You can use the measurement protocol, a language agnostic description of a http tracking request to the Google Analytics tracking server.
The problem in your case is that you do not have a script between the click and the download to send the tracking request. One possible workaround would be to use the server access log, provided you have some control over the server.
For example the Apache web server can user piped logs, e.g. instead if being written directly to a file the log entry is passed to a script or program. I'm reasonably sure that other servers have something similar.
You could pipe the logs to a script that evaluates if the log entry points at the URL of your pdf file, and if so breaks down the info into individual data fields and sends them via a programming language of your choice to the GA tracking server.
If you cannot control the server to that level you'd need to place a script with the same name and location as the original file on the server, map the pdf extension to a script interpreter of your choice (in apache via addType, which with many hosts can be done via a htaccess file) and have the script sending the tracking request before delivering the original file.
Both solutions require a modicum of programming practice (the latter much less than the former). Piping logs might be expensive, depending on the number of requests to your server (you might create an extra log file for downloadable files, though). An intermediary script would not be an expensive operation.
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