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}} ...

\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]] ...

{}+n(n+\alpha+\beta+1)y(x)=0 \sum_{n=0}^{\infty}\Jacobi{\alpha}{\beta}{n}@{x}t^n ...

<div align="right">Constraint(s): <math>{\displaystyle \lambda > 0\quad 0 < \phi < \cpi}</math></div><br /> {}+(n+2\lambda-1)\MeixnerPollaczek{\lambda}{n-1}@{x}{\phi}=0 ...

<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots,N}</math></div><br /> \qPochhammer{q^{-x},\gamma\delta q^{x+1}}{q}{k}=\prod_{j=0}^{k-1}\left(1-\mu(x)q^j+\gamma\delta q^{2j+1}\right) ...

<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots,N}</math></div><br /> \qPochhammer{q^{-x},\gamma\delta q^{x+1}}{q}{k}=\prod_{j=0}^{k-1}\left(1-\mu(x)q^j+\gamma\delta q^{2j+1}\right) ...

...D[w, {z, 2}]+(h - \[Nu]*(\[Nu]+ 1)*(k)^(2)* (JacobiSN[z, (k)^2])^(2))* w = 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-co ...^(- 2)- \[Nu]*(\[Nu]+ 1)* \[Xi],4*\[Xi]*(\[Xi]- 1)*(\[Xi]- (k)^(- 2))]*w = 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-co ...

\sum_{n=0}^\infty\frac{\pochhammer{\alpha+\beta+1}{n} n!}{\pochhammer{\alpha+1}{n}\po \sum_{n=0}^\infty\frac{2n+\alpha+\beta+1}{n+\alpha+\beta+1} ...

\sum_{k=0}^{\infty}\frac{\qPochhammer{bq}{q}{k}}{\qPochhammer{q}{q}{k}}(aq)^k\littleq ...0}{1}@@{-}{aq}{q}{aqxt}\,\qHyperrphis{2}{1}@@{x^{-1},0}{bq}{q}{xt}=\sum_{n=0}^{\infty} ...

...mes[Complex[-1, -1], Power[2, Rational[1, 2]]]]}</code><br><code>Complex[0.0, 3.141592653589793] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, ...egralEi[-(z)] + Ln[z] + EulerGamma = Integrate[Divide[1 - Exp[- t],t], {t, 0, z}]</code> || Error || Failure || - || Successful ...

\discrqHermiteII{n}@{x}{q}=\iunit^{-n}q^{-\binomial{n}{2}}\,\qHyperrphis{2}{0}@@{q^{-n},\iunit x}{-}{q}{-q^n} \qHyperrphis{2}{1}@@{q^{-n},q^{-n+1}}{0}{q^2}{-\frac{q^2}{x^2}} ...

<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots,N}</math></div><br /> (1+x^2)y''(x)+2\left(\nu-Nx\right)y'(x)-n(n-2N-1)y(x)=0 ...

<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots,N}</math></div><br /> \sum_{x=0}^N\frac{(2x+\gamma+\delta+1)\pochhammer{\gamma+1}{x}\pochhammer{-N}{x}N!}{( ...

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

...0^1\littleqLegendre{m}@{x}{q}\littleqLegendre{n}@{x}{q}\,d_qx=(1-q)\sum_{k=0}^{\infty}q^k\littleqLegendre{m}@{q^k}{q}\littleqLegendre{n}@{q^k}{q} ...is{0}{1}@@{-}{q}{q}{qxt}\,\qHyperrphis{2}{1}@@{x^{-1},0}{q}{q}{xt}=\sum_{n=0}^{\infty} ...

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

\sum_{x=0}^{\infty}\frac{\qPochhammer{bq}{q}{x}}{\qPochhammer{q,-bcq}{q}{x}}c^xq^{\bi ...s): <math>{\displaystyle 0\leq bq<1}</math> &<br /> <math>{\displaystyle c>0}</math></div><br /> ...

\sum_{x=0}^{\infty}\frac{\qPochhammer{bq}{q}{x}}{\qPochhammer{q,-bcq}{q}{x}}c^xq^{\bi ...s): <math>{\displaystyle 0\leq bq<1}</math> &<br /> <math>{\displaystyle c>0}</math></div><br /> ...

...heta},q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{-\iunit\theta}}{q^{\alpha+1},0}{q}{q} <math>{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) ...

