Hypertranscendence and linear difference equations

<p>After Hölder proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental (<italic>i.e.</italic>, they cannot be solution to an algebraic differential equation). In this paper, we obtain the first complete results for solutions to general linear difference equations associated with the shift operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x right-arrow from bar x plus h"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false">↦<!-- ↦ --></mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>h</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x\mapsto x+h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h element-of double-struck upper C Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">h\in \mathbb {C}^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>), the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-difference operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x right-arrow from bar q x"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false">↦<!-- ↦ --></mml:mo> <mml:mi>q</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x\mapsto qx</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q element-of double-struck upper C Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">q\in \mathbb {C}^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> not a root of unity), and the Mahler operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x right-arrow from bar x Superscript p"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false">↦<!-- ↦ --></mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">x\mapsto x^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p\geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> integer). The only restriction is that we constrain our solutions to be expressed as (possibly ramified) Laurent series in the variable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with complex coefficients (or in the variable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 slash x"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1/x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in some special case associated with the shift operator). Our proof is based on the parametrized difference Galois theory initiated by Hardouin and Singer. We also deduce from our main result a general statement about algebraic independence of values of Mahler functions and their derivatives at algebraic points.</p>

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