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)));

$${{2\,n^{2}+3\,n+1} \over {n}}$$

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

$${{2\,n^{2}+5\,n+3} \over {n+{{1} \over {2}}}}$$