...|| <code>Divide[t,Exp[t]- 1]= Sum[BernoulliB[n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}]</code> || Failure || Successful || Skip || - ...ath>]] || <code>bernoulli(2*n + 1)= 0</code> || <code>BernoulliB[2*n + 1]= 0</code> || Failure || Failure || Successful || <div class="toccolours mw-col ...

\BesselPoly{n}@{x}{a}=\HyperpFq{2}{0}@@{-n,n+a+1}{-}{-\frac{x}{2}} <div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots,N}</math></div><br /> ...

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

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

(-a)^nq^{-\binomial{n}{2}}\,\qHyperrphis{2}{0}@@{q^{-n},x}{-}{q}{\frac{q^n}{a}} \sum_{k=0}^{\infty}\frac{q^{k^2}a^k}{\qPochhammer{q}{q}{k}\qPochhammer{aq}{q}{k}} ...

\qHyperrphis{3}{2}@@{q^{-n},a\expe^{\iunit\theta},a\expe^{-\iunit\theta}}{ab,0}{q}{q} <math>{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) ...

...lon],z - a])* D[w, z]+Divide[\[Alpha]*\[Beta]*z - q,z*(z - 1)*(z - a)]*w = 0</code> || Failure || Failure || Skip || Successful ..., Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.1238316385762896, 0.11571250751855189] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2] ...

...292471861073, -1.3934110303935494] <- {Rule[k, 2]}</code><br><code>Complex[0.7299050661514341, -1.0197357312195425] <- {Rule[k, 3]}</code><br></div></di ...1532066437278] <- {Rule[k, 2]}</code><br><code>Complex[-2.044850831577895, 0.1768605538132445] <- {Rule[k, 3]}</code><br></div></div> ...

...ermiteI{n}@{x}{q}=q^{\binomial{n}{2}}\,\qHyperrphis{2}{1}@@{q^{-n},x^{-1}}{0}{q}{-qx} \discrqHermiteI{n}@{x}{q}=x^n\qHyperrphis{2}{0}@@{q^{-n},q^{-n+1}}{0}{q^2}{\frac{q^{2n-1}}{x^2}} ...

\HyperpFq{1}{1}@@{-n}{a}{x}=\frac{(-x)^n}{\pochhammer{a}{n}}\,\HyperpFq{2}{0}@@{-n,-a-n+1}{-}{-\frac{1}{x}} <div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br /> ...

<math>{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) \lambda_n{\tilde w}(x;q^{\alpha},q^{\beta}|q)y(x)=0 ...

...ta}\,\qHyperrphis{3}{2}@@{q^{-n},\beta,\beta\expe^{2\iunit\theta}}{\beta^2,0}{q}{q} <math>{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) ...

..., Power[2, Rational[1, 2]]]]}</code><br><code>Complex[-3.5355339059327373, 0.7071067811865475] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]] .../2)^{1/2}\BesselJ{\ell+\frac{1}{2}}@{\rho}</math>]] || <code>CoulombF(ell, 0, rho)=(Pi*rho/ 2)^(1/ 2)* BesselJ(ell +(1)/(2), rho)</code> || <code>Error< ...

{\scriptstyle k=0,1,2\ldots}\\ {\scriptstyle a+k<0}\end{array}} ...

...ide[1,2]*z)^(2*n),Gamma[n +Divide[3,2]]*Gamma[n + \[Nu]+Divide[3,2]]], {n, 0, Infinity}]</code> || Successful || Successful || - || - ...sible mw-collapsed">Fail<div class="mw-collapsible-content"><code>Complex[-0.09119395476353123, -212.19060110903138] <- {Rule[z, Times[Complex[-1, 1], P ...

<math>{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) {}+\lambda_n{\tilde w}(x;a,b,c,d;q)y(x)=0 ...

...|| <code>Zeta[s]=Divide[1,1 - (2)^(- s)]*Sum[Divide[1,(2*n + 1)^(s)], {n, 0, Infinity}]</code> || Successful || Successful || - || - ...s - 1]+ Sum[Divide[(- 1)^(n),(n)!]*Subscript[\[Gamma], n]*(s - 1)^(n), {n, 0, Infinity}]</code> || Failure || Failure || Skip || Skip ...

