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Journal articles on the topic 'Theta series'

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Published: 4 June 2021

Last updated: 25 July 2025

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1

von Köhler, G. "Theta series on the theta group." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 58, no. 1 (1988): 15–45. http://dx.doi.org/10.1007/bf02941367.

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2

Peters, M. "Jacobi theta series." Rocky Mountain Journal of Mathematics 19, no. 3 (1989): 863–70. http://dx.doi.org/10.1216/rmj-1989-19-3-863.

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3

Freitag, Eberhard, and Riccardo Salvati Manni. "Octavic theta series." Asian Journal of Mathematics 21, no. 3 (2017): 483–98. http://dx.doi.org/10.4310/ajm.2017.v21.n3.a4.

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4

Andrianov, A. N. "SPHERICAL THETA SERIES." Mathematics of the USSR-Sbornik 62, no. 2 (1989): 289–304. http://dx.doi.org/10.1070/sm1989v062n02abeh003241.

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5

Colombo, Fabrizio, Rolf Krausshar, and Irene Sabadini. "Slice monogenic theta series." Indiana University Mathematics Journal 73, no. 6 (2024): 2039–71. https://doi.org/10.1512/iumj.2024.73.60085.

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6

Chu, Wenchang, and Cangzhi Jia. "Quartic theta hypergeometric series." Ramanujan Journal 32, no. 1 (2013): 23–81. http://dx.doi.org/10.1007/s11139-012-9414-6.

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7

KRIEG, ALOYS. "THETA SERIES OVER THE HURWITZ QUATERNIONS." International Journal of Number Theory 06, no. 01 (2010): 25–36. http://dx.doi.org/10.1142/s1793042110002788.

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There are six theta constants over the Hurwitz quaternions on the quaternion half-space of degree 2. The paper describes the behavior of these theta constants under the transpose mapping, which can be derived from the Fourier expansions. The results are applied to the theta series of the first and second kind.

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8

Pál, Ambrus. "Theta Series, Eisenstein Series and Poincaré Series over Function Fields." Canadian Journal of Mathematics 56, no. 2 (2004): 406–30. http://dx.doi.org/10.4153/cjm-2004-019-1.

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9

Chen, Bin. "On the dual nature theory of bilateral series associated to mock theta functions." International Journal of Number Theory 14, no. 01 (2017): 63–94. http://dx.doi.org/10.1142/s1793042118500069.

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In recent work, Hickerson and Mortenson introduced a dual notion between Appell–Lerch sums and partial theta functions. In this sense, Appell–Lerch sums and partial theta functions appear to be dual to each other. In this paper, by making the substitution [Formula: see text] in the tail of the associated bilateral series of mock theta functions and universal mock theta functions, we demonstrate how to obtain duals of the second type in terms of Appell–Lerch sums defined by Mortenson for such functions. Then by using the substitution [Formula: see text] in duals of the second type of each bilat

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10

Duverney, Daniel, Keiji Nishioka, Kumiko Nishioka, and Iekata Shiokawa. "Transcendence of Jacobi's theta series." Proceedings of the Japan Academy, Series A, Mathematical Sciences 72, no. 9 (1996): 202–3. http://dx.doi.org/10.3792/pjaa.72.202.

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11

Friedberg, Solomon. "Differential operators and theta series." Transactions of the American Mathematical Society 287, no. 2 (1985): 569. http://dx.doi.org/10.1090/s0002-9947-1985-0768726-4.

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12

Eholzer, W., and N. P. Skoruppa. "Conformal characters and theta series." Letters in Mathematical Physics 35, no. 3 (1995): 197–211. http://dx.doi.org/10.1007/bf00761292.

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13

Deitmar, Anton, and Aloys Krieg. "Theta correspondence for Eisenstein series." Mathematische Zeitschrift 208, no. 1 (1991): 273–88. http://dx.doi.org/10.1007/bf02571525.

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14

Lee, Min Ho. "On generalized theta series liftings." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 62, no. 2 (1997): 229–38. http://dx.doi.org/10.1017/s1446788700000781.

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AbstractWe generalize dual reductive pairs by using reductive groups that are not necessarily subgroups of symplectic groups and construct the corresponding theta-series liftings for certain types of automorphic forms. We also discuss connections of such generalized theta-series liftings with families of abelian varieties parametrized by an arithmetic variety.

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15

Srivastava, Bhaskar. "Theta functions and Eisenstein series." Boletín de la Sociedad Matemática Mexicana 21, no. 2 (2014): 189–203. http://dx.doi.org/10.1007/s40590-014-0044-4.

