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? $$\pmb{\frac{(\frac12)_j^2}{j!^2}}\sum_{i=0}^{j-1}\frac4{2i+1} =\sum_{i=0}^{j-1}\pmb{\frac{(\frac12)_i^2}{i!^2}}\frac1{j-i}.$$

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