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Finding modular functions for Ramanujan-type identities. (English) Zbl 1433.05019

Summary: This paper is concerned with a class of partition functions \(a(n)\) introduced by C.-S. Radu [J. Symb. Comput. 68, Part 1, 225–253 (2015; Zbl 1356.11072); J. 20, No. 2, 215–251 (2009; Zbl 1204.11165)] and defined in terms of eta-quotients. By utilizing the transformation laws of M. Newman [Proc. Lond. Math. Soc. (3) 9, 373–387 (1959; Zbl 0178.43001); ibid. 7, 334–350 (1957; Zbl 0097.28701)], B. Schoeneberg [Elliptic modular functions. An introduction. Translated from the German by J. R. Smart and E. A. Schwandt. Springer, Berlin (1974; Zbl 0285.10016)] and S. Robins [Contemp. Math. 166, 119–128 (1994; Zbl 0808.11031)], and Radu’s algorithms, we present an algorithm to find Ramanujan-type identities for \(a(mn+t)\). While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for \(p(11n+6)\) with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions \(\overline{p}(5n+2)\) and \(\overline{p}(5n+3)\) and Andrews-Paule’s broken 2-diamond partition functions \(\triangle_2(25n+14)\) and \(\triangle_2(25n+24)\). It can also be extended to derive Ramanujan-type identities on a more general class of partition functions. For example, it yields the Ramanujan-type identities on G. E. Andrews’ singular overpartition functions \(\overline{Q}_{3,1}(9n+3)\) and \(\overline{Q}_{3,1}(9n+6)\) [Adv. Math. 41, 186–208 (1981; Zbl 0477.33009)] due to E. Y. Y. Shen [Int. J. Number Theory 12, No. 3, 841–852 (2016; Zbl 1337.05010)], the 2-dissection formulas of Ramanujan [loc. cit.], and the 8-dissection formulas due to M. D. Hirschhorn [Ramanujan J. 5, No. 4, 369–375 (2001; Zbl 0993.30003)].

MSC:

05A15 Exact enumeration problems, generating functions
11P83 Partitions; congruences and congruential restrictions
11P84 Partition identities; identities of Rogers-Ramanujan type
05A17 Combinatorial aspects of partitions of integers

Software:

4ti2
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References:

