Finding the Sum of Convergent Series t^(-n^2) for n=1 to ∞

[nipunmalhotra]

what is the sum of the following series? I know it's convergent (using ratio test) but I'm not able to work it out :(
S=t^(-1) + t^(-4)+t^(-9)...t^(k^2)...to ∞
where t>1
Thanks

 

[Norwegian]

S = [θ₃(1/t)-1]/2, where θ₃ is a Jacobi theta function. I strongly suspect S is transcendental for integer t, but couldn't easily find a good reference, nor a convincing argument.

 

[ZARman]

Hey everyone
Im a maths tutor and this question was given to one of my students. i may be thinking about it to cryptically so i thought id post it

5,9,13 ,27,,,217

i have posted it to various other pupils but no one has gotten it yet.