Pentecostals see themselves as continuing in the tradition of the early Church following the Day of Pentecost as described in Acts 2:
1 When the day of Pentecost had come, they were all together in one place. 2 And suddenly there came from heaven a noise like a violent rushing wind, and it filled the whole house where they were sitting. 3 And there appeared to them tongues as of fire distributing themselves, and they rested on each one of them. 4 And they were all filled with the Holy Spirit and began to speak with other tongues, as the Spirit was giving them utterance. (New American Standard)
Contemporary Pentecostals include baptism by the Holy Spirit as one of the three baptisms that is required for Christ's salvation. The church fathers of the Patristic period had differing views upon the ongoing administration of Pentecostal signs of the ministrations of the Holy Spirit, therefore, discussion of influence has to be limited to individual church fathers.
For example Irenaeus said that there were many in his time and experience who were practicing the spiritual gifts that were manifested by the Holy Spirit through the Apostles on the Day of Pentecost, whereas Augustine said that the Holy Spirit was not personally received as during the Apostles time and that manifestation like healing and tongues did not occur during his time and experience.
The Patristic Period represented the early church's first attempt to chart the work of the Holy Spirit and its role in the believer's life. Looking at the day of Pentecost, which in the Book of Acts relates the first descending of the Holy Spirit on the disciples and the way that they were empowered by the Holy Spirit to speak the news of God in different languages, we can see the seed of a doctrine that was later to define the Pentecostal movement.
The only thing I could find about Pneumatology, the Pentecostal faith, and the Patristic Period was the fact that the Pentecostals were able to be baptized by the Holy Spirit based upon the ideologies of the Patrisitc Period (based upon the fact that the Holy Spirit possessed divine powers).