[1] 4ti2 team: 4ti2 - A software package for algebraic, geometric and combinatorial problems on linear spaces software. Available at https://4ti2.github.io
[2] Andrews, Ge, Ramunujan’s “lost” notebook III. The Rogers-Ramanujan continued fraction., Adv. Math., 41, 186-208 (1981) · Zbl 0477.33009
[3] Andrews, Ge, Singular overpartitions, Int. J. Number Theory, 11, 5, 1523-1533 (2015) · Zbl 1325.11107
[4] Andrews, Ge; Paule, P., MacMahon’s partition analysis XI: Broken diamonds and modular forms, Acta Arith., 126, 3, 281-294 (2007) · Zbl 1110.05010
[5] Atkin, Aol; Swinnerton-Dyer, P., Some properties of partitions, Proc. London Math. Soc., 3, 4, 84-106 (1954) · Zbl 0055.03805
[6] Berndt, B.C.: Number Theory in the Spirit of Ramanujan. Student Mathematical Library, 34. Amer. Math. Soc., Providence, RI (2006) · Zbl 1117.11001
[7] Bilgici, G.; Ekin, Ab, Some congruences for modulus 13 related to partition generating function, Ramanujan J., 33, 2, 197-218 (2014) · Zbl 1308.11091
[8] Bilgici, G.; Ekin, Ab, \(11\)-Dissection and modulo \(11\) congruences properties for partition generating function, Int. J. Contemp. Math. Sci., 9, 1-4, 1-10 (2014)
[9] Chan, Sh, Some congruences for Andrews-Paule’s broken 2-diamond partitions, Discrete Math., 308, 23, 5735-5741 (2008) · Zbl 1206.05020
[10] Cho, B.; Koo, Jk; Park, Yk, Arithmetic of the Ramanujan-Göllnitz-Gordon continued fraction, J. Number Theory, 129, 4, 922-947 (2009) · Zbl 1196.11014
[11] Corteel, S.; Lovejoy, J., Overpartitions, Trans. Amer. Math. Soc., 356, 4, 1623-1635 (2004) · Zbl 1040.11072
[12] Diamond, F., Shurman, J.: A First Course in Modular Forms. Graduate Texts in Mathematics, 228. Springer, New York (2005) · Zbl 1062.11022
[13] Eichhorn, D.A.: Some results on the congruential and gap-theoretic study of partition functions. Ph.D. Thesis. University of Illinois at Urbana-Champaign (1999)
[14] Eichhorn, D.A., Ono, K.: Congruences for partition functions. In: Berndt, B.C., Diamond, H.G., Hildebrand, A.J. (eds.) Analytic Number Theory, Vol. 1 (Allerton Park, IL, 1995), pp. 309-321. Progr. Math., 138, Birkhäuser Boston, Boston, MA (1996) · Zbl 0852.11056
[15] Eichhorn, Da; Sellers, Ja, Computational proofs of congruences for 2-colored Frobenius partitions, Int. J. Math. Math. Sci., 29, 6, 333-340 (2002) · Zbl 0997.11085
[16] Gasper, G.; Rahman, M., Basic Hypergeometric Series. Second Edition. Encyclopedia of Mathematics and its Applications, 96 (2004), Cambridge: Cambridge University Press, Cambridge · Zbl 1129.33005
[17] Gordon, B., Some continued fractions of the Rogers-Ramanujan type, Duke Math. J., 32, 741-748 (1965) · Zbl 0178.33404
[18] Hardy, Gh, Note on Ramanujan’s arithmetic function \(\tau (n)\), Proc. Cambridge Philos. Soc., 23, 675-680 (1927) · JFM 53.0150.01
[19] Hardy, Gh, A further note on Ramanujan’s arithmetic function \(\tau (n)\), Proc. Cambridge Philos. Soc., 34, 309-315 (1938) · JFM 64.0099.02
[20] Hemmecke, R., Dancing samba with Ramanujan partition congruences, J. Symbolic Comput., 84, 14-24 (2018) · Zbl 1432.11146
[21] Hirschhorn, Md, On the expansion of Ramanujan’s continued fraction, Ramanujan J., 2, 4, 521-527 (1998) · Zbl 0924.11005
[22] Hirschhorn, Md, On the expansion of a continued fraction of Gordon, Ramanujan J., 5, 4, 369-375 (2001) · Zbl 0993.30003
[23] Hirschhorn, M.D., Roselin: On the 2-, 3-, 4- and 6-dissections of Ramanujan’s cubic continued fraction and its reciprocal. In: Baruah, N.D., Berndt, B.C., Cooper, S., Huber, T., Schlosser, M.J. (eds.) Ramanujan Rediscovered, pp. 125-138. Ramanujan Math. Soc. Lect. Notes Ser., 14, Ramanujan Math. Soc., Mysore (2010) · Zbl 1221.30011
[24] Hirschhorn, Md; Sellers, Ja, Arithmetic relations for overpartitions, J. Combin. Math. Combin. Comput., 53, 65-73 (2005) · Zbl 1086.11048
[25] Knopp, M.: Modular Functions in Analytic Number Theory. Second Edition. Amer. Math. Soc., Chelsea Publishing (1993) · Zbl 0997.11500
[26] Kolberg, O., Some identities involving the partition function, Math. Scand., 5, 77-92 (1957) · Zbl 0080.03303
