MOp¶

the Mahler operator¶

Calling sequence:¶

$\operatorname{MOp}(f, x, b)$

Parameters:¶

  • $f$, a Maple expression,
  • $x$, a name,
  • $b$, a positive integer.

Description:¶

  • $\operatorname{MOp}$ implements the Mahler operator which substitutes $x^b$ to $x$.
  • More generally, a linear Mahler operator, that is a polynomial in $M$ with rational functions in $x$ as coefficients, can be evaluated on an expression using $LMOpEval$. See LMOpEval.

Example:¶

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

$\operatorname{MOp}$ is expected to mimic $\operatorname{M}_b$ the substitution Mahler operator which replaces $x$ by $x^b$. The Mahler operator may be applied to rational functions.

In [3]:
 MOp((1+x)/(1-x), x, 3);
Out[3]:

$$-\frac{x^{3}+1}{x^{3}-1}$$

It can be applied to Laurent series

In [4]:
 y := series(sin(x)/x^3, x, 10);
Out[4]:

$$x^{-2}-\frac{1}{6}+\frac{1}{120} x^{2}-\frac{1}{5040} x^{4}+\frac{1}{362880} x^{6}+\mathrm{O}\! \left(x^{8}\right)$$

In [5]:
 My := MOp(y, x, 3);
Out[5]:

$$x^{-6}-\frac{1}{6}+\frac{1}{120} x^{6}-\frac{1}{5040} x^{12}+\frac{1}{362880} x^{18}+\mathrm{O}\! \left(x^{24}\right)$$

or Puiseux series.

In [6]:
 y := PuiseuxSeries(2, series(sin(x), x));
Out[6]:

$$\sqrt{x}-\frac{x^{\frac{3}{2}}}{6}+\frac{x^{\frac{5}{2}}}{120}+O\! \left(x^{\frac{7}{2}}\right)$$

In [7]:
 My := MOp(y, x, 3);
Out[7]:

$$x^{\frac{3}{2}}-\frac{x^{\frac{9}{2}}}{6}+\frac{x^{\frac{15}{2}}}{120}+O\! \left(x^{\frac{21}{2}}\right)$$

Purely logarithmic Mahler hypergeometric functions appear as eigenvectors for the Mahler operator.

In [8]:
 y := MHypergeom(5,x,3);
Out[8]:

$$\ln \! \left(\frac{1}{x}\right)^{\frac{\ln \left(5\right)}{\ln \left(3\right)}}$$

In [9]:
 My := MOp(y, x, 3);
Out[9]:

$$5 \ln \! \left(\frac{1}{x}\right)^{\frac{\ln \left(5\right)}{\ln \left(3\right)}}$$

It is equivalent to use $\operatorname{MOp}$ or $\operatorname{LMOpEval}$ with the operator $M$ as first argument.

In [10]:
 y := x/(1 + 3*x);
Out[10]:

$$\frac{x}{1+3 x}$$

In [11]:
 My1 := MOp(y, x, 2);
Out[11]:

$$\frac{x^{2}}{3 x^{2}+1}$$

In [12]:
 My2 := LMOpEval(M, y, x, M, 2);
Out[12]:

$$\frac{x^{2}}{3 x^{2}+1}$$