My library or movies and tv series I have bought are not showing up in my library on my Roku. They show up on my iphone but not my roku device.

The Apple TV app on Roku should have a Library tab (left side), where it would list the video media purchases for the Apple ID account. If the Apple ID account for the Roku and the iPhone match, then the purchases should match — if all is working correctly.

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For a misbehaving Apple TV app on Roku, try this:

Another user (mygreensocks) suggested this procedure from Roku Support staff, which reportedly helped several users.

1. Remove the Apple TV app,
2. Shut off and unplug the TV (or streaming device),
3. Wait 30 seconds,
4. Plug the TV (or streaming device) back in,
5. Reload the Apple TV app.

Else try the Network Connection Reset, as mentioned by user Jarieta.

1. Roku: Home﹥Settings﹥System﹥Advanced System Settings﹥Network Connection Reset

If the issue persists after that, then please contact the Apple Support staff for further assistance.