GosperPetkovsekForm¶

Gosper Petkovšek form in the Mahler operator context¶

Calling sequence:¶

$\operatorname{GosperPetkovsekForm}(u, x, b, s)$

Parameters:¶

  • $u$, a rational function in $x$
  • $x$, the variable
  • $b$, the radix
  • $s$, the order

Description:¶

  • $\operatorname{GosperPetkovsekForm}(u, x, b, s)$ computes a sequence $\zeta$, $A$, $B$, $C$, with
    • $\zeta$ a number in the ground field,
    • $A$, $B$, $C$ monic polynomials, such that $$ \operatorname{MOp}(u, x, b^s) = \zeta \frac{\operatorname{MOp}(C, x, b)}{C} \frac{\operatorname{MOp}(A, x, b^s)}{B}.$$
  • This procedure is only made for illustration of the cited article and never used in the package.
  • The procedure will be deleted in future versions.

References:¶

  • Frédéric Chyzak, Thomas Dreyfus, Philippe Dumas, and Marc Mezzarobba. First-order factors of linear Mahler operators. In preparation.

Example:¶

In [1]:
libname := libname, FileTools:-JoinPath(["maple","lib","dcfun.mla"],base=homedir):
In [2]:
 with(dcfun):
Out[2]:

We test the formula on an example.

In [3]:
 u := (1+x)/(1-x);
Out[3]:

$$\frac{1+x}{1-x}$$

In [4]:
 b := 2;
Out[4]:

$$2$$

In [5]:
 s := 3;
Out[5]:

$$3$$

In [6]:
 zeta, A, B, C := GosperPetkovsekForm((1+x)/(1-x), x, b, s);
Out[6]:

$$-1, 1, x^{4}-1, x^{4}+1$$

In [7]:
 u1 := zeta * MOp(C, x, b)/C * MOp(A, x, b^s)/B;
Out[7]:

$$-\frac{x^{8}+1}{\left(x^{4}+1\right) \left(x^{4}-1\right)}$$

In [8]:
 u2 := MOp(u, x, b^s);
Out[8]:

$$-\frac{x^{8}+1}{x^{8}-1}$$

In [9]:
 normal(u1 - u2);
Out[9]:

$$0$$