Theta Function Identities Research Articles

Arithmetic properties arising from Ramanujan’s theta functions

We prove some interesting arithmetic properties of theta function identities that are analogous to q-series identities obtained by Michael D. Hirschhorn. In addition, we find infinite family of congruences modulo powers of 2 for representations of a non-negative integer n as \(\triangle _1+4\triangle _2\) and \(\triangle +k\square \).

A note on some modular equations using Theta functions

In this paper, we have given simple proof to the modular equations using theta function identities.

Some congruences for 3-component multipartitions

AbstractLetp3(n) denote the number of 3-component multipartitions ofn. Recently, using a 3-dissection formula for the generating function ofp3(n), Baruah and Ojah proved that forn≥ 0,p3(9n+ 5) ≡ 0 (mod 33) andp3 (9n+ 8) ≡ 0 (mod 34). In this paper, we prove several congruences modulo powers of 3 forp3(n) by using some theta function identities. For example, we prove that forn≥ 0,p3 (243n+ 233) ≡p3 (729n+ 638) ≡ 0 (mod 310).

On Somos’s theta-function identities of level 14

M. Somos discovered around 6200 theta-function identities of different levels using the computer and offered no proof for them. The purpose of this paper is to prove twenty four of his identities of level 14.

New congruences modulo 5 for the number of 2-color partitions

Let pk(n) be the number of 2-color partitions of n where one of the colors appears only in parts that are multiples of k. In this paper, we find some interesting congruences modulo 5 for pk(n) for k∈{2,3,4} by employing Ramanujan's theta function identities and some identities for the Rogers–Ramanujan continued fraction. The congruence for p2(n) was earlier proved by Chen and Lin with the aid of modular forms.

A note for Ramanujan's circular summation formula

Recently, Chan and Liu give a new formula for circular summation of theta functions [see On a new circular summation of theta functions. J Number Theory. 2010;130:1190–1196]. In this note, we further extend their formula and derive the corresponding imaginary transformation and alternating summation formulas. As some applications, some new identities of theta functions are also shown.

The Bailey transform and Hecke–Rogers identities for the universal mock theta functions

Recently, Garvan obtained two-variable Hecke–Rogers identities for three universal mock theta functions g2(z;q), g3(z;q), K(z;q) by using basic hypergeometric functions, and he proposed a problem of finding direct proofs of these identities by using Bailey pair technology. In this paper, we give proofs of Garvan's identities by applying Bailey's transform with the conjugate Bailey pair of Warnaar and three Bailey pairs deduced from two special cases of ψ66 given by Slater. In particular, we obtain a compact form of two-variable Hecke–Rogers identity related to g3(z;q), which implies the corresponding identity given by Garvan. We also extend these two-variable Hecke–Rogers identities into infinite families.

On Ramanujan's modular equation of degree 7

In this paper, we give an alternative proof of a Ramanujan's modular equation of degree 7 by employing certain theta function identities.

Congruences for some l-regular partitions modulo l

Let bl(n) denote the number of l-regular partitions of n. Dandurand and Penniston found numerous congruences modulo l for bl(n), where l∈{5,7,11}. In this paper, employing some theta function identities due to Ramanujan, we show that bl(A(k)n+B(k))≡C(k)bl(n)(modl), where A(k), B(k) and C(k) are functions in k and l∈{13,17,19}. As corollaries of these congruences, several strange congruences for bl(n) modulo l are derived. For example, b19(123×510k−34)≡0(mod19).

Two theta-function identities for the Ramanujan–Selberg continued fraction and applications

We prove two theta-function identities for the Ramanujan–Selberg continued fraction which are analogous to those of the Rogers–Ramanujan continued fraction. These identities are then used to prove reciprocity formulas and general theorems for the explicit evaluations of the Ramanujan–Selberg continued fraction.

Seiberg-Witten curves and double-elliptic integrable systems

An old conjecture claims that commuting Hamiltonians of the double-elliptic integrable system are constructed from the theta-functions associated with Riemann surfaces from the Seiberg-Witten family, with moduli treated as dynamical variables and the Seiberg-Witten differential providing the pre-symplectic structure. We describe a number of theta-constant equations needed to prove this conjecture for the $N$-particle system. These equations provide an alternative method to derive the Seiberg-Witten prepotential and we illustrate this by calculating the perturbative contribution. We provide evidence that the solutions to the commutativity equations are exhausted by the double-elliptic system and its degenerations (Calogero and Ruijsenaars systems). Further, the theta-function identities that lie behind the Poisson commutativity of the three-particle Hamiltonians are proven.

