General coefficient-vanishing results associated with theta series. (English) Zbl 07904795
Summary: There are a number of sporadic coefficient-vanishing results associated with theta series, which suggest certain underlying patterns. By expanding theta powers as linear combinations of products of theta functions, we present two strategies that will provide a unified treatment. Our approaches rely on studying the behavior of products of two theta series under the action of the huffing operator. For this purpose, some explicit criteria are given. We may use the presented methods to not only verify experimentally discovered coefficient-vanishing results, but also to produce a series of general phenomena.
MSC:
| 11F27 | Theta series; Weil representation; theta correspondences |
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
| 11J13 | Simultaneous homogeneous approximation, linear forms |
| 11J20 | Inhomogeneous linear forms |
Keywords:
Ramanujan’s theta series; vanishing coefficient; huffing operator; linear congruence; sublattice of \(\mathbb{Z}^2\)References:
| [1] | Alladi, K.; Gordon, B., Vanishing coefficients in the expansion of products of Rogers-Ramanujan type, (The Rademacher Legacy to Mathematics. The Rademacher Legacy to Mathematics, University Park, PA, 1992, 1994, American Mathematical Society: American Mathematical Society Providence, RI), 129-139 · Zbl 0809.33009 |
| [2] | Andrews, G. E.; Bressoud, D. M., Vanishing coefficients in infinite product expansions, J. Aust. Math. Soc. Ser. A, 27, 2, 199-202, 1979 · Zbl 0397.10047 |
| [3] | Baruah, N. D.; Kaur, M., Some results on vanishing coefficients in infinite product expansions, Ramanujan J., 53, 3, 551-568, 2020 · Zbl 1476.33010 |
| [4] | Berndt, B. C., Ramanujan’s Notebooks. Part III, 1991, Springer-Verlag: Springer-Verlag New York · Zbl 0733.11001 |
| [5] | Borwein, J. M.; Borwein, P. B., Pi and the AGM. A Study in Analytic Number Theory and Computational Complexity, 1987, John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York · Zbl 0611.10001 |
| [6] | Chan, S. H.; Yesilyurt, H., The periodicity of the signs of the coefficients of certain infinite products, Pac. J. Math., 225, 1, 13-32, 2006 · Zbl 1118.30002 |
| [7] | Chern, S.; Tang, D., Vanishing coefficients in quotients of theta functions of modulus five, Bull. Aust. Math. Soc., 102, 3, 387-398, 2020 · Zbl 1475.11073 |
| [8] | Chern, S.; Tang, D., Vanishing coefficients and identities concerning Ramanujan’s parameters, Ramanujan J., 57, 4, 1367-1385, 2021 · Zbl 1492.11090 |
| [9] | Cooper, S., The quintuple product identity, Int. J. Number Theory, 2, 1, 115-161, 2006 · Zbl 1159.33300 |
| [10] | Ding, H.; Xia, E. X.W., Arithmetic properties for Appell-Lerch sums, Ramanujan J., 56, 3, 763-783, 2021 · Zbl 1503.11135 |
| [11] | Dou, D. Q.J.; Xiao, J., The 5-dissections of two infinite product expansions, Ramanujan J., 54, 3, 475-484, 2021 · Zbl 1466.33012 |
| [12] | Du, J. Q.D.; Tang, D., Vanishing coefficients in quotients of theta functions with odd moduli, Int. J. Number Theory, 19, 10, 2451-2481, 2023 · Zbl 1536.11075 |
| [13] | Hirschhorn, M. D., On the expansion of a continued fraction of Gordon, Ramanujan J., 5, 4, 369-375, 2001 · Zbl 0993.30003 |
| [14] | Hirschhorn, M. D., The Power of q. A Personal Journey, 2017, Springer: Springer Cham · Zbl 1456.11001 |
| [15] | Hirschhorn, M. D., Two remarkable q-series expansions, Ramanujan J., 49, 2, 451-463, 2019 · Zbl 1460.11059 |