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16

Kahl, Helmut, and Günter Köhler. "Components of Hecke Theta Series." Journal of Mathematical Analysis and Applications 232, no. 2 (1999): 312–31. http://dx.doi.org/10.1006/jmaa.1999.6274.

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17

Chu, Wenchang. "Twisted cubic theta hypergeometric series." Mathematical Methods in the Applied Sciences 44, no. 1 (2020): 239–52. http://dx.doi.org/10.1002/mma.6723.

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18

Han, Li, Yong Li, David Sauzin, and Shanzhong Sun. "Resurgence and Partial Theta Series." Functional Analysis and Its Applications 57, no. 3 (2023): 248–65. http://dx.doi.org/10.1134/s001626632303005x.

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19

Dummigan, Neil, and Pham Huu Tiep. "Congruences for Certain Theta Series." Journal of Number Theory 71, no. 1 (1998): 86–105. http://dx.doi.org/10.1006/jnth.1998.2234.

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20

Böcherer, Siegfried, Jens Funke, and Rainer Schulze-Pillot. "Trace Operator and Theta Series." Journal of Number Theory 78, no. 1 (1999): 119–39. http://dx.doi.org/10.1006/jnth.1999.2404.

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21

Hu, Qiuxia, Hanfei Song, and Zhizheng Zhang. "Third-order mock theta functions." International Journal of Number Theory 16, no. 01 (2019): 91–106. http://dx.doi.org/10.1142/s1793042120500050.

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In [G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook, Part II (Springer, New York, 2009), Entry 3.4.7, p. 67; Y.-S. Choi, The basic bilateral hypergeometric series and the mock theta functions, Ramanujan J. 24(3) (2011) 345–386; B. Chen, Mock theta functions and Appell–Lerch sums, J. Inequal Appl. 2018(1) (2018) 156; E. Mortenson, Ramanujan’s radial limits and mixed mock modular bilateral [Formula: see text]-hypergeometric series, Proc. Edinb. Math. Soc. 59(3) (2016) 1–13; W. Zudilin, On three theorems of Folsom, Ono and Rhoades, Proc. Amer. Math. Soc. 143(4) (2015) 1471–1476], the au

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22

Kuryliak, M. R., and O. B. Skaskiv. "On the domain of convergence of general Dirichlet series with complex exponents." Carpathian Mathematical Publications 15, no. 2 (2023): 594–607. http://dx.doi.org/10.15330/cmp.15.2.594-607.

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Let $(\lambda_n)$ be a sequence of the pairwise distinct complex numbers. For a formal Dirichlet series $F(z)=\sum\limits_{n=0}^{+\infty} a_ne^{z\lambda_n}$, $z\in\mathbb{C}$, we denote $G_{\mu}(F),$ $G_{c}(F),$ $G_{a}(F)$ the domains of the existence, of the convergence and of the absolute convergence of maximal term $\mu(z,F)=\max\big\{|a_n|e^{\Re(z\lambda_n)} : n\geq 0\big\}$, respectively. It is well known that $G_\mu(F), G_a(F)$ are convex domains. Let us denote $\mathcal{N}_1(z):=\{n : \Re(z\lambda_n)>0\}$, $\mathcal{N}_2(z):=\{n : \Re(z\lambda_n)<0\}$ and \[\alpha^{(1)}(\the

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23

Chen, Bin. "Bilateral series and Ramanujan radial limits of mock (false) theta functions." International Journal of Mathematics 30, no. 04 (2019): 1950023. http://dx.doi.org/10.1142/s0129167x1950023x.

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Ramanujan gave a list of seventeen functions which he called mock theta functions. For one of the third-order mock theta functions [Formula: see text], he claimed that as [Formula: see text] approaches an even order [Formula: see text] root of unity [Formula: see text], then [Formula: see text] He also pointed at the existence of similar properties for other mock theta functions. Recently, [J. Bajpai, S. Kimport, J. Liang, D. Ma and J. Ricci, Bilateral series and Ramanujan’s radial limits, Proc. Amer. Math. Soc. 143(2) (2014) 479–492] presented some similar Ramanujan radial limits of the fifth

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24

Kim, Dae-Yeoul, and Ja-Kyung Koo. "A REMARK OF EISENSTEIN SERIES AND THETA SERIES." Bulletin of the Korean Mathematical Society 39, no. 2 (2002): 299–307. http://dx.doi.org/10.4134/bkms.2002.39.2.299.

