Did you try using NIntegrate? You "saved previous values of k[t]=" for later access, since there is recursive definition founded in real numbers if 'x' is provided, but you didn't save equation parts of R1[t] for re-use.

I think you could post more information about what you are doing. Attempting to solve part of a PDE with another PDE? What is orig. formula? Without knowing that, suggesting any numerical approach isn't practical to suggest.

I tried to cut&paste the code (Mathematica 12.1.1 / win10) and summation limits were reversed during Paste. So far d=1 computed but not d>=2. I might try more some other time.