How to find Discrete-time Fourier Transform (DTFT) of Kaiser Window!

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I saw a post on Fourier transform of a continuous Kaiser window.

But I need a compact formula for DTFT of a Kaiser window. When I plotted the DTFT, I noticed that the nulls for DTFT occurs at slightly different frequencies in $[-\pi ,\pi]$ compared to continuous Fourier transform.

$w[n]=\frac{I_0 \left( \beta [1-(n/a)^2]^{1/2} \right) }{I_0(\beta)}, n=-a,-a+1...,a-1,a$, where a is a positive integer and $I_0(.)$ is the modified Bessel function of first kind.

DTFT: $W(e^{j\omega}) = \sum_{n=-a}^a w[n] e^{-j\omega n}$