# Voronoi Summation for $d_k$ where k>2

So, I'm looking at the Voronoi Summation formula for the sum of divisor function, given by

$\displaystyle\sum_{n \leq x}'d(n) = x(\log x + 2\gamma -1) + \frac{1}{4} -\frac{x^{\frac{1}{2}}}{2 \pi} \displaystyle\sum_{n=1}^{\infty}\frac{d(n)}{n^{\frac{1}{2}}}(K_1(4\pi(nx)^\frac{1}{2}) + \frac{1}{2}\pi Y_1(4\pi (nx)^\frac{1}{2})$

where $d(n)$ is the divisor function, the tick means the final value of n is multiplied by 1/2 if n is a whole number, and $K_1$ and $Y_1$ are Bessel functions. You can find a reference to this and more details here: Voronoi Summation Formula as found in Ivic.

Suppose that I define higher indexed versions of the divisor function in the usual way, with

$d_k(n) = \displaystyle\sum_{j | n} d_{k-1}(j)$ and $d_1(n) = 1$

Are there Voronoi Summation formulas for expressing $\displaystyle\sum_{n \leq x}'d_k(n)$ ? Any references for such things? Or, alternatively, can anyone show me the steps leading to this identity? Outside of Ivic, I'm having a hard time tracking down good information on this.

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