Two years back I was working with lacunary series. In my explorations I had derived that the following series is periodic with period 1:

$$ f(x)= \sum_{n=0}^{\infty} \left[ e^{2^{x+n}} \right] + x - \sum_{n=1}^{\infty}\left[ 1-e^{2^{x-n}}\right] $$

And that the constant term of its fourier expansion is equal to $\gamma$ which is the euler mascheroni constant.

I'm studying a similar such series today

$$f(x) = -e^x \sum_{n=-\infty}^{\infty} \left[ 2^n e^{-2^n e^x} \right] $$

Which is also periodic with period 1.

The constant term of this series' fourier expansion is

$$ C = \sum_{n=-\infty}^{\infty} \left[ (e^e)^{-2^n} - e^{-2^n} \right] $$

I am however unable to relate this to any previously known constants or hypergeometric functions etc... Can anyone relate this constant to something we already know?

Two years back I was working with lacunary series. In my explorations I had derived that the following series is periodic with period 1:

$$ f(x)= \sum_{n=0}^{\infty} \left[ e^{2^{x+n}} \right] + x - \sum_{n=1}^{\infty}\left[ 1-e^{2^{x-n}}\right] $$

And that the constant term of its fourier expansion is equal to $\frac{\gamma}{\ln(2)}$ where $\gamma$ is the euler mascheroni constant.

I'm studying a similar such series today

$$f(x) = -e^x \sum_{n=-\infty}^{\infty} \left[ 2^n e^{-2^n e^x} \right] $$

Which is also periodic with period 1.

The constant term of this series' fourier expansion is

$$ C = \sum_{n=-\infty}^{\infty} \left[ (e^e)^{-2^n} - e^{-2^n} \right] $$

I am however unable to relate this to any previously known constants or hypergeometric functions etc... Can anyone relate this constant to something we already know?