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

[Joerg Arndt]

* 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 )
equivalently
  (1+x)^(1+x) = 1F0( [-1-x] , [] , -x )
Also
  (1+x)^(1+x) = 1/(1 + 2*x) * 2F1( [-x-2, x+3] , [x+2] , -x )
Taken from my "matters Computational", section 36.5, "The function x^x"
Cf. A005727

Of course there is always
  Lisa Lorentzen, Haakon Waadeland:
  Continued Fractions and Applications,
  North-Holland, (1992)

Not sure which of the "massive Russian formula tomes"
are to be recommend here (anyone?).

Best regards,   jj