| [16] | Kuar Vandna, M., Results on vanishing coefficients in infinite q-series expansions for certain arithmetic progressions mod 7, Ramanujan J., 58, 1, 269-289, 2022 · Zbl 1510.11068 |
| [17] | Lin, B. L.S., On the expansion of a continued fraction of order twelve, Int. J. Number Theory, 9, 8, 2019-2031, 2013 · Zbl 1335.11008 |
| [18] | Mc Laughlin, J., Further results on vanishing coefficients in infinite product expansions, J. Aust. Math. Soc., 98, 1, 69-77, 2015 · Zbl 1318.33029 |
| [19] | Mc Laughlin, J., A generalization of Schröter’s formula, Ann. Comb., 23, 3-4, 889-906, 2019 · Zbl 1436.33015 |
| [20] | Mc Laughlin, J., New infinite q-product expansions with vanishing coefficients, Ramanujan J., 55, 2, 733-760, 2021 · Zbl 1495.11043 |
| [21] | Mc Laughlin, J., Some observations on Lambert series, vanishing coefficients and dissections of infinite products and series, (Analytic and Combinatorial Number Theory: The Legacy of Ramanujan—Contributions in Honor of Bruce C. Berndt, 2024, World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ) |
| [22] | Mc Laughlin, J.; Zimmer, P., Further results on vanishing coefficients in infinite products of the form \(( q^b , q^{p - b} ; q^p )_\infty^3 ( q^{j b} , q^{2 p - j b} ; q^{2 p} )_\infty \), Int. J. Number Theory, 18, 8, 1863-1885, 2022 · Zbl 1495.11044 |
| [23] | Ramanujan, S., Notebooks. Vol. II, 1957, Tata Institute of Fundamental Research: Tata Institute of Fundamental Research Bombay · Zbl 0138.24201 |
| [24] | Richmond, B.; Szekeres, G., The Taylor coefficients of certain infinite products, Acta Sci. Math. (Szeged), 40, 3-4, 347-369, 1978 · Zbl 0397.10046 |
| [25] | Schröter, H., De aequationibus modularibus, 1854, Albertina Litterarum Universitate: Albertina Litterarum Universitate Regiomonti, Dissertatio Inauguralis |
| [26] | Tang, D., Vanishing coefficients in some q-series expansions, Int. J. Number Theory, 15, 4, 763-773, 2019 · Zbl 1459.11117 |
| [27] | Tang, D., Vanishing coefficients on four quotients of infinite product expansions, Bull. Aust. Math. Soc., 100, 2, 216-224, 2019 · Zbl 1469.11081 |
| [28] | Tang, D., On 5- and 10-dissections for some infinite products, Ramanujan J., 56, 2, 425-450, 2021 · Zbl 1478.11067 |
| [29] | Tang, D., Vanishing coefficients in two families of quotients of theta functions, Bull. Malays. Math. Sci. Soc., 46, Article 28 pp., 2023 · Zbl 1510.11102 |
| [30] | Tang, D., Vanishing coefficients in three families of products of theta functions, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., 117, Article 36 pp., 2023 · Zbl 1523.11077 |
| [31] | Tang, D., Vanishing coefficients in three families of products of theta functions. II, Results Math., 78, 4, Article 124 pp., 2023 · Zbl 1537.11058 |
| [32] | Tang, D., Vanishing coefficients in powers of theta functions with odd moduli, Rocky Mt. J. Math., 54, 1, 261-267, 2024 · Zbl 07823201 |
| [33] | Tang, D.; Xia, E. X.W., Several q-series related to Ramanujan’s theta functions, Ramanujan J., 53, 3, 705-724, 2020 · Zbl 1467.33013 |
| [34] | Vandna; Kaur, M., Vanishing coefficients of \(q^{5 n + r}\) and \(q^{11 n + r}\) in certain infinite q-product expansions, Ann. Comb., 26, 3, 533-557, 2022 · Zbl 1520.33012 |
| [35] | Xia, E. X.W.; Zhao, A. X.H., Generalizations of Hirschhorn’s results on two remarkable q-series expansions, Exp. Math., 31, 3, 878-882, 2022 · Zbl 1516.33013 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
