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Quasi-periodic solutions of the KP hierarchy acted by vertex operators are studied. We show, with the aid of the Sato Grassmannian, that solutions thus constructed correspond to torsion free rank one sheaves on some singular algebraic curves whose normalizations are the non-singular curves corresponding to the seed quasi-periodic solutions. It means that the action of the vertex operator has an effect of creating singular points on an algebraic curve. We further check, by examples, that solutions obtained here can be considered as solitons on quasi-periodic backgrounds, where the soliton matrices are determined by parameters in the vertex operators. [ABSTRACT FROM AUTHOR]

For an ample line bundle L on an Abelian variety M, we study the theta functions associated with the family of line bundles L ⊗ T on M indexed by T ∈ Pic 0 (M) . Combined with an appropriate differential geometric setting, this leads to an explicit curvature computation of the direct image bundle E on Pic 0 (M) , whose fiber E T is the vector space spanned by the theta functions for the line bundle L ⊗ T on M. Some algebro-geometric properties of E are also remarked. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sums of four squares of integers whose prime factorizations are restricted, making progress towards a conjecture of Sun that states that two of the integers may be restricted to the forms 2 a 3 b and 2 c 5 d. We obtain an ineffective generalization of results of Gauss and Legendre on sums of three squares and an effective generalization of Lagrange's four-square theorem. [ABSTRACT FROM AUTHOR]

With the Hirota's bilinear method, N-periodic wave solutions to the coupled nonlinear Klein-Gordon equation are constructed in terms of theta function. The asymptotic behaviors under small amplitude limits are also deduced strictly for the one and two-periodic waves. Moreover, a numerical scheme is presented for the case of N\geq 3. Some three-periodic wave solutions are calculated numerically as examples. [ABSTRACT FROM AUTHOR]

In this article, we discuss and survey the recent progress towards the Schottky problem, and make some comments on the relations between the André-Oort conjecture, Okounkov convex bodies, Coleman's conjecture, stable modular forms, Siegel-Jacobi spaces, stable Jacobi forms and the Schottky problem. [ABSTRACT FROM AUTHOR]

In the eleventh paper in the series on MacMahon's partition analysis, Andrews and Paule introduced the k -elongated partition diamonds. Recently, they revisited the topic. Let d k (n) count the partitions obtained by adding the links of the k -elongated plane partition diamonds of length n. Andrews and Paule obtained several generating functions and congruences for d 1 (n) , d 2 (n) , and d 3 (n). Da Silva et al. further found many congruences modulo 4, 5, 7, 8, 9, and 11 for d k (n). In this paper, we extend some individual congruences found by Andrews and Paule and da Silva et al. to their respective families as well as find new families of congruences for d k (n). We also present a refinement in an existence result for congruences of d k (n) found by da Silva et al. and prove some new individual as well as a few families of congruences modulo 5, 7, 8, 11, 13, 16, 17, 19, 23, 25, 32, 49, 64, and 128. [ABSTRACT FROM AUTHOR]

The aim of the research presented in this paper is to derive the systems of ordinary differential equations (ODEs) satisfied by certain modular forms of level six and to construct extensions of the differential field of the cubic theta functions, generalizing the classical Ramanujan and Halphen fields. We treat both modular forms that appear in the literature and others that do not. We explore applications to number theory and find new relations among the Eisenstein series. Moreover, we find Riccati equations satisfied by modular functions of level six. The ODEs have relationship with some theta constant identities. [ABSTRACT FROM AUTHOR]

In this work, we extend two modular identities due to Cooper and Ye [4] in weight aspect, and obtain two new Ramanujan-Mordell type formulas, which can be viewed as analogues of work of Chan and Cooper [2] in level aspect. [ABSTRACT FROM AUTHOR]

