Is there a specific supplication that I should recite when slaughtering the sacrificial animal?

It is Sunnah (recommended) for the one intending to slaughter the sacrificial animal to say: "Bismillahi, Walllahu Akbar, Allahumma Hadha Minka Walaka, Hadha Anni (or 'An Fulan' if slaughtering on behalf of someone else), Allahumma Taqabbal Min Fulan wa Aal Fulan (and he names himself)." The obligatory part of this is saying "Bismillah" (in the name of Allah), and anything extra is recommended. This is supported by prophetic hadiths, such as the Prophet's sacrifice of two rams, where he mentioned Allah's name, proclaimed His greatness, and placed his foot on their sides. Also, his saying at the time of slaughter: "In the name of Allah, O Allah, accept from Muhammad, and the family of Muhammad, and from the Ummah of Muhammad," and his saying: "In the name of Allah, Allah is the Greatest, this is on my behalf and on behalf of those of my Ummah who have not sacrificed." The meaning of "Allahumma Minka" (O Allah, from You) is that the sacrifice is a gift and provision from Allah, and "Walaka" (and for You) means it is purely for Him.