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sections


Usage

sections(f, m)


Parameter Type Description
L ${\mathbb{Q}}[n,E]$ A difference operator
m ${\mathbb{Z}}$ A positive integer


Returns

Returns $[L_0,\dots,L_{m-1}]$ of minimal orders such that for any solution $y = (y_0,y_1,y_2,\dots)$ of $L y = 0$, the sequences $y^{(i)} = (y_i, y_{i+m}, y_{i+2m},\dots)$ for $0 \le i < m$ satisfy $L_i y^{(i)} = 0$.


Example

The $2$-sections of

\begin{eqnarray*}
L := (n+2)(n^5+2n^4+n^3-1)y(n+2) &+& (n+1)(5n^2+8n+4)y(n+1)\\
&-& n^2 (n^5+7n^4+19n^3+25n^2+16n+3) y(n) = 0
\end{eqnarray*}



are computed as follows:
1 --> L := (n+2)*(n^5+2*n^4+n^3-1)*E^2 + (n+1)*(5*n^2+8*n+4)*E
           - n^2 * (n^5+7*n^4+19*n^3+25*n^2+16*n+3);
2 --> L2 := sections(L,2);
3 --> tex(L2);

\begin{eqnarray*}[
\left(40\,n^{3}+112\,n^{2}+72\,n+16\right)\,E^{2}&-&
\left(16...
...
&+& 160\,n^{5}+928\,n^{4}+2000\,n^{3}+1968\,n^{2}+882\,n+146
]
\end{eqnarray*}



Those turn out to have hypergeometric solutions, which yields a Liouvillian solution of $L y = 0$:
4 --> tex(hyper(element(L2,1)));

\begin{displaymath}
{{2\,n^{2}+3\,n+1} \over {n}}
\end{displaymath}

5 --> tex(hyper(element(L2,2)));

\begin{displaymath}
{{2\,n^{2}+5\,n+3} \over {n+{{1} \over {2}}}}
\end{displaymath}



Manuel Bronstein 2002-09-04