I am running and underground cable from a power pole to a house, about 1500 feet. What size cable do I need to buy for 100 amp service?

Has the utility approved this plan? That's where I would start first.

The transformer's secondary run is way to long.

Don is absolutely correct. 1500 feet is much too long to deliver 120/240 volts.

The utility, or a local contractor, should be able to offer a line extension, to deliver the high voltage distribution much closer to the home. This can be underground direct burial cable, using DB conduit at least is better, to a ground mounted transformer, the closer to the home the better.