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\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}} ...

\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}} ...

<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots,N}</math></div><br /> \sum_{x=0}^N\binomial{N}{x}p^x(1-p)^{N-x} \Krawtchouk{m}@{x}{p}{N}\Krawtchouk{n}@{x}{ ...

\littleqLaguerre{n}@{x}{a}{q}=\qHyperrphis{2}{1}@@{q^{-n},0}{aq}{q}{qx} ...e{n}@{x}{a}{q}=\frac{1}{\qPochhammer{a^{-1}q^{-n}}{q}{n}}\,\qHyperrphis{2}{0}@@{q^{-n},x^{-1}}{-}{q}{\frac{x}{a}} ...

...}{n} = 0</math>]] || <code>binomial(m,n)= 0</code> || <code>Binomial[m,n]= 0</code> || Failure || Failure || Skip || Successful ...^(n), n = 0..m)=(1 + x)^(m)</code> || <code>Sum[Binomial[m,n]*(x)^(n), {n, 0, m}]=(1 + x)^(m)</code> || Successful || Successful || - || - ...

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

<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots,N}</math></div><br /> \sum_{x=0}^N\binomial{\alpha +x}{x}\binomial{\beta+N-x}{N-x}\Hahn{m}@{x}{\alpha}{\bet ...

\RiemannZeta@{s} = \frac{1}{1 - 2^{-s}} \sum_{n=0}^\infty \frac{1}{(2n+1)^s} <div align="right">Constraint(s): <math>{\displaystyle \realpart{s} > 0}</math></div><br /> ...

<div align="right">Constraint(s): <math>{\displaystyle 0<q<1}</math></div><br /> <div align="right">Constraint(s): <math>{\displaystyle k=0,1,2,\ldots,n}</math></div><br /> ...

...houk{n}@{q^{-x}}{p}{N}{q}=\qHyperrphis{3}{2}@@{q^{-n},q^{-x},-pq^n}{q^{-N},0}{q}{q} \sum_{x=0}^N\frac{\qPochhammer{q^{-N}}{q}{x}}{\qPochhammer{q}{q}{x}}(-p)^{-x}\qKrawtc ...

...houk{n}@{q^{-x}}{p}{N}{q}=\qHyperrphis{3}{2}@@{q^{-n},q^{-x},-pq^n}{q^{-N},0}{q}{q} \sum_{x=0}^N\frac{\qPochhammer{q^{-N}}{q}{x}}{\qPochhammer{q}{q}{x}}(-p)^{-x}\qKrawtc ...

\ctsqHermite{n}@{x}{q}=\expe^{\iunit n\theta}\,\qHyperrphis{2}{0}@@{q^{-n},0}{-}{q}{q^n\expe^{-2\iunit\theta}} <math>{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) ...

\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\qHyperrphis{3}{2}@@{q^{-n},0,q^{-x}}{pq,q^{-N}}{q}{q} <div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots,N}</math></div><br /> ...

\qCharlier{n}@{q^{-x}}{a}{q}=\qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{0}{q}{-\frac{q^{n+1}}{a}} \sum_{x=0}^{\infty}\frac{a^x}{\qPochhammer{q}{q}{x}}q^{\binomial{x}{2}}\qCharlier{m}@ ...

\qCharlier{n}@{q^{-x}}{a}{q}=\qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{0}{q}{-\frac{q^{n+1}}{a}} \sum_{x=0}^{\infty}\frac{a^x}{\qPochhammer{q}{q}{x}}q^{\binomial{x}{2}}\qCharlier{m}@ ...

...e>EllipticTheta[1, 0, q]= (D[EllipticTheta[2, temp, q], {temp, 1}]/.temp-> 0)</code> || Successful || Successful || - || - ...], {temp, 1}]/.temp-> 0)= (D[EllipticTheta[3, temp, q], {temp, 1}]/.temp-> 0)</code> || Failure || Successful || Error || - ...

\Hermite{n}@{x}=(2x)^n\,\HyperpFq{2}{0}@@{-n/2,-(n-1)/2}{-}{-\frac{1}{x^2}} \Hermite{n+1}@{x}-2x\Hermite{n}@{x}+2n\Hermite{n-1}@{x}=0 ...

