What you are looking to do is not possible at this time because it is dependent on functionality that is not currently in our API. Our REST API currently supports the creation/modification/deletion of contacts and lists, as well as the ability to access reporting data from email campaigns. While you can use our API to locate the contacts with upcoming birthdays you would not be able to create an email dynamically using the API.
The functionality that you require is something that people have requested and may be implemented in a future update to our API.
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