To a positive-definite even lattice Q, one can associate the lattice vertex algebra VQ, and any automorphism σ of Q lifts to an automorphism of VQ. In this paper, we investigate the orbifold vertex algebra VQσ, which consists of the elements of VQ fixed under σ, in the case when σ has prime order. We describe explicitly the irreducible VQσ-modules, compute their characters, and determine the modular transformations of characters. As an application, we find the asymptotic and quantum dimensions of all irreducible VQσ-modules. We consider in detail the cases when the order of σ is 2 or 3, as well as the case of permutation orbifolds. [ABSTRACT FROM AUTHOR]

Recently, da Silva and Sellers [11] studied the arithmetical properties of the coefficients p ξ (n) of the third order mock theta function ξ (q) , which was introduced by Gordon and McIntosh and is defined by ξ (q) = 1 + 2 ∑ n = 1 ∞ q 6 n 2 - 6 n + 1 q ; q 6 n q 5 ; q 6 n = : ∑ n = 0 ∞ p ξ (n) q n. Da Silva and Sellers discovered several congruences modulo 3, 4, 5, 8, and 9 for p ξ (n) . In this paper, we find unexpected congruences modulo some multiples of 5 and 7 for the coefficients of ξ (q) , namely, for all n ≥ 0 , p ξ 1800 n + r ≡ 0 (m o d 24 · 5) for r ∈ { 435 , 795 , 1155 , 1515 } , p ξ 504 n + r ≡ 0 (m o d 576 · 7) for r ∈ { 219 , 291 , 435 }. We also confirm all the conjectural congruences modulo 25 and 125 posed by Zhang and Shi [23] for the coefficients of the sixth order mock theta function β (q) , which is defined by β (q) = ∑ n = 0 ∞ q 3 n 2 + 3 n + 1 q ; q 3 n + 1 q 2 ; q 3 n + 1 . [ABSTRACT FROM AUTHOR]

The main purpose of this article is to prove two Rogers–Ramanujan–Slater type theta function identities related to φ(q) and ϕ0(q), which were earlier investigated by two legendary mathematicians of their time. The results presented in this paper are motivated essentially by recent works of Cao et al. (see [5]). [ABSTRACT FROM AUTHOR]

If t is a positive integer, then a partition of a non-negative integer n is a t-core if none of the hook numbers of the associated Ferrers-Young diagram is divisible by t. Let a t (n) denote the number of t-core partitions of n. The purpose of this article is to prove some infinite families of congruences modulo 3 for the 5-core partition function a 5 (n) and parity results for the 7-core partition function a 7 (n) by using elementary generating functions and q-series manipulations. We further verify the congruences for the 5-core partition function a 5 (n) using the theory of modular forms. [ABSTRACT FROM AUTHOR]

The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan's 1 ψ 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. [ABSTRACT FROM AUTHOR]

The solutions of multi-breather and high-order rogue waves for the Hirota equation are systematically constructed on the elliptic function background, which were not previously studied. The spectral problem with the elliptic function potential is solved by the modified squared wave function approach and the traveling wave transformation. The fundamental solution of spectral problem is expressed in terms of theta functions. By the elementary Darboux transformation, the multi-breather solutions on different elliptic function backgrounds are constructed including the general breathers, Kuznetsov–Ma breathers and Akhmediev breathers, and their interaction patterns are exhibited by several examples. At branch points for breather solutions, higher-order rogue wave solutions on the elliptic function background are presented by the general Darboux transformation and some limit techniques. It is expected that these nonlinear dynamics of the breathers and rogue waves on the periodic background may help explain the isolated steep wave events in deep ocean or other nonlinear dispersive systems. [ABSTRACT FROM AUTHOR]

The Riemann–Wirtinger integral is an analogue of the hypergeometric integral, which is defined on a one-dimensional complex torus. We study the intersection forms on the twisted homology and cohomology groups associated with the Riemann–Wirtinger integral. We derive explicit formulas of some intersection numbers, and apply them to studies of the monodromy representation, connection problems, and contiguity relations. [ABSTRACT FROM AUTHOR]

In this paper, we establish several expansion formulas for products of the Jacobi theta functions. As applications, we derive some expressions of the powers of (q ; q) ∞ by using these expansion formulas. [ABSTRACT FROM AUTHOR]