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25

Jennings-Shaffer, Chris, and Antun Milas. "On q-series identities for false theta series." Advances in Mathematics 375 (December 2020): 107411. http://dx.doi.org/10.1016/j.aim.2020.107411.

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26

WALLING, LYNNE H. "ACTION OF HECKE OPERATORS ON SIEGEL THETA SERIES, II." International Journal of Number Theory 04, no. 06 (2008): 981–1008. http://dx.doi.org/10.1142/s1793042108001845.

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We apply the Hecke operators T(p)2 and [Formula: see text](1 ≤ j ≤ n ≤ 2k) to a degree n theta series attached to a rank 2k ℤ-lattice L equipped with a positive definite quadratic form in the case that L/pL is regular. We explicitly realize the image of the theta series under these Hecke operators as a sum of theta series attached to certain sublattices of [Formula: see text], thereby generalizing the Eichler Commutation Relation. We then show that the average theta series (averaging over isometry classes in a given genus) is an eigenform for these operators. We explicitly compute the eigenval

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27

Xu, Xiaoping. "Theta Series of Unimodular Lattices, Combinatorial Identities and Weighted Symmetric Polynomials." Algebra Colloquium 13, no. 01 (2006): 67–86. http://dx.doi.org/10.1142/s1005386706000101.

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Hecke proved that the theta series of a positive definite even unimodular lattice is a polynomial of the well-known Essenstein series E4(z) and the Ramanujan series Δ24(z). A natural question is what kind of polynomials in E4(z) and Δ24(z) could be the theta series of positive definite even unimodular lattices. In this paper, we find two combinatorial identities on the theta series of the root lattices of the finite-dimensional simple Lie algebras of type D4n and the cosets in their integral duals, in terms of E4(z) and Δ24(z). Using these two identities, we prove that three families of weight

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28

Mortenson, Eric. "Ramanujan's Radial Limits and Mixed Mock Modular Bilateralq-Hypergeometric Series." Proceedings of the Edinburgh Mathematical Society 59, no. 3 (2015): 787–99. http://dx.doi.org/10.1017/s0013091515000425.

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AbstractUsing results from Ramanujan's lost notebook, Zudilin recently gave an insightful proof of a radial limit result of Folsomet al.for mock theta functions. Here we see that Mortenson's previous work on the dual nature of Appell–Lerch sums and partial theta functions and on constructing bilateralq-series with mixed mock modular behaviour is well suited for such radial limits. We present five more radial limit results, which follow from mixed mock modular bilateralq-hypergeometric series. We also obtain the mixed mock modular bilateral series for a universal mock theta function of Gordon a

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29

Hu, Qiuxia, Bilal Khan, Serkan Araci, and Mehmet Acikgoz. "New double-sum expansions for certain Mock theta functions." AIMS Mathematics 7, no. 9 (2022): 17225–35. http://dx.doi.org/10.3934/math.2022948.

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<abstract><p>The study of expansions of certain mock theta functions in special functions theory has a long and quite significant history. Motivated by recent correlations between $ q $-series and mock theta functions, we establish a new $ q $-series transformation formula and derive the double-sum expansions for mock theta functions. As an application, we state new double-sum representations for certain mock theta functions.</p></abstract>

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30

YAMAUCHI, ATSUO, and HIRO-AKI NARITA. "SOME VECTOR-VALUED SINGULAR AUTOMORPHIC FORMS ON U(2, 2) AND THEIR RESTRICTION TO Sp(1, 1)." International Journal of Mathematics 23, no. 10 (2012): 1250104. http://dx.doi.org/10.1142/s0129167x12501042.

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In this paper we provide a construction of theta series on the real symplectic group of signature (1,1) or the 4-dimensional hyperbolic space. We obtain these by considering the restriction of some vector-valued singular theta series on the unitary group of signature (2,2) to this indefinite symplectic group. Our (vector-valued) theta series are proved to have algebraic Fourier coefficients, and lead to a new explicit construction of automorphic forms generating quaternionic discrete series representations and automorphic functions on the hyperbolic space.

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31

Koizumi, Shoji. "Theta series and the Poincaré divisor." Proceedings of the Japan Academy, Series A, Mathematical Sciences 65, no. 7 (1989): 263–67. http://dx.doi.org/10.3792/pjaa.65.263.

