The LREtools[mhypergeomsols] command computes $m$-fold hypergeometric term solutions of holonomic recurrence equations, i.e. linear homogeneous recurrence equation with polynomial coefficients, where $m$ is a positive integer. An $m$-fold hypergeometric term $a\left(n\right)$ is such that $\frac{a\left(n+m\right)}{a\left(n\right)}$ is a rational function of $n$ over the considered field.

All appearances of the dependent variable a(.) in the rec must be of the form $a\left(n+k\right)$, where $k$ is an integer.

The output is a list of pairs $\left[m\,H_m\right]$, where $m$ is a positive integer and $H_m$is a nonempty set of hypergeometric terms, or, when mj=[m,j] is specified, just a single set $H_m$ of hypergeometric terms.

When mj is not specified, each $h\in H_m$ corresponds to an $m$-fold hypergeometric term $t$ via $t\left(\mathrm{nm}\right)=h\left(n\right)$ for $n\ge 0$ and $t\left(k\right)=0$ when $m$ does not divide $k\>0$.

When mj=[m,j] is given, each $h\in H_m$ corresponds to an $m$-fold hypergeometric term $t$ via $t\left(\mathrm{nm}+j\right)=h\left(n\right)$ for $n\ge 0$ and $t\left(k\right)=0$ when $k\ge 0$ and $m$ does not divide $k-j$.

In the former case, when mj is not given, for all positive integers $m$ occurring in the result, the set of corresponding $m$-fold hypergeometric terms constitutes a basis of the solution space for $j=0$. The other solutions can be computed afterwards by calling mhypergeomsols with mj=[m,j], where $m$ is a value taken from the result and $0\le j<m$.

The field of computation is controlled by the field parameter. When complex is supplied, the computation considers algebraic extension fields; this includes solutions containing symbols. By default, field equals rational and the corresponding field is the rational field $\mathbb{Q}$ without any symbols.

$\mathrm{with}\left(\mathrm{LREtools}\right)\&colon;$

$\mathrm{RE}≔\left(n+1\right)\left(n+2\right)\left(n+3\right)\left(n+4\right)a\left(n+4\right)+10\left(n+1\right)\left(n+2\right)a\left(n+2\right)+9a\left(n\right)=0$

$\mathrm{RE}≔\left(n+1\right)\left(n+2\right)\left(n+3\right)\left(n+4\right)a\left(n+4\right)+10\left(n+1\right)\left(n+2\right)a\left(n+2\right)+9a\left(n\right)=0$

$\mathrm{sol}≔\mathrm{mhypergeomsols}\left(\mathrm{RE}\&comma;a\left(n\right)\right)$

$\mathrm{sol}≔\left[\left[2\&comma;\left\{\frac{{\left(\mathrm{-9}\right)}^n}{\left(2n\right)!}\&comma;\frac{{\left(\mathrm{-1}\right)}^n}{\left(2n\right)!}\right\}\right]\right]$

$\mathrm{sol1}≔\mathrm{unapply}\left(\mathrm{eval}\left(\mathrm{op}\left(\left[1\&comma;2\&comma;1\right]\&comma;\mathrm{sol}\right)\&comma;n=\frac{n}{2}\right)\&comma;n\right)$

$\mathrm{sol1}≔n\mapsto \frac{{\left(\mathrm{-9}\right)}^{\frac{n}{2}}}{n!}$

$\mathrm{normal}\left(\mathrm{expand}\left(\mathrm{eval}\left(\mathrm{RE}\&comma;a=\mathrm{sol1}\right)\right)\right)$

$0=0$

$\mathrm{sol2}≔\mathrm{unapply}\left(\mathrm{eval}\left(\mathrm{op}\left(\left[1\&comma;2\&comma;2\right]\&comma;\mathrm{sol}\right)\&comma;n=\frac{n}{2}\right)\&comma;n\right)$

