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The Pochhammer symbol is notation used for both rising and falling factorials, e.g. in defining basic hypergeometric series and related special functions. This tag is also appropriate for questions about the $q$-Pochhammer symbol, which plays a similar role in defining $q$-hypergeometric series, etc.
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Asymptotic of inverse q-Pochhammer symbol
Let $(x,q)_k$ denote the $q$-Pochhammer symbol and $\mathrm{Coeff}_n\hspace{0.1em} f(q)$ denote the coefficient of $q^n$ in $f(q)$. What I want to know is the large-$n$ asymptotic of \begin{align} ... asymptotics pochhammer-symbol- 111
Solving an infinite sum of incomplete gamma with integer parameter
In trying to simplify a distribution function, I stumbled upon this infinite sum involving an (upper) Gamma function. I would believe it can be simplified further, but can't find how. $$1- \frac{(1-\... sequences-and-series gamma-function pochhammer-symbol- 149
Simplify multiple sum involving rising factorials
In the course of a calculation, I arrived at the quantity $$ f(x,y,a,b)= \sum_{n,m,i,r,q,l\ge 0}\sum_{k=0}^{n+m} K_{n,m,i,r,l,q,k}\frac{(x)^{(i)}(y)^{(r)}(x)_{(l)}(y)_{(q)}}{(x+y+1)^{(n+m+r+i-k)}(x+y-... sequences-and-series summation pochhammer-symbol- 1,523
The special case of Pochhammer Symbol at Zero?
I am interested in a property of Pochhammer Symbol. So I need an information about it. Let $a^{\bar{n}}$ Pochhammer symbol or rising factorial. As you know in the literature $a^{\bar{0}}=1.$ I ... reference-request soft-question special-functions hypergeometric-function pochhammer-symbol- 411
Evaluating $\sum_{n=0}^\infty\frac{(1/2)_n}{n!}(H_n-H_{n-1/2})$
I am seeking a closed-form (a form in terms of known special functions) to the sum $$ \sum_{n=0}^\infty\frac{(1/2)_n}{n!}(H_n-H_{n-1/2}). $$ Context: I am searching for closed-forms to special cases ... sequences-and-series special-functions hypergeometric-function harmonic-numbers pochhammer-symbol- 4,081
Identity involving double sum with factorials
In the course of a calculation, I have met a complicated identity, which I want to prove. Let $m>0$ and $0<\ell<m$ be integers. Let $(x)^{(n)}=x(x+1)\cdots (x+n-1)$ be the rising factorial. ... summation binomial-coefficients factorial pochhammer-symbol- 1,523
Prove that: $r^{\underline{k}}\:\cdot \:\left(r-\frac{1}{2}\right)^{\underline{k}}\:=\:\frac{\left(2r\right)^{\underline{2k}}}{2^{2k}}$
I have to prove for $r \in \mathbb R$ and $k \in \mathbb N$, that: $r^{\underline{k}}\:\cdot \:\left(r-\frac{1}{2}\right)^{\underline{k}}\:=\:\frac{\left(2r\right)^{\underline{2k}}}{2^{2k}}$ I tried ... combinatorics factorial pochhammer-symbol- 59
The Product of two rising factorials
Let $(x)_{2 n}$ is a rising factorial of 2n terms ,$\left(\frac{x}{2}\right)_{n}$ and $\left(\frac{1+x}{2}\right)_{n}$ are rising factorial of n terms. From special-functions factorial pochhammer-symbol- 411
q-Pochhammer summation identity
Due to a long-term interest in the theory of modular forms, I am reading through Cohen and Strömberg's Modular Forms: A Classical Approach and I came across an exercise that asked me to prove a ... combinatorics power-series modular-forms q-series pochhammer-symbol- 1
Proof the equality $\prod_{r=1}^{mn}\left(x+mn -r\right)=\prod_{k=1}^{m}\prod_{l=1}^{n}\left(x+mn-(1+ml-k) \right)$
I came across a proof of Gauss multiplication formula for the Gamma function which relies on the following indentity (without a proof) $$\frac{\Gamma \left(x+mn \right)}{\Gamma \left(x \right)}=m^{mn}\... combinatorics induction special-functions gamma-function pochhammer-symbol- 2,159
On the decimal expansion of $\prod_{k=1}^{\infty} \sum_{\ell =1}^{k} \frac{9}{10^\ell}$
Consider the following expression: $$\prod_{k=1}^{\infty} \sum_{\ell =1}^{k} \frac{9}{10^\ell} = 0.9\times0.99\times0.999\times0.9999\times...$$ I have a curiosity about the decimal expansion of this ... decimal-expansion pochhammer-symbol- 1,902
Looking for a Hypergeometric Function Related to Appell Series
The Appell-Series $F_1$ is given by \begin{equation} F_1[a;b_1, b_2; c;x,y] = \sum_{m = 0}^\infty \sum_{n = 0}^\infty \frac{ (a)_{m+n} (b_1)_{m} (b_2)_n}{(c)_{m+n}} \frac{x^m}{m!} \frac{y^n}{n!}\,, \... sequences-and-series hypergeometric-function pochhammer-symbol- 139
Convergence of asymptotic series for $z^k/(z)_k$, $z\to\infty$
Let $z>0$, $k\in\Bbb Z$, and $(s)_n=\Gamma(s+n)/\Gamma(s)$ denote the Pochhammer symbol. According to DLMF 5.11.13 as $z\to\infty$: $$ \frac{z^k}{(z)_k}\sim\sum_{\ell=0}^\infty\binom{-k}{\ell}B_\... sequences-and-series convergence-divergence asymptotics gamma-function pochhammer-symbol- 152
Asymptotic expansion of q-Pochhammer symbol near q = 1
I'd like to understand the asymptotics of the q-Pochhammer symbol $(a;q)_\infty$ as $q \to 1^-$ with $a$ complex, where $$(a;q)_\infty = \prod_{n = 0}^\infty (1- aq^n).$$ More specifically, I'm ... complex-analysis asymptotics special-functions infinite-product pochhammer-symbol- 518
Integrating the Pochhammer symbol. $\int_{-1}^{1} \prod_{j=0}^{\infty} (1-x^{j+1})$
I had a question regarding the Pochhammer symbol, specifically integrating $[x;x]_{\infty} = \prod_{j=0}^{\infty} (1-x^{j+1})$. Firstly, is there a closed form for the $n^{th}$ partial product of this,... pochhammer-symbol- 1,902
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