The wpdb object can be used to access any database and query any table. Absolutely no need to be WordPress related, which is very interesting.

Simply create a new wpdb object:

  $wpdb_mydb = new wpdb('username','password','database','localhost');

And use it in your functions as you do for the wp database:

  global $wpdb_mydb;

  $sql = $wpdb_mydb->prepare("SELECT * FROM {$wpdb->prefix}my_table_name WHERE FileName = %s", $FileName);
  if (current_user_can('administrator'))
    $wpdb_mydb->show_errors();
  $Results = $wpdb_mydb->get_results($sql, ARRAY_A);
  if (count($Results) > 0)
    return(true);
  else
    return(false);