$\mathrm{sol2}≔n\mapsto \frac{{\left(\mathrm{-1}\right)}^{\frac{n}{2}}}{n!}$

$\mathrm{normal}\left(\mathrm{expand}\left(\mathrm{eval}\left(\mathrm{RE}\&comma;a=\mathrm{sol2}\right)\right)\right)$

$0=0$

$\mathrm{mhypergeomsols}\left(\mathrm{RE}\&comma;a\left(n\right)\&comma;\mathrm{complex}\right)$

$\left[\left[1\&comma;\left\{\frac{{\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)}^n}{n!}\&comma;\frac{{\mathrm{RootOf}\left({\mathrm{\_Z}}^2+9\right)}^n}{n!}\right\}\right]\&comma;\left[2\&comma;\left\{\frac{{\left(\mathrm{-9}\right)}^n}{\left(2n\right)!}\&comma;\frac{{\left(\mathrm{-1}\right)}^n}{\left(2n\right)!}\right\}\right]\right]$

$\mathrm{map}\left(\mathrm{allvalues}\&comma;\mathrm{mhypergeomsols}\left(\mathrm{RE}\&comma;a\left(n\right)\&comma;\mathrm{complex}\&comma;\mathrm{mj}=\left[1\&comma;0\right]\right)\right)$

$\left\{\frac{{\left(-3\mathrm{I}\right)}^n}{n!}\&comma;\frac{{\left(\mathrm{-I}\right)}^n}{n!}\&comma;\frac{{\mathrm{I}}^n}{n!}\&comma;\frac{{\left(3\mathrm{I}\right)}^n}{n!}\right\}$

$\mathrm{hypergeomsols}\left(\mathrm{RE}\&comma;a\left(n\right)\&comma;\varnothing \&comma;\mathrm{output}=\mathrm{basis}\right)$

$\left[\frac{{\left(-3\mathrm{I}\right)}^n}{\mathrm{\Gamma }\left(n+1\right)}\&comma;\frac{{\left(3\mathrm{I}\right)}^n}{\mathrm{\Gamma }\left(n+1\right)}\&comma;\frac{{\mathrm{I}}^n}{\mathrm{\Gamma }\left(n+1\right)}\&comma;\frac{{\left(\mathrm{-I}\right)}^n}{\mathrm{\Gamma }\left(n+1\right)}\right]$

$\mathrm{sol}≔\mathrm{mhypergeomsols}\left(\mathrm{RE}\&comma;a\left(n\right)\&comma;\mathrm{complex}\&comma;\mathrm{mj}=\left[2\&comma;1\right]\right)$

$\mathrm{sol}≔\left\{\frac{{\left(\mathrm{-9}\right)}^n}{\left(2n+1\right)\left(2n\right)!}\&comma;\frac{{\left(\mathrm{-1}\right)}^n}{\left(2n+1\right)\left(2n\right)!}\right\}$

$\mathrm{sol1}≔\mathrm{unapply}\left(\mathrm{eval}\left(\mathrm{sol}\left[1\right]\&comma;n=\frac{n-1}{2}\right)\&comma;n\right)$

$\mathrm{sol1}≔n\mapsto \frac{{\left(\mathrm{-9}\right)}^{\frac{n}{2}-\frac{1}{2}}}{n\cdot \left(n-1\right)!}$

$\mathrm{normal}\left(\mathrm{expand}\left(\mathrm{eval}\left(\mathrm{RE}\&comma;a=\mathrm{sol1}\right)\right)\right)$

$0=0$

$\mathrm{sol2}≔\mathrm{unapply}\left(\mathrm{eval}\left(\mathrm{sol}\left[2\right]\&comma;n=\frac{n-1}{2}\right)\&comma;n\right)$

$\mathrm{sol2}≔n\mapsto \frac{{\left(\mathrm{-1}\right)}^{\frac{n}{2}-\frac{1}{2}}}{n\cdot \left(n-1\right)!}$

