This is the simply way to connect to an external database in phpBB (we go to use mysql lib here, but you could use another of these available in /phpbb32/phpbb/db/driver folder changing namespace (if on php7> you need to use mysqli and NOT mysql or you’ll get error) using phpBB db classes

$w3dbhost = 'localhost';
$w3dbuser = 'root';
$w3dbpass = 'apass';
$w3dbname = 'mydbname';
$phpbb_c = new \phpbb\db\driver\mysql();
$phpbb_c->sql_connect($w3dbhost, $w3dbuser, $w3dbpass, $w3dbname, $port = false, $persistency = false, $new_link = false);
$sql = "SELECT * FROM myTable";
$result = $phpbb_c->sql_query($sql);
while ($row = $phpbb_c->sql_fetchrow($result))
{
	print_r($row);
}
$phpbb_c->sql_close($result);

Note: if on Php7 > change line

$phpbb_c = new \phpbb\db\driver\mysql();
into
$phpbb_c = new \phpbb\db\driver\mysqli();

to switch and use mysqli.