...qHyperrphis{3}{2}@@{q^{-n},a\expe^{\iunit\theta},a\expe^{-\iunit\theta}}{0,0}{q}{q} \ctsbigqHermite{n}@{x}{a}{q}=\expe^{\iunit n\theta}\,\qHyperrphis{2}{0}@@{q^{-n},a\expe^{\iunit\theta}}{-}{q}{q^n\expe^{-2\iunit\theta}} ...

\bigqLaguerre{n}@{x}{a}{b}{q}=\qHyperrphis{3}{2}@@{q^{-n},0,x}{aq,bq}{q}{q} \qPochhammer{bqt}{q}{\infty}\cdot\qHyperrphis{2}{1}@@{aqx^{-1},0}{aq}{q}{xt} ...

<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots,N}</math></div><br /> \sum_{x=0}^N\frac{\qPochhammer{\alpha q,q^{-N}}{q}{x}}{\qPochhammer{q,\beta^{-1}q^{-N ...

<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots,N}</math></div><br /> \sum_{x=0}^N\frac{\qPochhammer{\alpha q,q^{-N}}{q}{x}}{\qPochhammer{q,\beta^{-1}q^{-N ...

\sum_{x=0}^{\infty}\frac{\pochhammer{\beta}{x}}{x!}c^x\Meixner{m}@{x}{\beta}{c}\Meixn <div align="right">Constraint(s): <math>{\displaystyle \beta > 0\quad 0 < c < 1}</math></div><br /> ...

...n+\alpha+1-x)\Laguerre[\alpha]{n}@{x}+(n+\alpha)\Laguerre[\alpha]{n-1}@{x}=0 xy''(x)+(\alpha+1-x)y'(x)+ny(x)=0 ...

<div align="right">Constraint(s): <math>{\displaystyle c<0}</math></div><br /> \qHyperrphis{2}{1}@@{qx^{-1},0}{q}{q}{xt}\,\qHyperrphis{1}{1}@@{c^{-1}x}{q}{q}{cqt} ...

...de>Sum[Divide[Pochhammer[a, k],Pochhammer[b, k]]*Divide[(z)^(k),(k)!], {k, 0, m}]=Divide[Pochhammer[a, m]*(z)^(m),Pochhammer[b, m]*(m)!]*HypergeometricP ...[Complex[1, 1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.0, 4.0] <- {Rule[n, 2], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}< ...

...n}@{x}{q}=\frac{1}{\qPochhammer{q}{q}{n}}\,\qHyperrphis{2}{1}@@{q^{-n},-x}{0}{q}{q^{n+\alpha+1}} \int_{0}^{\infty}\frac{x^{\alpha}}{\qPochhammer{-x}{q}{\infty}}\qLaguerre[\alpha]{m ...

...n}@{x}{q}=\frac{1}{\qPochhammer{q}{q}{n}}\,\qHyperrphis{2}{1}@@{q^{-n},-x}{0}{q}{q^{n+\alpha+1}} \int_{0}^{\infty}\frac{x^{\alpha}}{\qPochhammer{-x}{q}{\infty}}\qLaguerre[\alpha]{m ...

...s{3}{2}@@{q^{-n},a\expe^{\iunit(\theta+2\phi)},a\expe^{-\iunit\theta}}{a^2,0}{q}{q} <div align="right">Constraint(s): <math>{\displaystyle 0<a<1}</math></div><br /> ...

...s{3}{2}@@{q^{-n},a\expe^{\iunit(\theta+2\phi)},a\expe^{-\iunit\theta}}{a^2,0}{q}{q} <div align="right">Constraint(s): <math>{\displaystyle 0<a<1}</math></div><br /> ...

\Charlier{n}@{x}{a}=\HyperpFq{2}{0}@@{-n,-x}{-}{-\frac{1}{a}} \sum_{x=0}^{\infty}\frac{a^x}{x!}\Charlier{m}@{x}{a}\Charlier{n}@{x}{a}= ...

...|| <code>(Subscript[S, m*n])^(1)*(\[Gamma], 0)=(- 1)^(m)* LegendreP[n, m, 0]</code> || Failure || Failure || Error || Error ..., Power[2, Rational[1, 2]]]]}</code><br><code>Complex[33.660899653979236, -0.8858131487889274] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]] ...