about a summary problem from middle school

Question

if we have a function $I_a=1+2^{a}x^{1}+3^{a}x^2+……+p^{a}x^{p-1}$ how can i write the general term of $I_a$? i think it must be fantastc.i havent use this before,please forgive me about grammar mistakes and writing.thank you for answering!

Answer 1

As @ K.defaoite commented, $$I_a=\sum_{k=1}^{p}k^a \,x^{k-1}=\frac{\text{Li}_{-a}(x)}{x}-x^p\, \Phi (x,-a,p+1)$$ $$I_\infty=\frac{\text{Li}_{-a}(x)}{x}$$