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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.

98 questions

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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} ... user avatar Lightcone

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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-\... user avatar Denis Cousineau

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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-... user avatar Marcel

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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 ... user avatar user1062

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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 ... user avatar Aaron Hendrickson

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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. ... user avatar Marcel

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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 ... user avatar Meto ballaes

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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 user avatar user1062

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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 ... user avatar Brandon Epstein

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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}\... user avatar Ricardo770

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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 ... user avatar Luna145

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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!}\,, \... user avatar horropie

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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_\... user avatar epsilonz3ro

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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 ... user avatar Sebastian

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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,... user avatar Luna145

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