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Ramanujan's Identities and Representation of Integers by Certain Binary and Quaternary Quadratic Forms
Alexander Berkovich, Hamza Yesilyurt
We revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x^2+5y^2. Making use of Ramanujan's_1ψ_1 summation formula we establish…
K. Saito's Conjecture for Nonnegative Eta Products
Alexander Berkovich, Frank G. Garvan
We prove that the Fourier coefficients of a certain general eta product considered by K. Saito are nonnegative. The proof is elementary and depends on a multidimensional theta func…
New Identities for 7-cores with prescribed BG-rank
Alexander Berkovich, Hamza Yesilyurt
A q-series with nonnegative power series coefficients is called positive. The partition statistics BG-rank is defined as an alternating sum of parities of parts of a partition. It…
The BG-rank of a partition and its applications
Alexander Berkovich, Frank G. Garvan
Let πbe a partition. In [2] we defined BG-rank(π) as an alternating sum of parities of parts. This statistic was employed to generalize and refine the famous Ramanujan modulo 5 par…