Abstract
In this paper, we study the partition function \(p_{[c^{l}d^{m}]}(n)\) defined by \(\sum_{n=0}^{\infty}p_{[c^{l}d^{m}]}(n)q^{n}=(q^{c};q^{c})_{\infty}^{-l}(q^{d};q^{d})_{\infty}^{-m}\) and prove some analogues of Ramanujan’s partition identities. We also deduce some interesting partition congruences.
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Atkin, A.O.L.: Proof of a conjecture of Ramanujan. Glasg. Math. J. 8, 14–32 (1967)
Baruah, N.D.: A few theta function identities and some of Ramanujan’s modular equations. Ramanujan J. 4, 239–250 (2000)
Baruah, N.D., Berndt, B.C.: Partition identities and Ramanujan’s modular equations. J. Comb. Theory, Ser. A 114, 1024–1045 (2007)
Baruah, N.D., Berndt, B.C.: Partition identities arising from theta function identities. Acta Math. Sin. Engl. Ser. 24, 955–970 (2008)
Baruah, N.D., Bora, J.: Some new proofs of Ramanujan’s modular equations of degree 9. Indian J. Math. 47, 99–122 (2005)
Baruah, N.D., Bora, J., Saikia, N.: Some new proofs of modular relations for the Göllnitz–Gordon functions. Ramanujan J. 15, 281–301 (2008)
Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991)
Berndt, B.C.: Partition-theoretic interpretations of certain modular equations of Schröter, Russel, and Ramanujan. Ann. Comb. 11, 115–125 (2007)
Chan, H.-C.: Ramanujan’s cubic continued fraction and a generalization of his “Most beautiful identity”. Int. J. Number Theory 6, 673–680 (2010)
Chan, H.-C.: Ramanujan’s cubic continued fraction and Ramanujan type congruences for a certain partition function. Int. J. Number Theory 6, 819–834 (2010)
Chan, H.-C., Cooper, S.: Congruences modulo powers of 2 for a certain partition function. Ramanujan J. 22, 101–117 (2010)
Chan, H.H., Toh, P.C.: New analogues of Ramanujan’s partition identities. J. Number Theory 129, 1898–1913 (2010)
Hirschhorn, M.D., Hunt, D.C.: A simple proof of the Ramanujan conjecture for powers of 5. J. Reine Angew. Math. 326, 1–17 (1981)
Ramanujan, S.: Some properties of p(n), the number of partitions of n. Proc. Camb. Philos. Soc. 19, 207–210 (1919)
Ramanujan, S.: Collected Papers. Cambridge University Press, Cambridge (1927); reprinted by Chelsea, New York (1962); reprinted by the American Mathematical Society, Providence (2000)
Yan, Q.: Several identities for certain products of Theta functions. Ramanujan J. 19, 79–94 (2009)
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Baruah, N.D., Ojah, K.K. Analogues of Ramanujan’s partition identities and congruences arising from his theta functions and modular equations. Ramanujan J 28, 385–407 (2012). https://doi.org/10.1007/s11139-011-9296-z
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DOI: https://doi.org/10.1007/s11139-011-9296-z