Generate hash value using public key along with message
Actually, the notation of the authors is not complete. They simply do not define what it means to hash several values at once. However, what one would do in practice is appending everything. So $H_i$ would be
$$H_i = \mathcal{H}(m\|Y_1\|Y_2\|\ldots\|Y_i)$$

where $\|$ denotes string concatenation.