The sum of A^(x^2)

Question

Is there a formula for the sum of $A^{x^2}$ If not, can it be approximated somehow? I tried finding a formula for it using the sum of $A^x$..I tried differentiating $A^x$ with respect to another function to get just $A$ and then $nA$ which is its sum and then found with respect to what function I will integrate $nA$ to get the standard formula for the sum of $A^x$ .

What I mean is that I tried to find the formula for the sum of $A^x$ by differentiating it with respect to some function to reduce it into a form I can sum and then I integrate the sum of the function I get from differentation with respect to another function such that the result is the sum of the series I wanted to sum which is $A^x$

I Tried doing the same with $A^{x^2}$ so I diffrentiated it with respect to some function to get $A^x$ and then I integrated the sum of $A^x$ with respect to the same function that turned the sum $nA$ into the formula for the sum of $A^x$ but I ended up with an integral that can't be written in closed form..sorry for the ambiguity

I tried many many different methods but none worked.

Answer 1

The sum of $$f(n)=\sum_{x=1}^{n} a^{x^{2}}$$ is $$a_1+...+a_n^{n^{2}}$$

Basically, the formula for the $n$th term of the series is one with $n$ terms and each exponent is the $n$th perfect square. Example:

$$f(10) = a^{100}+a^{81}+a^{64}+a^{49}+a^{36}+a^{25}+a^{16}+a^9+a^4+a$$

Answer 2

The wording of the question is very ambiguous. They are some known equations : $$\sum_{n=1}^\infty A^{(n^2)}=\frac12\left(\vartheta_3(0,A)-1 \right)\qquad A<1$$ Jacobi theta function, see : http://mathworld.wolfram.com/JacobiThetaFunctions.html $$\int A^{(x^2)}dx=\frac12\sqrt{\frac{\pi}{\ln(A)}}\text{ erfi}\left(x\ln(A)\right)+\text{constant}$$ Function erfi , see : http://mathworld.wolfram.com/Erfi.html

As far as I know, there is no special name for $\sum_{k=1}^n A^{(k^2)}$.