Theta Function Identities Research Articles

New truncated theorems for three classical theta function identities

In 2012, Andrews and Merca derived a truncated version of Euler’s pentagonal number theorem. Their work inspired several mathematicians to work on truncated theta series including Guo and Zeng, who examined two other classical theta series identities of Gauss. In this paper, revisiting these three theta series identities of Euler and Gauss, we obtain new truncated theorems. As corollaries of our results, we obtain infinite families of linear inequalities involving the partition function, the overpartition function and the pod function. These inequalities yield the positivity result of Andrews and Merca on the partition function as well as a conjecture on the overpartition function, which was posed by Andrews–Merca and Guo–Zeng, and proved independently by Mao and Yee. We will also provide a unified combinatorial treatment for our results.

New congruence properties for Ramanujan’s 𝜙 function

In 2012, Chan proved a number of congruences for the coefficients of Ramanujan’s ϕ \phi function. In this paper, we prove some new congruences modulo powers of 2 and 3 for Ramanujan’s ϕ \phi function by employing Newman’s identities and theta function identities.

Ramanujan-type congruences modulo m for (l, m)-regular bipartitions

Let \(B_{l,m}(n)\) denote the number of (l, m)-regular bipartitions of n. Recently, many authors proved several infinite families of congruences modulo 3, 5 and 11 for \(B_{l,m}(n)\). In this paper, we use theta function identities to prove infinite families of congruences modulo m for (l, m)-regular bipartitions, where \(m\in \{7,3,11,13,17\}\).

Relations between Rα, Rβ and Rm functions related to Jacobi’s triple-product identity and the family of theta-function identities

In this paper, the author establishes a set of three new theta-function identities involving Rα, Rβ and Rm functions which are based upon a number of q-product identities and Jacobi’s celebrated triple-product identity. These theta-function identities depict the inter-relationships that exist among theta-function identities and combinatorial partition-theoretic identities. Here, in this paper we answer a open question of Srivastava et al [33], and established relations in terms of Rα, Rβ and Rm (for m = 1, 2, 3), and q-products identities. Finally, we choose to further emphasize upon some close connections with combinatorial partition-theoretic identities.

A universal identity for theta functions of degree eight and applications

Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we will use this identity to reexamine our work in theta function identities in the past two decades. Hundreds of results about elliptic modular functions, both classical and new, are derived from this identity with ease. Essentially, this general theta function identity is a theta identities generating machine. Our investigation shows that many well-known results about elliptic modular functions with different appearances due to Jacobi, Kiepert, Ramanujan and Weierstrass among others, actually share a common source. This paper can also be seen as a summary of my past work on theta function identities. A conjecture is also proposed.

Ranks, cranks for overpartitions and Appell–Lerch sums

The definitions of the rank and crank for overpartitions were given by Bringmann, Lovejoy and Osburn. Let $$\overline{N}(s,l;n)$$ (resp. $$\overline{M}(s,l;n)$$ , $$\overline{M2}(s,l;n)$$ ) denote the number of overpartitions of n with rank (resp. the first residual crank, the second residual crank) congruent to s modulo l. The rank differences of overpartitions modulo 3, 5, 6, 7 and 10 were determined. In this paper, we establish the generating functions for $$\overline{N}(s,l;n)$$ , $$\overline{M}(s,l;n)$$ and $$\overline{M2}(s,l;n)$$ with $$l=4, 8$$ by utilizing Appell–Lerch sums and theta function identities. Moreover, in light of these generating functions, we obtain some equalities and inequalities on ranks and cranks of overpartitions modulo 4 and 8.

Hecke-Rogers double-sums and false theta functions

We develop a setting in which one can evaluate certain Hecke-Rogers series in terms of false theta functions. We apply our setting to recent false theta function identities of Chan and Kim as well as Andrews and Warnaar.

The Kronecker theta function and a decomposition theorem for theta functions I

The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan’s \({}_{1}\psi _{1}\) summation. Using the Kronecker theta function as building blocks, we prove a decomposition theorem for theta functions. This decomposition theorem is the common source of a large number of theta function identities. Many striking theta function identities, both classical and new, are derived from this decomposition theorem with ease. A new addition formula for theta functions is established. Several known results in the theory of elliptic theta functions due to Ramanujan, Weierstrass, Kiepert, Winquist and Shen among others are revisited. A curious trigonometric identity is proved.

Evaluations of the cubic continued fraction by some theta function identities: Revisited

In this paper, we exploit some known theta function identities involving two parameters lk,n and l’k,n for the theta function □ to find about 54 new values of the Ramanujan''s cubic continued fraction.

Some modular equations analogous to Ramanujan’s identities

Srinivasa Ramanujan (1887–1920) recorded many modular equations in his notebooks. In this paper, we establish some new modular equations of signature two and three by using the theta-function identities of composite degrees.

On Ramanujan’s Incomplete Elliptic Integral Identities

In his lost notebook, S. Ramanujan recorded incomplete elliptic integral identities of the first kind. In this paper, we give new proofs to Ramanujan’s incomplete elliptic integrals of the first kind for level 5 using the parameter k(q) = R(q)R2(q2), where R(q) is Rogers-Ramanujan continued fraction. Also, we construct new identities for level 5 similar to ones found in Ramanujan’s work. Further, we prove related identities of level 7, using theta function identities.

