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In this note, we will give a short proof of an identity for cubicpartition function. 1. Introduction∑Let p ( n ) denote the number of the unrestricted partitions of n , de ned by 1n =0 p ( n ) q n =∏ 1n =011 q n . One of the celebrated results about p ( n ) is thetheorem which was proved by Watson [8]: if k 1, then for every nonnegativeinteger n (1) p (5 k n + r k ) 0 (mod 5 k ) ; where r k is the reciprocal modulo 5 k of 24. Recently, the notion of cubicpartitions of a natural number n , named by Kim [5], was introduced by Chan[1] in connection with Ramanujan’s cubic continued fraction. By de ntion, thegenerating function of the number of cubic partitions of n is(2)∑ 1n =0 a ( n ) q n =∏ 1n =1 1(1 q n )(1 q 2 n ) : Chan [1] from the Ramanujan’s cubic continued fraction v ( q ) := q 13 1 + q + q 3 1 + q 2 + q 4 1 + ::: jqj 1derived an elegant identity: let x ( q ) = q 13 v ( q ), then1 x ( q ) q 13 2 q 23 x ( q ) =( q 13 ; q 13 ) 1 ( q 23 ; q 23 ) 1 ( q 3 ; q 3 ) 1 ( q |
