On Christol’s conjecture
Copyright policy )On Christol’s conjecture
We show that the unresolved examples of Christol's conjecture $ \, _3F_{2}\left([2/9,5/9,8/9],[2/3,1],x\right)$ and $_3F_{2}\left([1/9,4/9,7/9],[1/3,1],x\right)$, are indeed diagonals of rational functions. We also show that other $\, _3F_2$ and $\, _4F_3$ unresolved examples of Christol's conjecture are diagonals of rational functions. Finally we give two arguments that show that it is likely that the $\, _3F_2([1/9, 4/9, 5/9], \, [1/3,1], \, 27 \cdot x)$ function is a diagonal of a rational function.
13 pages
AMS Classification scheme numbers: 33C05, 68W30 Key-words: Christol's conjecture, inverse creative telescoping, diagonals of rational functions, Christol's conjecture, FOS: Physical sciences, 33F10, creative telescoping, Modular and Shimura varieties, Classical Analysis and ODEs (math.CA), FOS: Mathematics, D-finite series, Number Theory (math.NT), Mathematical Physics, Generalized hypergeometric series, \({}_pF_q\), Mathematics - Number Theory, globally bounded series, Shimura curves, Mathematical Physics (math-ph), [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph], 33C20, telescopers, Mathematics - Classical Analysis and ODEs
AMS Classification scheme numbers: 33C05, 68W30 Key-words: Christol's conjecture, inverse creative telescoping, diagonals of rational functions, Christol's conjecture, FOS: Physical sciences, 33F10, creative telescoping, Modular and Shimura varieties, Classical Analysis and ODEs (math.CA), FOS: Mathematics, D-finite series, Number Theory (math.NT), Mathematical Physics, Generalized hypergeometric series, \({}_pF_q\), Mathematics - Number Theory, globally bounded series, Shimura curves, Mathematical Physics (math-ph), [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph], 33C20, telescopers, Mathematics - Classical Analysis and ODEs
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