Given the complex signal
$s[n]=A\cdot e^{j(2\pi f_0 nT_s+\theta)}$
Which is zero-padded to length $M$, I'm trying to compute the steps to obtain its discrete time fourier transform (DTFT).
I'm considering the $k$-th element as:
$S[k]=\sum_{n=0}^{N-1}s[n]e^{-j2\pi kn/N}$
Additionally, I'm basing my considerations on the results shown in [1], where it is defined that the following is valid:
$S[k]=\sum_{n=0}^{N-1}s[n]e^{-j2\pi kn/N}$=$\frac {A\cdot sin(\pi (f_0 T_s-k/N)M} {sin(\pi(f_0T_s-k/N))} e^{j\pi(f_0T_s-k/N)(M-1)+\theta}$
What I want to do, is understand the mathematical steps between the summation and the explicit form.
References:
[1] XIANG, J., Wei, C. U. I., & Qing, S. H. E. N. (2018). Flexible and accurate frequency estimation for complex sinusoid signal by interpolation using dft samples. Chinese Journal of Electronics, 27(1), 109-114.