[27] Lewis, R.; Liu, Z-G, A conjecture of Hirschhorn on the 4-dissection of Ramanujan’s continued fraction, Ramanujan J., 4, 4, 347-352 (2000) · Zbl 1023.11005
[28] Newman, M., Construction and application of a class of modular functions, Proc. London. Math. Soc., 3, 7, 334-350 (1957) · Zbl 0097.28701
[29] Newman, M., Construction and application of a class of modular functions II, Proc. Lond. Math. Soc., 3, 9, 373-387 (1959) · Zbl 0178.43001
[30] Paule, P., Radu, C.-S.: A unified algorithmic framework for Ramanujan’s congruences modulo powers of 5, 7, and 11. Preprint (2018)
[31] Paule, P.; Radu, C.-S; Andrews, G. E.; Garvan, F., A new witness identity for \(11|p(11n + 6)\), Analytic Number Theory, Modular Forms and \(q\)-Hypergeometric Series, Springer Proc. Math. Stat., 221, 625-639 (2017), Cham: Springer, Cham · Zbl 1416.11155
[32] Paule, P.; Radu, S.; Beveridge, A.; Griggs, J. R.; Hogben, L.; Musiker, G.; Tetali, P., Partition analysis, modular functions, and computer algebra, Recent Trends in Combinatorics, IMA Vol. Math. Appl., 159, 511-543 (2016), Cham: Springer, Cham · Zbl 1354.05010
[33] Rademacher, H., The Ramanujan identities under modular substitutions, Trans. Amer. Math. Soc., 51, 609-636 (1942) · Zbl 0060.10006
[34] Rademacher, H., Topics in Analytic Number Theory. Die Grundlehren der mathematischen Wissenschaften, Band 169 (1973), New York-Heidelberg: Springer-Verlag, New York-Heidelberg · Zbl 0253.10002
[35] Radu, C-S, An algorithmic approach to Ramanujan-Kolberg identities, J. Symbolic Comput., 68, 1, 225-253 (2015) · Zbl 1356.11072
[36] Radu, S.: An algorithmic approach to Ramanujan’s congruences and related problems. Ph.D. Thesis. Research Institute for Symbolic Computation Johannes Kepler University, Linz (2009) · Zbl 1204.11165
[37] Radu, S., An algorithmic approach to Ramanujan’s congruences, Ramanujan J., 20, 2, 215-251 (2009) · Zbl 1204.11165
[38] Ramanujan, S., Some properties of \(p(n)\), the number of partitions of \(n\), Proc. Cambridge Philos. Soc., 19, 207-210 (1919) · JFM 47.0885.01
[39] Ramanujan, S., On certain arithmetical functions, Trans. Cambridge Philos. Soc., 22, 159-184 (1916) · Zbl 07426016
[40] Ramanujan, S., The Lost Notebook and Other Unpublished Papers (1988), New Delhi: Narosa Publishing House, New Delhi · Zbl 0639.01023
[41] Robins, S.: Generalized Dedekind \(\eta \)-products. In: Andrews, G.E., Bressoud, D.M., Parson, L.A. (eds.) The Rademacher Legacy to Mathematics (University Park, PA, 1992), Contemp. Math., 166, pp. 119-128. Amer. Math. Soc., Providence, RI (1994) · Zbl 0808.11031
[42] Rogers, Lj, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc., 25, 318-343 (1894)
[43] Schoeneberg, B., Elliptic Modular Functions: An Introduction. Die Grundlehren der mathematischen Wissenschaften, Band 203 (1974), New York-Heidelberg: Springer-Verlag, New York-Heidelberg · Zbl 0285.10016
[44] Schrijver, A., Theory of Linear and Integer Programming (1986), Chichester: Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons Ltd, Chichester · Zbl 0665.90063
[45] Shen, Eyy, Arithmetic properties of \(l\)-regular overpartitions, Int. J. Number Theory, 12, 3, 841-852 (2016) · Zbl 1337.05010
[46] Smoot, N.A.: An implementation of Radu’s Ramanujan-Kolberg algorithm. RISC Technical Report (2019)
[47] Srivastava, B., On 2-dissection and 4-dissection of Ramanujan’s cubic continued fraction and identities, Tamsui Oxf. J. Math. Sci., 23, 3, 305-315 (2007) · Zbl 1219.11011
[48] Stein, W.: Modular Forms, A Computational Approach. Graduate Studies in Mathematics, 79. Amer. Math. Soc., Providence, RI (2007) · Zbl 1110.11015
[49] Xia, Exw; Yao, Xm, The 8-dissection of the Ramanujan-Göllnitz-Gordon continued fraction by an iterative method, Int. J. Number Theory, 7, 6, 1589-1593 (2011) · Zbl 1231.11011
[50] Zuckerman, Hs, Identities analogous to Ramanujan’s identities involving the partition function, Duke Math. J., 5, 1, 88-110 (1939) · JFM 65.0160.02
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