We study solutions to the Kadomtsev-Petviashvili equation whose underlying algebraic curves undergo tropical degenerations. Riemann's theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the Hirota variety which parametrizes all tau functions arising from such a sum. We compute tau functions from points on the Sato Grassmannian that represent Riemann-Roch spaces and we present an algorithm that finds a soliton solution from a rational nodal curve. [ABSTRACT FROM AUTHOR]

We evaluate the classic sum ∑ n ∈ Z e - π n 2 . The novelty of our approach is that it does not require any prior knowledge about modular forms, elliptic functions or analytic continuation. Even the Γ function, in terms of which the result is expressed, only appears as a complex function in the computation of a real integral by the residue theorem. Another contribution of this note is to provide a very simple proof of the Kronecker limit formula. Finally, employing the evaluation of the sum and some other ideas, we also obtain an undemanding proof of one of the most emblematic formulas of Ramanujan. [ABSTRACT FROM AUTHOR]

We show that the series expansions of certain q-products have matching coefficients with their reciprocals. Several of the results are associated to Ramanujan's continued fractions. For example, let R(q) denote the Rogers–Ramanujan continued fraction having the well-known q-product representation: R (q) = (q ; q 5) ∞ (q 4 ; q 5) ∞ (q 2 ; q 5) ∞ (q 3 ; q 5) ∞ . If ∑ n = 0 ∞ α (n) q n = 1 R 5 q = ∑ n = 0 ∞ α ′ (n) q n - 1 , ∑ n = 0 ∞ β (n) q n = R (q) R q 16 = ∑ n = 0 ∞ β ′ (n) q n - 1 , then α (5 n + r) = - α ′ (5 n + r - 2) , r ∈ { 3 , 4 } , β (10 n + r) = - β ′ (10 n + r - 6) , r ∈ { 7 , 9 }. [ABSTRACT FROM AUTHOR]

Let N (c 1 , c 2 , ... , c k ; n) , r (c 1 , c 2 , ... , c k ; n) , and t (c 1 , c 2 , ... , c k ; n) count the representations of a positive integer n in the quadratic forms c 1 x 1 2 + x 1 x 2 + x 2 2 + c 2 x 3 2 + x 3 x 4 + x 4 2 + ⋯ + c k (x 2 k - 1 2 + x 2 k - 1 x 2 k + x 2 k 2) , c 1 x 1 2 + c 2 x 2 2 + ⋯ + c k x k 2 , c 1 x 1 (x 1 + 1) / 2 + c 2 x 2 (x 2 + 1) / 2 + ⋯ + c k x k (x k + 1) / 2 , respectively, where c i 's and x i 's are integers. Using Ramanujan's theta functions φ (q) , ψ (q) , Borweins' cubic theta function a(q), and simple dissecting techniques, we find relations satisfied by N (c 1 , c 2 , ... , c k ; n) , r (c 1 , c 2 , ... , c 2 k ; n) , and t (c 1 , c 2 , ... , c 2 k ; n) when 1 < 2 k ≤ 8 . We also deduce some new explicit formulas for r (c 1 , c 2 , c 3 , c 4 ; n) for some particular n when c i ∈ { 1 , 2 , 3 , 6 , 8 , 9 , 16 , 18 , 24 , 48 }. [ABSTRACT FROM AUTHOR]

Let B ¯ ℓ (n) denote the number of ℓ - regular overpartition pairs of n. In this paper, we find some Ramanujan like congruences and infinite families of congruences for B ¯ ℓ (n) for ℓ ∈ { 3 , 4 , 5 , 8 } . For example, for n ≥ 0 and α ≥ 0 , B ¯ 3 (4 α + 1 (6 n + 5)) ≡ 0 (mod 9) , B ¯ 3 (12 n + 10) ≡ 0 (mod 9) , B ¯ 5 (4 n + 3) ≡ 0 (mod 8) and B ¯ 8 (16 n + 12) ≡ 0 (mod 16) . [ABSTRACT FROM AUTHOR]

