On a dividend let's say it pays .05 per share for example and the stock closes the day before @ .25 the opening trade on dividend day will trade at .20. But you have to be on record the day before the dividend date. Then the company will state the date you receive the money, you can choose to reinvest the dividends and get more shares or opt to take the cash which is deposited in your stock account.
Whatever the dividend per share is paid out it is reduced at the open.
A special dividend is different I am not sure it applies I dont think it does.