Some Ramanujan-type circular summation formulas

In this paper, we give two Ramanujan-type circular summation formulas by applying the way of elliptic functions and the properties of theta functions. As applications, we obtain the corresponding imaginary transformation formulas for Ramanujan-type circular summations and some theta function identities.

A mock theta function identity related to the partition rank modulo 3 and 9

We prove a new mock theta function identity related to the partition rank modulo 3 and 9. As a consequence, we obtain the [Formula: see text]-dissection of the rank generating function modulo [Formula: see text]. We also evaluate all of the components of the rank–crank differences modulo [Formula: see text]. These are analogous to conjectures of Lewis [The generating functions of the rank and crank modulo 8, Ramanujan J. 18 (2009) 121–146] on rank–crank differences modulo [Formula: see text], first proved by Mortenson [On ranks and cranks of partitions modulo 4 and 8, J. Combin. Theory Ser. A 161 (2019) 51–80].

Derivable quadratic forms and representation numbers

Let Fm denote the set of positive-definite primitive integral quadratic forms in m variables. Let f,g∈Fm. In this paper we introduce a new concept, namely that of g being derivable from f. This concept is based on a certain theta function identity being valid. A consequence of this concept is that if g is derivable from f then the representation number of g can be given in terms of that of f. Many examples are given, especially for diagonal ternary quadratic forms.

On q-series identities for false theta series

We prove several infinite families of q-series identities for false theta functions and related series. These identities are motivated by considerations of characters of modules of vertex operator superalgebras and of quantum dilogarithms. We also obtain closely related modular identities of the Göllnitz-Gordon-Andrews type. As a byproduct of our identities, we establish several identities for the Rogers dilogarithm function coming from multi q-hypergeometric series with “double poles”.

A general q-expansion formula based on matrix inversions and its applications

In this paper, by the technique of matrix inversions, we establish a general q-expansion formula of arbitrary formal power series F(z) with respect to the base $$\begin{aligned} \left\{ z^n\frac{(az)_{n}}{(bz)_{n}}\bigg |n=0,1,2,\ldots \right\} . \end{aligned}$$ Some concrete expansion formulas and their applications to q-series identities are presented, including Carlitz’s q-expansion formula and a new partial theta function identity as well as a coefficient identity for Ramanujan’s $${}_1\psi _1$$ summation formula as special cases.

Elliptic classes, McKay correspondence and theta identities

We revisit the construction of elliptic class given by Borisov and Libgober for singular algebraic varieties. Assuming torus action we adjust the theory to the equivariant local situation. We study theta function identities having a geometric origin. In the case of quotient singularities {mathbb {C}}^n/G, where G is a finite group the theta identities arise from McKay correspondence. The symplectic singularities are of special interest. The Du Val surface singularity A_n leads to a remarkable formula.

New theta function identities for a continued fraction of Ramanujan and their applications

We prove three new theta function identities for the continued fraction H(q) defined by $$\begin{aligned} H(q):=q^{1/8}-\frac{q^{7/8}}{1-q}_{+}\frac{q^2}{1+q^2}_{-}\frac{q^3}{1-q^3}_{+}\frac{q^4}{1+q^4}_{- \cdots }, \vert q\vert <1. \end{aligned}$$ The theta-function identities are then used to prove integral representations for the continued fraction H(q). We also prove general theorems and reciprocity formulas for the explicit evaluation of the continued fraction H(q). The results are analogous to those of the famous Rogers-Ramanujan continued fraction.

Proofs for certain Πq-identities of W. Gosper

In 2001 W. Gosper introduced constant Πq, and conjectured many intriguing identities on this constant. In this paper we employ some knowledge of modular equations with degree 5 to confirm several of Gosper's Πq-identities. Two strange q-identities involving Πq and a Lambert series conjectured by W. Gosper are proved. As a consequence, a q-identity involving Πq and two Lambert series, which was conjectured by Gosper, is proved. As an application, we confirm an interesting q-trigonometric identity of Gosper, which can be viewed as a theta function analogue for the related well-known trigonometric identity. Our proof relies on an interesting general theta function identity of Zhi-Guo Liu.

TWO THETA-FUNCTION IDENTITIES OF LEVEL 10

In this paper, we prove two theta-function identities using modular equation of degree 3. Furthermore, as an application of this we establish combinatorial interpretations of colored partitions. 1. INTRODUCTION Ramanujan’s theta function f(x, y) is defined as f(x, y) := X∞ n=−∞ x n(n+1)/2 y n(n−1)/2 |xy| < 1. The function f(x, y) enjoys the well-known Jacobi’s triple-product identity [5, p. 35] given by f(x, y) = (−x; xy)∞(−y; xy)∞(xy; xy)∞ , where here and throughout the paper, we assume |q| < 1 and employ the standard notation (x; q)∞ := Y∞ n=0 (1 − xqn ).