Questions tagged [pochhammer-symbol]

The Pochhammer symbol is the notation used for rising and falling factorials. The $q$-Pochhammer symbol is the $q$-analog.

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What is the limit of the series (summation) of the q-Pochhammer symbol or the ~q-Pochhammer symbol?

I am interested in knowing if the following series converges or not: $$\sum_{n=1}^{\infty} \prod_{i=1}^n \left(1-e^{-\sqrt{i}} \right) \qquad Expression \; 1$$ If that is ...
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Are there other definitions of Jacobi polynomials?

While I am reading "On some dual integral equations occurring in potential problems with axial symmetry" by C. J. Tranter, Quarterly Journal of Mechanics & Applied Math (1950) p. 414, the author ...
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Is $(7,4)$ the only non-trivial integer solution for $(n)_k=n!$?

I accidentally noticed that: $$(7)_4=7 \cdot 8 \cdot 9 \cdot 10=2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7=7!$$ Here $(n)_k$ is the Pochhammer symbol. I wonder, are there any other non-...
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Identity using q-Pochhammer symbols

Prove - $$∑_{n=0}^{∞} \frac{(a;q)_n}{(q;q)_n} q^{n\choose 2} q^n={(−q;q)_∞}{(aq;q^2)_∞}.$$ where $(a;q)$ are the q-Pochhammer symbols. I know that the RHS is the product of generating functions of ...
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Falling Factorial Notation

If $(x)_{n}$ refers to $$x(x-1)\cdots(x-n+1)$$ then what does $(xy;x)_{n}$ refer to? Is it $$xy(xy-1)\ldots(xy-n+1)?$$ Thanks. The notation in question is used on page two of this paper.
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Product identity for $n^n$

I came across the rather nice identity \begin{align} &&\frac{(-n)^{n-1} \Gamma (n+1)}{(1-n)_{n-1}}&&\tag{1}&\\ \\ &=&\prod _{k=1}^{n-1} \frac{(k+1) n^2}{n^2-k n}&&\...
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Mathematical expressions for binomial coefficient and Pochhammer’s Symbol with negative values

I have two questions regarding the binomial coefficient and Pochhammer’s Symbol when they contain negative value; In the following example $\sum\limits_{k=0}^{-n} \binom{-n}{k} \left(a\right)_{-n}$. ...
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Integral of binomial coefficients

Let the integral in question be given by \begin{align} f_{n}(x) = \int_{1}^{x} \binom{t-1}{n} \, dt. \end{align} The integral can also be seen in the form \begin{align} f_{n}(x) = \frac{1}{n!} \, \...
I am trying to come up with an expression for $(x+y)^{\underline{n}}$ in terms of $x^{\underline{r}}$ and $y^{\underline{r}}$. I tried for $n=2$ and $n=3$ and it looks like binomial expansion holds, ...