Search results

\int_a^bp_m(x)p_n(x)w(x)\,dx=0,\quad m\neq n <div align="right">Constraint(s): <math>{\displaystyle m,n\in\{0,1,2,\ldots\}}</math></div><br /> ...

[0]=0,\quad [n]=\frac{1-q^n}{1-q}=\sum_{k=0}^{n-1}q^k \qPochhammer{a}{q}{0}:=1\quad\textrm{and}\quad \qPochhammer{a}{q}{k}:=\prod_{n=1}^k(1-aq^{n-1}), ...

{ \LerchPhi@{z}{s}{a} = \sum_{n=0}^\infty \frac{z^n}{(a+n)^s} } <div align="right">Constraint(s): <math>{\displaystyle a \neq 0,-1,-2,\dots, |z| < 1}</math> &<br /> <math>{\displaystyle \realpart{s} > 1, ...

...j), j = 0..n)=(1 - z)^(n)</code> || <code>Sum[Binomial[n,j]*(- z)^(j), {j, 0, n}]=(1 - z)^(n)</code> || Successful || Successful || - || - ...

\qexpKLS{q}@{z}:=\qHyperrphis{1}{0}@@{0}{-}{q}{z}=\sum_{n=0}^{\infty}\frac{z^n}{\qPochhammer{q}{q}{n}} =\frac{1}{\qPochhammer{z}{q}{\infty}},\quad 0<|q|<1 ...

\qHyperrphis{1}{0}@@{a}{-}{q}{z}=\sum_{n=0}^{\infty}\frac{\qPochhammer{a}{q}{n}}{\qPochhammer{q}{q}{n}}z^n= \frac{\qPochhammer{az}{q}{\infty}}{\qPochhammer{z}{q}{\infty}},\quad 0<|q|<1 ...

\sum\limits_{k=0}^{\infty}\frac{\pochhammer{a_1,\ldots,a_r}{k}}{\pochhammer{b_1,\ldots,b_s}{ \displaystyle 0 & \quad\textrm{if}\quad r > s+1.\end{array}\right. ...

...lign="right">Constraint(s): <math>{\displaystyle z \in \Complex \setminus [0,\infty)}</math></div><br /> = m \sum_{k=0}^{m-1} \Dilogarithm@{z \expe^{2 \cpi \iunit k/m}} ...

\pochhammer{a}{0}:=1\quad\textrm{and}\quad \pochhammer{a}{k}:=\prod_{n=1}^k(a+n-1),\quad k=1 <div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br /> ...

\HurwitzZeta@{s}{a} = \sum_{n=0}^\infty \frac{1}{(n+a)^s} ...\displaystyle \realpart{s} > 1}</math> &<br /> <math>{\displaystyle a \neq 0,-1,-2,\dots}</math></div><br /> ...

\displaystyle \frac{f(z)-f(qz)}{(1-q)z}, & z\neq 0\\[5mm] f'(0), & z=0.\end{array}\right. ...

...\displaystyle x}</math> is an integer, <math>{\displaystyle \realpart{s} > 0}</math> otherwise</div><br /> <div align="right">Constraint(s): <math>{\displaystyle 0 < x < 1}</math> &<br /> <math>{\displaystyle \realpart{s} > 1}</math></div> ...

<div align="right">Constraint(s): <math>{\displaystyle s \neq 0,1}</math></div><br /> \sum_{m=0}^k \sum_{r=0}^m ...

\sum_{k \hiderel{=} 0}^\infty <div align="right">Constraint(s): <math>{\displaystyle s \neq 1,0,-1,-2,\dots}</math></div><br /> ...

\RiemannZeta@{0} = - \frac{1}{2} \RiemannZeta@{-2n} = 0 ...

<div align="right">Constraint(s): <math>{\displaystyle \realpart{s} > 0}</math> &<br /> <math>{\displaystyle s \neq 1}</math></div><br /> <div align="right">Constraint(s): <math>{\displaystyle \realpart{s} > 0}</math> &<br /> <math>{\displaystyle s \neq 1}</math></div><br /> ...

\DirichletL@{-2n}{\chi} \hiderel{=} 0 \text{\quad if\quad} \chi(-1) \hiderel{=} 1 <div align="right">Constraint(s): <math>{\displaystyle n = 0,1,2,\dots}</math></div><br /> ...

...rt{n}@{x}{q}=\frac{1}{\qPochhammer{q}{q}{n}}\,\qHyperrphis{1}{1}@@{q^{-n}}{0}{q}{-q^{n+1}x} \int_{0}^{\infty}\frac{\StieltjesWigert{m}@{x}{q}\StieltjesWigert{n}@{x}{q}}{\qPoch ...

\HyperpFq{1}{0}@@{a}{-}{z}=\sum_{n=0}^{\infty}\frac{\pochhammer{a}{n}}{n!}z^n=(1-z)^{-a} \HyperpFq{1}{0}@@{-n}{-}{z}=\sum_{k=0}^n\frac{\pochhammer{-n}{k}}{k!}z^k ...

\qBesselPoly{n}@{x}{a}{q}=\qHyperrphis{2}{1}@@{q^{-n},-aq^n}{0}{q}{qx} ...n}@{x}{a}{q}=\left(-aq^nx\right)^n\cdot\qHyperrphis{2}{1}@@{q^{-n},x^{-1}}{0}{q}{-\frac{q^{-n+1}}{a}} ...