Powers of Euler’s Product and Related Identities
نویسندگان
چکیده
Ramanujan’s partition congruences can be proved by first showing that the coefficients in the expansions of (q; q)∞ satisfy certain divisibility properties when r = 4, 6 and 10. We show that much more is true. For these and other values of r , the coefficients in the expansions of (q; q)∞ satisfy arithmetic relations, and these arithmetic relations imply the divisibility properties referred to above. We also obtain arithmetic relations for the coefficients in the expansions of (q; q)∞(q ; qt )∞, for t = 2, 3, 4 and various values of r and s. Our proofs are explicit and elementary, and make use of the Macdonald identities of ranks 1 and 2 (which include the Jacobi triple product, quintuple product and Winquist’s identities). The paper concludes with a list of conjectures.
منابع مشابه
Bijections and Congruences for Generalizations of Partition Identities of Euler and Guy
In 1958, Richard Guy proved that the number of partitions of n into odd parts greater than one equals the number of partitions of n into distinct parts with no powers of 2 allowed, which is closely related to Euler’s famous theorem that the number of partitions of n into odd parts equals the number of partitions of n into distinct parts. We consider extensions of Guy’s result, which naturally l...
متن کاملMultiplicative Relations in Powers of Euler’s Product
In a recent paper, Cooper and Hirschhorn conjecture relations among the coefficients of certain products of powers of Euler’s product. Here we use the theory of modular forms with complex multiplication to prove these conjectures.
متن کاملVirasoro Algebra, Dedekind η-function and Specialized Macdonald’s Identities
We motivate and prove a series of identities which form a generalization of the Euler’s pentagonal number theorem, and are closely related to specialized Macdonald’s identities for powers of the Dedekind η–function. More precisely, we show that what we call “denominator formula” for the Virasoro algebra has “higher analogue” for all cs,t-minimal models. We obtain one identity per series which i...
متن کاملq-HYPERGEOMETRIC PROOFS OF POLYNOMIAL ANALOGUES OF THE TRIPLE PRODUCT IDENTITY, LEBESGUE’S IDENTITY AND EULER’S PENTAGONAL NUMBER THEOREM
X iv :m at h/ 02 03 22 9v 1 [ m at h. C O ] 2 2 M ar 2 00 2 2000]Primary 05A19, 33D15 q-HYPERGEOMETRIC PROOFS OF POLYNOMIAL ANALOGUES OF THE TRIPLE PRODUCT IDENTITY, LEBESGUE’S IDENTITY AND EULER’S PENTAGONAL NUMBER THEOREM S. OLE WARNAAR Abstract. We present alternative, q-hypergeometric proofs of some polynomial analogues of classical q-series identities recently discovered by Alladi and Berk...
متن کاملWeighted forms of Euler's theorem
In answer to a question of Andrews about finding combinatorial proofs of two identities in Ramanujan’s “lost” notebook, we obtain weighted forms of Euler’s theorem on partitions with odd parts and distinct parts. This work is inspired by the insight of Andrews on the connection between Ramanujan’s identities and Euler’s theorem. Our combinatorial formulations of Ramanujan’s identities rely on t...
متن کامل