$\mathrm{normal}\left(\mathrm{expand}\left(\mathrm{eval}\left(\mathrm{RE}\&comma;a=\mathrm{sol2}\right)\right)\right)$

$0=0$

$\mathrm{RE}≔\left(n+1\right)a\left(n-4\right)-p\left(n+1\right)a\left(n-2\right)-q\left(n+1\right)a\left(n-1\right)+pq\left(n+1\right)a\left(n+1\right)=0$

$\mathrm{RE}≔\left(n+1\right)a\left(n-4\right)-p\left(n+1\right)a\left(n-2\right)-q\left(n+1\right)a\left(n-1\right)+pq\left(n+1\right)a\left(n+1\right)=0$

$\mathrm{mhypergeomsols}\left(\mathrm{RE}\&comma;a\left(n\right)\&comma;\mathrm{complex}\right)$

$\left[\left[1\&comma;\left\{{\mathrm{RootOf}\left(p{\mathrm{\_Z}}^2-1\right)}^n\&comma;{\mathrm{RootOf}\left(q{\mathrm{\_Z}}^3-1\right)}^n\right\}\right]\&comma;\left[2\&comma;\left\{{\left(\frac{1}{p}\right)}^n\right\}\right]\&comma;\left[3\&comma;\left\{{\left(\frac{1}{q}\right)}^n\right\}\right]\right]$

$\mathrm{RE}≔-9{\left(n-13\right)}^{2}a\left(n-13\right)+3\left(n-19\right){\left(n-10\right)}^{2}a\left(n-10\right)-18\left(n-9\right)\left(n-11\right)a\left(n-9\right)+3\left(n-10\right){\left(n-7\right)}^{2}a\left(n-7\right)+6\left(n-6\right)\left(n^2-21n+116\right)a\left(n-6\right)-9\left(n-5\right)\left(n-9\right)a\left(n-5\right)+2\left(n-6\right)\left(n-17\right)\left(n-3\right)a\left(n-3\right)+3\left(n-2\right)\left(n-6\right)\left(n-7\right)a\left(n-2\right)-\left(n-2\right)\left(n-3\right)\left(n+1\right)a\left(n+1\right)=0$

$\mathrm{RE}≔-9{\left(n-13\right)}^{2}a\left(n-13\right)+3\left(n-19\right){\left(n-10\right)}^{2}a\left(n-10\right)-18\left(n-9\right)\left(n-11\right)a\left(n-9\right)+3\left(n-10\right){\left(n-7\right)}^{2}a\left(n-7\right)+6\left(n-6\right)\left(n^2-21n+116\right)a\left(n-6\right)-9\left(n-5\right)\left(n-9\right)a\left(n-5\right)+2\left(n-6\right)\left(n-17\right)\left(n-3\right)a\left(n-3\right)+3\left(n-2\right)\left(n-6\right)\left(n-7\right)a\left(n-2\right)-\left(n-2\right)\left(n-3\right)\left(n+1\right)a\left(n+1\right)=0$

$\mathrm{mhypergeomsols}\left(\mathrm{RE}\&comma;a\left(n\right)\right)$

$\left[\left[3\&comma;\left\{\frac{1}{n!}\right\}\right]\&comma;\left[4\&comma;\left\{\frac{{\left(\mathrm{-1}\right)}^n}{n}\right\}\right]\right]$

