Confluent A-hypergeometric functions and rapid decay homology cycles

We study confluent $A$-hypergeometric functions introduced by Adolphson. In particular, we give their integral representations by using rapid decay homology cycles of Hien. The method of toric compactifications introduced in our previous works will be used to prove our main theorem. Moreover we apply it to obtain a formula for the asymptotic expansions at infinity of confluent $A$-hypergeometric functions.