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:

libname := libname, FileTools:-JoinPath(["maple","lib","dcfun.mla"],base=homedir):

 with(dcfun):

We test the formula on an example.

 u := (1+x)/(1-x);

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

 b := 2;

$$2$$

 s := 3;

$$3$$

 zeta, A, B, C := GosperPetkovsekForm((1+x)/(1-x), x, b, s);

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

 u1 := zeta * MOp(C, x, b)/C * MOp(A, x, b^s)/B;

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

 u2 := MOp(u, x, b^s);

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

 normal(u1 - u2);

$$0$$