We show that the unresolved examples of Christol’s conjecture ##IMG## [http://ej.iop.org/images/1751-8121/53/20/205201/aab82dcieqn3.gif] {${\;}_{3}{F}_{2}\left(\left[2/9,5/9,8/9\right],\left[2/3,1\right],x\right)$} and ##IMG## [http://ej.iop.org/images/1751-8121/53/20/205201/aab82dcieqn4.gif] {${\;}_{3}{F}_{2}\left(\left[1/9,4/9,7/9\right],\left[1/3,1\right],x\right)$} , are indeed diagonals of rational functions. We also show that other 3 F 2 and 4 F 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 3 F 2 ([1/9, 4/9, 5/9], [1/3, 1], 27 ⋅ x ) function is a diagonal of a rational function.

中文翻译：

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关于克里斯托尔的猜想

我们显示了克里斯托尔猜想的未解决示例## IMG ## [http://ej.iop.org/images/1751-8121/53/20/205201/aab82dcieqn3.gif] {$ {\;} _ {3 } {F} _ {2} \ left（\ left [2 / 9,5 / 9,8 / 9 \ right]，\ left [2 / 3,1 \ right]，x \ right）$}和## IMG ## [http://ej.iop.org/images/1751-8121/53/20/205201/aab82dcieqn4.gif] {$ {\;} _ {3} {F} _ {2} \ left（ \ left [1 / 9,4 / 9,7 / 9 \ right]，\ left [1 / 3,1 \ right]，x \ right）$}确实是有理函数的对角线。我们还表明，克里斯托猜想的其他3 F 2和4 F 3未解决的例子是有理函数的对角线。最后，我们给出两个参数，表明3 F 2（[1/9，4/9，5/9]，[1/3，1]，27⋅x）函数可能是有理数的对角线功能。