$\mathrm{RE}≔5\left(n-6\right){\left(n-11\right)}^{2}a\left(n-11\right)-5\left(n-5\right){\left(n-10\right)}^{2}a\left(n-10\right)-3\left(2n-19\right)\left(n-4\right)\left(n-9\right)a\left(n-9\right)+3\left(2n-17\right)\left(n-3\right)\left(n-8\right)a\left(n-8\right)-3\left(n-2\right)\left(n-7\right)a\left(n-7\right)-\left(n-6\right)\left(97n-597\right)a\left(n-6\right)+\left(n-5\right)\left(97n-500\right)a\left(n-5\right)-\left(n-4\right)\left(2n^2-115n+513\right)a\left(n-4\right)+\left(n-3\right)\left(2n^2-111n+400\right)a\left(n-3\right)-3\left(n-2\right)\left(n-7\right)a\left(n-2\right)-\left(n-1\right)\left(5n-8\right)\left(n-6\right)a\left(n-1\right)+n\left(5n-3\right)\left(n-5\right)a\left(n\right)+4n\left(n+1\right)\left(n-4\right)a\left(n+1\right)-4\left(n+1\right)\left(n+2\right)\left(n-3\right)a\left(n+2\right)=0$

$\mathrm{RE}≔5\left(n-6\right){\left(n-11\right)}^{2}a\left(n-11\right)-5\left(n-5\right){\left(n-10\right)}^{2}a\left(n-10\right)-3\left(2n-19\right)\left(n-4\right)\left(n-9\right)a\left(n-9\right)+3\left(2n-17\right)\left(n-3\right)\left(n-8\right)a\left(n-8\right)-3\left(n-2\right)\left(n-7\right)a\left(n-7\right)-\left(n-6\right)\left(97n-597\right)a\left(n-6\right)+\left(n-5\right)\left(97n-500\right)a\left(n-5\right)-\left(n-4\right)\left(2n^2-115n+513\right)a\left(n-4\right)+\left(n-3\right)\left(2n^2-111n+400\right)a\left(n-3\right)-3\left(n-2\right)\left(n-7\right)a\left(n-2\right)-\left(n-1\right)\left(5n-8\right)\left(n-6\right)a\left(n-1\right)+n\left(5n-3\right)\left(n-5\right)a\left(n\right)+4n\left(n+1\right)\left(n-4\right)a\left(n+1\right)-4\left(n+1\right)\left(n+2\right)\left(n-3\right)a\left(n+2\right)=0$

$\mathrm{mhypergeomsols}\left(\mathrm{RE}\&comma;a\left(n\right)\right)$

$\left[\left[1\&comma;\left\{{\left(\mathrm{-1}\right)}^n\right\}\right]\&comma;\left[2\&comma;\left\{\frac{{n!}^2{4}^n}{n^2\left(2n\right)!}\right\}\right]\&comma;\left[5\&comma;\left\{{\left(\mathrm{-1}\right)}^n\right\}\right]\right]$

$\mathrm{RE}≔-2{\left(\exp \left(x\right)\right)}^2{\ln \left(x\right)}^{12}{\left(n-15\right)}^{2}a\left(n-15\right)+\exp \left(x\right){\ln \left(x\right)}^{12}\left(n-19\right){\left(n-13\right)}^{2}a\left(n-13\right)+{\ln \left(x\right)}^{12}\left(n-13\right){\left(n-11\right)}^{2}a\left(n-11\right)-4{\left(\exp \left(x\right)\right)}^2{\ln \left(x\right)}^6\left(n-9\right)\left(n-12\right)a\left(n-9\right)+2\exp \left(x\right){\ln \left(x\right)}^6\left(n-7\right)\left(n^2-20n+127\right)a\left(n-7\right)-{\ln \left(x\right)}^6\left(n-5\right)\left(n+19\right)\left(n-7\right)a\left(n-5\right)-2{\left(\exp \left(x\right)\right)}^2\left(n-3\right)\left(n-9\right)a\left(n-3\right)+\exp \left(x\right)\left(n-7\right){\left(n-1\right)}^{2}a\left(n-1\right)-2\left(n-5\right)\left(n-1\right)\left(n+1\right)a\left(n+1\right)=0$

