The phrases are correct as stated (and the rec function
has been tested to work). In that version of APL
a⍺b ←→ a↑b⍴1
See Table 2 in the document.
In Dyalog v12.1, for a random matrix m, rec m was
compared against ⌹m, with function alpha←{⍺↑⍵⍴1}
substituting for ⍺ .