I have heard that there are ways to express sums of rational functions in terms of polygamma functions, and I would like to read more about it. However, I don't know the literature about special functions very well. Can anybody suggest possible references? Thank you.
a journal publication with many generalisations is
Infinite sums as linear combinations of polygamma functions, Kh. Hessami Pilehrood, T. Hessami Pilehrood, Acta Arithmetica 130 (2007) 231-254.