I have been a huge fan of your mother for years and have owned and read all her books. The trilogy that I particularly loved has been the Crystal Singer.
I am a writer and wanted to asked permission to continue the series with a new novel. I have also written to your publishing house as well.
Thank you for your time.
The answer (at least for the foreseeable future) is no.
I have no doubt the publisher will tell you the same.
However, Mum understood that fans might want to write fan fiction, and established fan fiction rules. You might want to look at them and consider writing fan fiction.
Mum’s final word on fan fiction was that she felt people should write their own books on their own worlds.
Also, if you’re planning on becoming a full-time author, you might want to consider Writers of the Future. Mum was a judge (as am I).