...0..infinity)</code> || <code>Gamma[z]= Integrate[Exp[- t]*(t)^(z - 1), {t, 0, Infinity}]</code> || Successful || Successful || - || - ...0} = 1</math>]] || <code>pochhammer(a, 0)= 1</code> || <code>Pochhammer[a, 0]= 1</code> || Successful || Successful || - || - ...

...}{n}@{x}{q}=(-a)^nq^{\binomial{n}{2}}\,\qHyperrphis{2}{1}@@{q^{-n},x^{-1}}{0}{q}{\frac{qx}{a}} <div align="right">Constraint(s): <math>{\displaystyle a<0}</math></div><br /> ...

= x - \frac{\RiemannZeta'@{0}}{\RiemannZeta@{0}} ...\displaystyle x \to \infty}</math> &<br /> <math>{\displaystyle \epsilon > 0}</math></div><br /> ...

...| <code>S*Subscript[E*, n]*(s , z)=Divide[MathieuC[n, q, z],MathieuC[n, q, 0]]</code> || Failure || Failure || Error || Error ...s , z)=Divide[MathieuS[n, q, z],D[MathieuS[n, q, temp], {temp, 1}]/.temp-> 0]</code> || Failure || Failure || Skip || Error ...

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

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

...^(2)+ a)* w = 0</code> || <code>D[w, {z, 2}]-(Divide[1,4]*(z)^(2)+ a)* w = 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-co ...^(2)- a)* w = 0</code> || <code>D[w, {z, 2}]+(Divide[1,4]*(z)^(2)- a)* w = 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-co ...

{\scriptstyle k=0,1,2\ldots}\\ {\scriptstyle a+k<0}\end{array}} ...

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

\qHyperrphis{2}{1}@@{aqx^{-1},0}{aq}{q}{xt}\,\qHyperrphis{1}{1}@@{bc^{-1}x}{bq}{q}{cqt} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{cq}{q}{n}}{\qPochhammer{bq,q}{q}{n}}\bigqJacobi ...

...z)</code> || <code>Erf[z]=Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, 0, z}]</code> || Successful || Successful || - || - ...<code>Exp[- (z)^(2)]*(1 +Divide[2*I,Sqrt[Pi]]*Integrate[Exp[(t)^(2)], {t, 0, z}])= Exp[- (z)^(2)]*Erfc[- I*z]</code> || Successful || Successful || - | ...

...0..z)</code> || <code>Gamma[a, 0, z]= Integrate[(t)^(a - 1)* Exp[- t], {t, 0, z}]</code> || Failure || Successful || Skip || - ...<code>GAMMA(a)-GAMMA(a, z)+ GAMMA(a, z)= GAMMA(a)</code> || <code>Gamma[a, 0, z]+ Gamma[a, z]= Gamma[a]</code> || Successful || Successful || - || - ...

...r)^(2)+ (3)^(- 1/ 3)* Exp[- I*Divide[Pi,3]]*(Divide[1,3]*(z)^(2)- x)], {r, 0, Infinity}]</code> || Error || Failure || - || Error ...r)^(2)+ (3)^(- 1/ 3)* Exp[+ I*Divide[Pi,3]]*(Divide[1,3]*(z)^(2)- x)], {r, 0, Infinity}]</code> || Error || Failure || - || Error ...

...Subscript[f, 1]*(Subscript[f, 2]- Subscript[f, 3])+ 2*Subscript[\[Mu], 1]= 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-co ...Subscript[f, 2]*(Subscript[f, 3]- Subscript[f, 1])+ 2*Subscript[\[Mu], 2]= 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-co ...

...|| <code>AiryAi(0)=(1)/((3)^(2/ 3)* GAMMA((2)/(3)))</code> || <code>AiryAi[0]=Divide[1,(3)^(2/ 3)* Gamma[Divide[2,3]]]</code> || Successful || Successfu ...1/ 3)* GAMMA((1)/(3)))</code> || <code>(D[AiryAi[temp], {temp, 1}]/.temp-> 0)= -Divide[1,(3)^(1/ 3)* Gamma[Divide[1,3]]]</code> || Successful || Success ...

...vide[Pochhammer[a, s]*Pochhammer[b, s],Pochhammer[c, s]*(s)!]*(z)^(s), {s, 0, Infinity}]</code> || Failure || Successful || Skip || - ...m[Divide[Pochhammer[a, s]*Pochhammer[b, s],Gamma[c + s]*(s)!]*(z)^(s), {s, 0, Infinity}]</code> || Successful || Failure || - || Skip ...