BIg-IP DNS Listener question

You need to delegate the zone for production.company.com from company.com

So the zone file for company.com contains the following records:

production.company.com NS ns1.production.company.com
production.company.com NS ns2.production.company.com
ns1.production.company.com A 1.1.1.1 
ns2.production.company.com A 2.1.1.1

So when someone queries www.production.company.com, they will send a NS query to a .com nameserver for company.com, and get a reply.

They then send a NS query to the company.com nameserver asking for a production.company.com nameserver.

The company.com nameserver replies that an NS record for production.company.com is at ns1.production.company.com, and to help you out, the A record for ns1.production.company.com is 1.1.1.1.

The client then sends a www.production.company.com A record request to 1.1.1.1 which resolves the WideIP for 1.1.1.1.

This is a standard DNS zone delegation, and your DNS provider or manager should be able to set this up without any difficulty at all.