Identities Relating the Number of Partitions into an Even and Odd Number of Parts

If / > 0 and n > 1, let qf (n) denote the number of partitions of/? into an even number of parts, where each part occurs at most / times and Set qf(n) denote the number of partitions of n into an odd number of parts, where each part occurs at most/times. \ii>0, \etqf(0)= 1 andqf(0) = 0. For/ >Oandn > 0, \v\Aj(n) = qJ(n)-q°(n). For/= 1, it is well known [1] that , A (n) = I ~ i f n = f ^ orsome j = 0, 1,2,-, 1 I 0 otherwise.