Special products

Apart from the well-known Euler function, q-functions and q-analogues such as f(z) = (1 - a z)(1 - a z^2)(1 - a z^3)(1 - a z^4)... ,

https://mathworld.wolfram.com/q-PochhammerSymbol.html

https://mathworld.wolfram.com/q-Series.html

https://mathworld.wolfram.com/DedekindEtaFunction.html

https://mathworld.wolfram.com/q-Product.html

https://mathworld.wolfram.com/EulerFunction.html

Or the lesser known

f(x) = f(0) (1  + x/t_1) (1 + x/t_2) (1 + x/t_3) ...

What even works when f(x) = exp(x) 

 exp(x) = (1+z)(1 + (1/2) x^2)...

see

https://www.go.helms-net.de/math/musings/dreamofasequence.pdf

***

btw an interesting somewhat sum analogue of q-series and taylor expansion is this

 https://mathworld.wolfram.com/LambertSeries.html

***

there are other interesting infinite products.

Such as this 

For real x > 0

ln(x) = (x - 1) [2 / (x^(1/2) + 1)] [2 / (x^(1/4) + 1)][2 / (x^(1/8) + 1)][2 / (x^(1/16) + 1)][2 / (x^(1/32) + 1)]...)

Notice this is problematic if x would be on the unit circle, therefore we take x real > 0.

Replacing x with z such that Re(z) > 1 however also works.

Notice how this is a telescoping identity that follows from (x+1)(x-1) = x^2 -1.

Also notice that by telescoping it is clear that f(x^2) =  2 f(x) is satisfied here.

Notice the equation f(x^2) = 2 f(x) makes alot of sense if x is not on the unit circle.

Lets generalize

 For real x > 0

T(x) = (x - 1) [2 / (x^(1/a_1) + 1)] [2 / (x^(1/a_2) + 1)][2 / (x^(1/a_3) + 1)][2 / (x^(1/a_4) + 1)][2 / (x^(1/a_5) + 1)]...)

And 

1/a_1 + 1/a_2 + 1/a_3 + ... = 1.

Now clearly this function does not grow like any power.

T(x) =/= O( x^a)

precisely because 1/a_1 + 1/a_2 + 1/a_3 + ... = 1.

The log is an example of it since 1/2 + 1/4 + 1/8 + ... = 1 and indeed log(x) grows slower than any x^a.

so how fast does T(x) grow for a given sequence a_n ?

a_n = n(n+1) ?

( that is a telescoping sum to 1 btw) 

or

a_n being sylvester's sequence ? 

https://mathworld.wolfram.com/SylvestersSequence.html

Worth investigating.

Update :

See these posts by my friend.

https://math.stackexchange.com/questions/2651623/why-do-we-have-lnz-fracz-1e-1-prod-n-1-infty-frac-exp2

 The integral proof comes from me, I am his mentor.

https://math.stackexchange.com/questions/4977205/asymptotics-for-fz-z-1-prod-n-frac2z1-a-n1-with-1-a-1-1-a-2

Basically the same question on mathstack.

Also slightly related

https://math.stackexchange.com/questions/2653643/3x-3-prod-n-1%e2%88%9e-frac-exp2-n1x2-n1-x3-1-prod-n-1

Update 2 : 

Additional links by my friend :

https://math.stackexchange.com/questions/4982893/is-fx-d-x-1-prod-n0-frac1dx1-2nd-close-to-lnxgd