Bell’s Ternary Quadratic Forms and Tunnel’s Congruent Number Criterion Revisited

Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an integer as a sum of three squares. We present a modern elementary proof of Bell’s theorem that is based on three standard Ramanujan theta function identities and a set of five so-called three-square identities by Hurwitz. We use Bell’s theorem and a slight extension of it to find explicit and finite computable expressions for Tunnel’s congruent number criterion. It is known that this criterion settles the congruent number problem under the weak Birch-Swinnerton-Dyer conjecture. Moreover, we present for the first time an unconditional proof that a square-free number n 3(mod 8) is not congruent.

Infinite families of arithmetic identities and congruences for bipartitions with 3-cores

Let A3(n) denote the number of bipartitions of n that are 3-cores. By employing Ramanujan's simple theta function identities, we prove that A3(2n+1)=13 σ(6n+5), where σ(n) denotes the sum of the positive divisors of n. We also find several infinite families of arithmetic identities and congruences for A3(n), which include generalizations of some recent results on A3(n) by B.L.S. Lin (2014) [6].

A theta function identity and its applications

Liu (Adv Math 195:1–23, 2005) derived a theta function identity. In this paper, we will give an equivalent form of Liu’s identity, from which some nontrivial identities on circular summation of theta functions are deduced. Using these identities and the method of asymptotic analysis, we also derive some nontrivial finite trigonometric sum identities.

Hecke-type double sums, Appell-Lerch sums, and mock theta functions, I

By developing a connection between partial theta functions and Appell-Lerch sums, we find and prove a formula which expresses Hecke-type double sums in terms of Appell-Lerch sums and theta functions. Not only does our formula prove classical Hecke-type double sum identities such as those found in work Kac and Peterson on affine Lie Algebras and Hecke modular forms, but once we have the Hecke-type forms for Ramanujan's mock theta functions our formula gives straightforward proofs of many of the classical mock theta function identities. In particular, we obtain a new proof of the mock theta conjectures. Our formula also applies to positive-level string functions associated with admissable representations of the affine Lie Algebra $A_1^{(1)}$ as introduced by Kac and Wakimoto.

Elementary proofs of Radu and Sellers’ results for broken 2-diamond partitions

In 2007, Andrews and Paule introduced a new class of combinatorial objects called broken k-diamonds. Let Δk(n) be the number of broken k-diamonds partitions of n. By using MacMahon’s partition analysis, they found that the generating function of Δk(n) is an infinite product, which attracted many mathematicians to study its congruence properties. Recently, Radu and Sellers found infinitely many Ramanujan-like congruences modulo 3 for Δ2(n). In fact, they obtained 3-dissections of the generating function for Δ2(n) modulo 3 by using the theory of modular forms. In this brief note, we aim to present elementary proofs of Radu and Sellers’ results by using several theta function identities due to Ramanujan.

INFINITE FAMILIES OF ARITHMETIC IDENTITIES FOR DOUBLED DISTINCT t-CORES FOR t = 3, 4, …, 10

Relations between doubled distinct and self-conjugate t-cores for some small values of t are derived. By employing Ramanujan's simple theta function identities, several infinite families of arithmetic identities for doubled distinct t-cores for t = 3, 4, …, 10 are found.

Infinite families of arithmetic identities for self-conjugate 5-cores and 7-cores

We find infinite families of arithmetic identities for self-conjugate 5-cores and 7-cores by employing Ramanujan’s simple theta function identities.

New Types of Doubly Periodic Standing Wave Solutions for the Coupled Higgs Field Equation

Based on the Hirota bilinear method and theta function identities, we obtain a new type of doubly periodic standing wave solutions for a coupled Higgs field equation. The Jacobi elliptic function expression and long wave limits of the periodic solutions are also presented. By selecting appropriate parameter values, we analyze the interaction properties of periodic-periodic waves and periodic-solitary waves by some figures.

The t-coefficient method II: A new series expansion formula of theta function products and its implications

By means of Jacobiʼs triple product identity and the t-coefficient method, we establish a general series expansion formula with five free parameters for the product of arbitrary two Jacobi theta functions. It embodies the triple, quintuple, sextuple and septuple theta function product identities and the generalized Schröter formula. As further applications, we also set up a series expansion formula for the product of three theta functions. It not only generalizes Ewellʼs and Chen–Chen–Huangʼs octuple product identities, but also contains three cubic theta function identities due to Farkas–Kra and Ramanujan respectively and the Macdonald identity for the root system A2 as special cases. In the meantime, many other new identities including a new short expression of the triple theta series of Andrews are also presented.