Examine how the doctrinal developments of the Patristic Period (c.100-500 CE) are reflected in the faith and worship of pentecostal church tradition, and in worldwide Christianity, today with focus on the doctrine of the Holy Spirit.
Yes, you are picking a very specific period from Church History that is separated from the emergence of the Pentecostal movement by a vast span of years. The Patristic Period, however, did see early attempts of beginning to think about the role of the Holy Spirit and its function in our world and in the lives of Christians.
The only real impact the Partistic Period had on the Pentecostal church was the idea that the Holy Spirit could baptize the Pentecostals. It was not until much later, during the Reformation Period, that Pneumatology really had an impact on the religion. The person who was responsible for this was Phoebe Palmer (1807-1874).
Her impact on the Pentecostal church can be found in the link provided below.