golden ratio phi (or tau) = (1 + sqrt(5))/2 = Equals Product_{k>=0} ((5^(k+1) + 1)*(5^(k-1/2) + 1))/((5^k + 1)*(5^(k+1/2) + 1))

[Davide Rotondo]

Hi to all dear seq fans, I am analyzing formulas from Antonio Graciá Llorente in A001622 and find that a lot of his formula like Product_{k>=0} ((5*k + 2)*(5*k + 3))/((5*k + 1)*(5*k + 4)) and the subject cited Product_{k>=0} ((5^(k+1) + 1)*(5^(k-1/2) + 1))/((5^k + 1)*(5^(k+1/2) + 1)) give interesting results  changing 5 with other values, in particular I noticed that if we use Product_{k>=0} ((x^(k+1) + 1)*(x^(k-1/2) + 1))/((x^k + 1)*(x^(k+1/2) + 1) where x is are the natural numbers we obtain (sqrt(x)+1))/2.

Examples

Product_{k>=0} ((3^(k+1) + 1)*(3^(k-1/2) + 1))/((3^k + 1)*(3^(k+1/2) + 1)) =

1 + sqrt(3))/2

What do you think?

Davide

[jp allouche math]

Hi

First I am afraid that a factor of 2 is missing in the first formula you write
(but I might be wrong).
Second all the kinds of relations similar to the first relation 
you give are essentially trivial once you know

1. infinite products as, e.g., in 5.8.5 at https://dlmf.nist.gov/5.8
2. the Euler reflection formula (see, e.g., https://en.wikipedia.org/wiki/Reflection_formula

For the second family of relations, it looks like it is nothing but telescopic stuff

best
jp

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[Davide Rotondo]

Thanks.

Davide

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