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32

Niwa, Shinji. "On theta correspondences for Eisenstein series." Proceedings of the Japan Academy, Series A, Mathematical Sciences 83, no. 9-10 (2007): 161–66. http://dx.doi.org/10.3792/pjaa.83.161.

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33

Lee, Min Ho. "Theta series associated to symmetric matrices." Complex Variables and Elliptic Equations 61, no. 1 (2015): 15–22. http://dx.doi.org/10.1080/17476933.2015.1051478.

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34

Li, Jian-Shu. "Distinguished cusp forms are theta series." Duke Mathematical Journal 59, no. 1 (1989): 175–89. http://dx.doi.org/10.1215/s0012-7094-89-05905-x.

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35

Garcia, Luis E. "Superconnections, theta series, and period domains." Advances in Mathematics 329 (April 2018): 555–89. http://dx.doi.org/10.1016/j.aim.2017.12.021.

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36

Hung, D. C. "Theta series of ternary quadratic forms." Journal of Number Theory 26, no. 1 (1987): 1–7. http://dx.doi.org/10.1016/0022-314x(87)90091-6.

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37

Bröker, Reinier, and Jeff Hoffstein. "Fourier coefficients of sextic theta series." Mathematics of Computation 85, no. 300 (2015): 1901–27. http://dx.doi.org/10.1090/mcom3044.

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38

Schulze-Pillot, Rainer. "Some congruences for Siegel theta series." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 85, no. 2 (2015): 181–85. http://dx.doi.org/10.1007/s12188-015-0108-z.

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39

Sun, Qingfeng. "Shifted convolution sums involving theta series." Ramanujan Journal 44, no. 1 (2017): 13–36. http://dx.doi.org/10.1007/s11139-017-9900-y.

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40

Freitag, E. "Siegel eisenstein series of arbitrary level and theta series." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 66, no. 1 (1996): 229–47. http://dx.doi.org/10.1007/bf02940806.

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41

Rudd-Barnard, Alexandra, Sarah Jarvandi, Roxanne Rapoport, Sue Smith, and Natalia Witkowska. "A-79 Case Series Evaluating Quantitative Electroencephalograph and Neuropsychological Function in Neurological Lyme Disease." Archives of Clinical Neuropsychology 36, no. 6 (2021): 1123–24. http://dx.doi.org/10.1093/arclin/acab062.97.

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Abstract Objectives The purpose of this study was to investigate the characteristics of physician diagnosed Neurological Lyme disease (NLD) using Quantitative EEG and the Repeatable Battery for the Assessment of Neuropsychological Status (RBANS). We hypothesize that findings would include more slow wave (Delta/Theta) activity that is consistent with the severity reported dysfunction. Methods Subjects consisted of four adult females with a physician provided diagnosis of NLD. EEG was recorded from 21 sites during an eyes open and eyes-closed resting conditions. Raw EEG data was made quantifiabl

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42

Patane, Frank. "A proof of Hecke’s formula for binary quadratic forms." International Journal of Number Theory 16, no. 02 (2019): 233–40. http://dx.doi.org/10.1142/s179304212050013x.

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In Mathematische Werke, Hecke defines the operator [Formula: see text] and describes their utility in conjunction with theta series of quadratic forms. In particular, he shows that the image of theta series associated to classes of binary quadratic forms in CL[Formula: see text] is again a theta series associated to a collection of forms in CL[Formula: see text]. We state and prove an explicit formula for the action of [Formula: see text] on a binary quadratic form of negative discriminant.

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43

Kuryliak, A. O., and V. L. Tsvigun. "Wiman's inequality for analytic functions in $\mathbb{D}\times\mathbb{C}$ with rapidly oscillating coefficients." Carpathian Mathematical Publications 10, no. 1 (2018): 133–42. http://dx.doi.org/10.15330/cmp.10.1.133-142.

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Let $\mathcal{A}^2$ be a class of analytic functions $f$ represented by power series of the from $$ f(z)=f(z_1,z_2)=\sum^{+\infty}_{n+m=0}a_{nm}z_1^nz^m_2$$ with the domain of convergence $\mathbb{T}=\{ z\in \mathbb{C}^2 \colon |z_1|<1, |z_2|<+\infty \} $ such that $\frac{\partial}{\partial z_2}f(z_1,z_2)\not\equiv0$ in $\mathbb{T}$ and there exists $r_0=(r^0_1, r^0_2)\in [0,1)\times[0,+\infty)$ such that for all $r\in(r^0_1,1)\times(r^0_2,+\infty)$ we have $ r_1\frac{\partial}{\partial r_1}\ln M_f(r)+\ln r_1>1, \ $ where $M_f(r)=\sum_{n+m=0}^{+\infty}|a_{nm}|r_1^nr_2^m.$ Let $K(f,\th

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44

Xia, Ernest X. W., and Olivia X. M. Yao. "Eisenstein Series Identities Involving the Borweins' Cubic Theta Functions." Journal of Applied Mathematics 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/181264.

