we are planning to install JIRA in Test and production, how will a ticket opened in production also show up in Development for a tester to view without logging into the production environment?
It won't. You're describing two separate systems.
I'd look at why you're "synchronising" first. There are good reasons that dev test and production are separate systems. What is the real need to synchronise things here?
If you do need to do it, then you could write stuff, but bear in mind that dev and test could be torn down or rebuilt without it. I'd reach for K15t's Backbone or iDalko's Exalate products instead of trying to write my own, but legally, if you're synchronising production to other systems, you're using all of them as though they're production, so you need three full licences for them.
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