_{\infty }} is the (infinite) q-Pochhammer symbol. The invertible matrix products on the right-hand-side of the previous...

exists 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 (...

polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by ： P n ( x ; a ∣ q ) = a − n e i n ϕ ( a 2 ; q ) n ( q ; q ) n 3...

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

i\tau /12}\eta (\tau ).} The Euler function may be expressed as a q-Pochhammer symbol: ϕ ( q ) = ( q ; q ) ∞ . {\displaystyle \phi (q)=(q;q)_{\infty }.}...

polynomials are given in terms of basic hypergeometric 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 −...