MHypergeom

definition of a Mahlerian hypergeometric function

Calling sequence:

$\operatorname{MHypergeom}(g, x, b)$

Parameters:

• $g$, a nonzero ramified rational function
  
• $x$, the variable,
  
• $b$, the radix of the Mahler operator

Description:

• $\operatorname{MHypergeom}(g, x, b)$ returns the Mahlerian hypergeometric solution $y$ of the first order linear Mahler equation $My = g y$ normalized by the condition that its leading coefficient (that is its first nonzero coefficient in the series expansion at $0$ with respect to the scale $x^{\alpha} \ln(1/x)^{\beta})$ is equal to $1$.
  
• With the input $g = 0$, the output NULL is returned (and not $0$, even it is a hypergeometric element).
  
• $\operatorname{MHypergeom}$ is the right way to define a hypergeometric element.

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):

Mahlerian hypergeometric elements look like infinite products, because from $$y = \frac{1}{g}My$$ we wrongly deduce $$y = \frac{1}{g} \frac{1}{Mg}\frac{1}{M^2g}\dots = \prod_{k\geq 0}\frac{1}{M^kg}.$$ To give a meaning to the infinite product a sort of normalization is necessary. It generally makes appear a power of a logarithm and a power of the variable as factors of the infinite product.

 y := MHypergeom(6*x^11*( 1+ 4*x), x, 5);

$$\ln \! \left(\frac{1}{x}\right)^{\frac{\ln \left(6\right)}{\ln \left(5\right)}} x^{\frac{11}{4}} \left(\overset{\infty}{\underset{\textit{\_k0} =0}{\textcolor{gray}{\prod}}}\! \frac{1}{1+4 x^{5^{\textit{\_k0}}}}\right)$$