$\mathrm{RE}≔-2{\left({\&ExponentialE;}^x\right)}^2{\ln \left(x\right)}^{12}{\left(n-15\right)}^{2}a\left(n-15\right)+{\&ExponentialE;}^x{\ln \left(x\right)}^{12}\left(n-19\right){\left(n-13\right)}^{2}a\left(n-13\right)+{\ln \left(x\right)}^{12}\left(n-13\right){\left(n-11\right)}^{2}a\left(n-11\right)-4{\left({\&ExponentialE;}^x\right)}^2{\ln \left(x\right)}^6\left(n-9\right)\left(n-12\right)a\left(n-9\right)+2{\&ExponentialE;}^x{\ln \left(x\right)}^6\left(n-7\right)\left(n^2-20n+127\right)a\left(n-7\right)-{\ln \left(x\right)}^6\left(n-5\right)\left(n+19\right)\left(n-7\right)a\left(n-5\right)-2{\left({\&ExponentialE;}^x\right)}^2\left(n-3\right)\left(n-9\right)a\left(n-3\right)+{\&ExponentialE;}^x\left(n-7\right){\left(n-1\right)}^{2}a\left(n-1\right)-2\left(n-5\right)\left(n-1\right)\left(n+1\right)a\left(n+1\right)=0$

$\mathrm{mhypergeomsols}\left(\mathrm{RE}\&comma;a\left(n\right)\&comma;\mathrm{complex}\right)$

$\left[\left[1\&comma;\left\{\frac{{\left(\mathrm{-I}\ln \left(x\right)\right)}^n}{n}\&comma;\frac{{\left(\mathrm{I}\ln \left(x\right)\right)}^n}{n}\&comma;\frac{{\left(-\frac{\left(-\mathrm{I}+\sqrt{3}\right)\ln \left(x\right)}{2}\right)}^n}{n}\&comma;\frac{{\left(\frac{\left(-\mathrm{I}+\sqrt{3}\right)\ln \left(x\right)}{2}\right)}^n}{n}\&comma;\frac{{\left(-\frac{\left(\sqrt{3}+\mathrm{I}\right)\ln \left(x\right)}{2}\right)}^n}{n}\&comma;\frac{{\left(\frac{\left(\sqrt{3}+\mathrm{I}\right)\ln \left(x\right)}{2}\right)}^n}{n}\right\}\right]\&comma;\left[2\&comma;\left\{\frac{{\left(-{\ln \left(x\right)}^2\right)}^n}{n}\&comma;\frac{{\left(\frac{\left(1+\mathrm{I}\sqrt{3}\right){\ln \left(x\right)}^2}{2}\right)}^n}{n}\&comma;\frac{{\left(-\frac{\left(\mathrm{I}\sqrt{3}-1\right){\ln \left(x\right)}^2}{2}\right)}^n}{n}\&comma;\frac{{\left({\&ExponentialE;}^x\right)}^n}{n!}\right\}\right]\&comma;\left[3\&comma;\left\{\frac{{\left(\mathrm{-I}{\ln \left(x\right)}^3\right)}^n}{n}\&comma;\frac{{\left(\mathrm{I}{\ln \left(x\right)}^3\right)}^n}{n}\right\}\right]\&comma;\left[6\&comma;\left\{\frac{{\left(-{\ln \left(x\right)}^6\right)}^n}{n}\right\}\right]\right]$

Bertrand Teguia Tabuguia and Wolfram Koepf. Symbolic conversion of holonomic functions to hypergeometric type power series. Computer Algebra issue of the Journal of Programming and Computer Software. February 2022.

Bertrand Teguia Tabuguia. Power Series Representations of Hypergeometric Type and Non-Holonomic Functions in Computer Algebra. Ph.D. thesis. University of Kassel, Germany. May 2020.

Bertrand Teguia Tabuguia. A variant of van Hoeij's algorithm to compute hypergeometric term solutions of holonomic recurrence equations. arXiv:2012.11513 [cs.SC]. December 2020.

The LREtools[mhypergeomsols] command was introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.