At the start of the Bleach manga, Rukia stabs Ichigo in the heart with her zanpakuto in order to give him her shinigami powers, which allows him to protect his family from the Hollows.
After that, Ichigo obtains a sealed zanpakuto that he uses (alongside Rukia in a gigai) to fight Hollows around Karakura.
Later, Rukia returns to Seireitei and regains her shinigami powers, which robs Ichigo of his.
This implies that Ichigo had borrowed his powers from Rukia for the run of the Karakura Arc. This is further supported when Ichigo gains his own zanpakuto, Zangetsu, and his sword is unsealed rather than remaining in its more benign state.
Because Ichigo is borrowing Rukia's spirit powers, and because Zangetsu's form appears to be more volatile and less prone to remaining sealed, is the first zanpakuto that Ichigo carries Rukia's zanpakuto Sode no Shiraiyuki?