This came out of some work on the digamma function.

Let $(x)_k=x(x+1)\cdots(x+k-1)$ denote the Pochhammer symbol. Then,

Question. Can you prove/disprove this identity? $$\frac{(\frac12)_j^2}{j!^2}\sum_{i=0}^{j-1}\frac4{2i+1} =\sum_{i=0}^{j-1}\frac{(\frac12)_i^2}{i!^2}\frac1{j-i}.$$

I found this fascinating in view of fact that some factors are able "go in and out" of the sum.