Generalizations of Eulefts Recurrence Formula for Partitions
نویسنده
چکیده
where k is a positive integer and the left-hand side of (1) is the generating function for the number of partitions into parts & 0, ±k (mod Ik + 1), while the left-hand side of (2) is the generating function for the number of partitions into parts ^ 0, +1 (mod 2k + 1). As Aider remarks, when/r = 2, identities (Hand (2) reduce to the Rogers-Ramanujan identities for which £2,/?(x)=x . Alder showed that identities similar to (1) and (2) exist for the generating function for the number of partitions into parts £ 0, ±(k r) (mod 2k + 1) for all r with 0 < r < / r 1, so that, for a given modulus 2/r + 1, there exist k such identities. We shall show in this paper that a similar generalization is possible for recursion formulae for the number of unrestricted or restricted partitions of n. The best known of these is the Euler identity for the number of unrestricted partitions of n:
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