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Based on the theories of Ramanujan's elliptic functions and the (p,k)-parametrization of theta functions due to Alaca et al. (2006, 2007, 2006) we derive certain Eisenstein series identities involving the Borweins' cubic theta functions with the help of the computer. Some of these identities were proved by Liu based on the fundamental theory of elliptic functions and some of them may be new. One side of each identity involves Eisenstein series, the other products of the Borweins' cubic theta functions. As applications, we evaluate some convolution sums. These evaluations are different from the

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45

WALLING, LYNNE H. "ACTION OF HECKE OPERATORS ON SIEGEL THETA SERIES I." International Journal of Number Theory 02, no. 02 (2006): 169–86. http://dx.doi.org/10.1142/s1793042106000516.

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We apply the Hecke operators T(p) and [Formula: see text] to a degree n theta series attached to a rank 2k ℤ-lattice L, n ≤ k, equipped with a positive definite quadratic form in the case that L/pL is hyperbolic. We show that the image of the theta series under these Hecke operators can be realized as a sum of theta series attached to certain closely related lattices, thereby generalizing the Eichler Commutation Relation (similar to some work of Freitag and of Yoshida). We then show that the average theta series (averaging over isometry classes in a given genus) is an eigenform for these opera

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46

Kumar, B. R. Srivatsa, Shruthi, and D. Anu Radha. "Relation between Borweins’ Cubic Theta Functions and Ramanujan’s Eisenstein Series." Journal of Applied Mathematics 2021 (May 8, 2021): 1–6. http://dx.doi.org/10.1155/2021/6614572.

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Two-dimensional theta functions were found by the Borwein brothers to work on Gauss and Legendre’s arithmetic-geometric mean iteration. In this paper, some new Eisenstein series identities are obtained by using ( p , k )-parametrization in terms of Borweins’ theta functions.

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47

Funke, Jens, and Stephen Kudla. "On some incomplete theta integrals." Compositio Mathematica 155, no. 9 (2019): 1711–46. http://dx.doi.org/10.1112/s0010437x19007504.

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In this paper we construct indefinite theta series for lattices of arbitrary signature $(p,q)$ as ‘incomplete’ theta integrals, that is, by integrating the theta forms constructed by the second author with J. Millson over certain singular $q$-chains in the associated symmetric space $D$. These chains typically do not descend to homology classes in arithmetic quotients of $D$, and consequently the theta integrals do not give rise to holomorphic modular forms, but rather to the non-holomorphic completions of certain mock modular forms. In this way we provide a general geometric framework for the

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48

Shavgulidze, K. "On the Dimension of Some Spaces of Generalized Ternary Theta-Series." gmj 9, no. 1 (2002): 167–78. http://dx.doi.org/10.1515/gmj.2002.167.

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Abstract The upper bound of dimension of vector spaces of generalized theta-series corresponding to some ternary quadratic forms is established. In a number of cases, the dimension of vector spaces of generalized theta-series is established and bases of these spaces are constructed.

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49

LIU, ZHI-GUO. "SOME INVERSE RELATIONS AND THETA FUNCTION IDENTITIES." International Journal of Number Theory 08, no. 08 (2012): 1977–2002. http://dx.doi.org/10.1142/s1793042112501126.

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Two pairs of inverse relations for elliptic theta functions are established with the method of Fourier series expansion, which allow us to recover many classical results in theta functions. Many nontrivial new theta function identities are discovered. Some curious trigonometric identities are derived.

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50

Srivastava, Bhaskar. "A Mock Theta Function of Second Order." International Journal of Mathematics and Mathematical Sciences 2009 (2009): 1–15. http://dx.doi.org/10.1155/2009/978425.

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We consider the second-order mock theta function (), which Hikami came across in his work on mathematical physics and quantum invariant of three manifold. We give their bilateral form, and show that it is the same as bilateral third-order mock theta function of Ramanujan. We also show that the mock theta function () outside the unit circle is a theta function and also write as a coefficient of of a theta series. First writing as a coefficient of a theta function, we prove an identity for .

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