[seqfan] Re: A nice continued fraction for φᵠ (and other constants)

[Paolo Bonzini]

On 14/04/2017 17:56, Joerg Arndt wrote:
> * Thomas Baruchel <baruchel at gmx.com> [Apr 13. 2017 09:17]:
>> Hi,
>>
>> I just published on math.stackexchange.com some continued fractions for
>> various constants like \sqrt{3}, \sqrt[3]{4}, \sqrt[4]{5}, etc. and
>> more importantly (at least to my eyes) for for φᵠ  and φ²′ᵠ
>>
>> I have no idea whether they are abolutely new or not, but I thought
>> members of the seqfan mailing list could be interested:
>> http://math.stackexchange.com/questions/2230651/a-conjectured-continued-fraction-for-phi-phi
>>
>> Regards,
>>
>> -- 
>> Thomas Baruchel
>>
> 
> Maybe the following hypergeometric expressions for x^x are useful:
>   x^x = 1F0( [-x] , [], -x + 1 )

Probably, since this gives phi^phi = 1F0([-phi], [], -1/phi)

http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F0/10/
has continued fraction expansions of 1F0.

Paolo