If you purchased a download from Adobe, then the download link should appear in the order history of your account. You can then download the software on your new computer. Do you by chance have multiple accounts set up under multiple email addresses? Do you also have a copy of the receipt for your purchase? If you ordered online, you'll have an email confirmation. (The serial number is required for installation and when registered is proof of ownership, but isn't a proof of purchase.)
Here's another thread that addresses how to transfer the existing installation to your new computer: http://forums.adobe.com/message/4146532#4146532
Thanks for your prompt response. Yes – my Mastercard account confirms a payment of $NZ161.49 on July 17, 2010, to Adobe Systems Dublin Gbr, in the name of Richard Rayner. Is that enough? The email confirmation has been long since deleted.
Please ignore my last message ... I’ve just realised I’d ordered under a different email address! I knew I’d changed it some time ago, but thought it was well before my Adobe purchase. All now looks to be clear. Sorry for wasting your time, and many thanks.