The mirror dual of a smooth toric Fano surface X equipped with an anticanonical divisor E is a Landau–Ginzburg model with superpotential, W. Carl–Pumperla–Siebert give a definition of the superpotential in terms of tropical disks [13] using a toric degeneration of the pair (X , E). When E is smooth, the superpotential is proper. We show that this proper superpotential equals the open mirror map for outer Aganagic–Vafa branes in the canonical bundle K X , in framing zero. As a consequence, the proper Landau–Ginzburg potential is a solution to the Lerche–Mayr Picard–Fuchs equation. Along the way, we prove a generalization of a result about relative Gromov–Witten invariants by Cadman–Chen to arbitrary genus using the multiplication rule of quantum theta functions. In addition, we generalize a theorem of Hu that relates Gromov–Witten invariants of a surface under a blow-up from the absolute to the relative case. One of the two proofs that we give introduces birational modifications of a scattering diagram. We also demonstrate how the Hori–Vafa superpotential is related to the proper superpotential by mutations from a toric chamber to the unbounded chamber of the scattering diagram. [ABSTRACT FROM AUTHOR]

The classical Weierstrass theta identity plays a central role in the study of modular, elliptic and theta hypergeometric series. In this paper, we establish a multi-parameter Weierstrass theta identity with the use of the Cauchy determinant and an anti-symmetric difference of a product of two theta functions. Some applications of this generalized Weierstrass theta identity to basic and elliptic hypergeometric series are also discussed. [ABSTRACT FROM AUTHOR]

Based on the values of the Weierstrass elliptic function ℘ (z | τ) at z = π τ / 2 , (π + π τ) / 2 , (π + π τ) / 4 , (π + 2 π τ) / 4 and the theory of modular forms on the arithmetic group Γ 0 (2) , we decompose θ 2 2 n (τ) as sum of Eisenstein series and cusp forms. Using the recurrence relation of ℘ (2 n) (z | τ) , we provide an algorithm to determine the exact form of these cusp forms. [ABSTRACT FROM AUTHOR]

We prove a formula, which, given a principally polarized abelian variety $(A,\lambda)$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the Néron–Tate height of a symmetric theta divisor on $A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our formula can be expressed as the tropical moments of the tropicalizations of $(A,\lambda)$. [ABSTRACT FROM AUTHOR]

In this paper, we establish several expansion formulas for products of the Jacobi theta functions. As applications, we derive some expressions of the powers of (q;q)∞ by using these expansion formulas.

The notion of duality is a key element in understanding the interplay between the stability and chromatic numbers of a graph. This notion is a central aspect in the celebrated theory of perfect graphs and is further and deeply developed in the context of the Lovász theta function and its equivalent characterizations and variants. The main achievement of this paper is the introduction of a new family of norms, providing upper bounds for the stability number, that are obtained through duality from the norms motivated by Hoffman's lower bound for the chromatic number, and which achieve the (complementary) Lovász theta function at their optimum. As a consequence, our norms make it formal that Hoffman's bound for the chromatic number and the Delsarte-Hoffman ratio bound for the stability number are indeed dual. Further, we show that our new bounds strengthen the convex quadratic bounds for the stability number studied by Luz and Schrijver, and which achieve the Lovász theta function at their optimum. One of the key observations regarding weighted versions of these bounds is that, for any upper bound for the stability number of a graph which is a positive definite monotone gauge function, its gauge dual is a lower bound on the fractional chromatic number, and conversely. Our presentation is elementary and accessible to a wide audience. [ABSTRACT FROM AUTHOR]

A differential-difference operator is used to model the heat equation on a finite graph analogue of Poincaré's upper half-plane. Finite analogues of the classical theta functions are shown to be solutions to the heat equation in this setting. [ABSTRACT FROM AUTHOR]

In this paper, we consider adiabatic deformations of Riemann surfaces that preserve the integrability of the corresponding dynamic system, which leads to the appearance of modulated quasiperiodic motions, similar to the effect of parametric resonance. We show that in this way it is possible to control the amplitude and frequency of nonlinear modes. We consider several examples of the dynamics of top-type systems. [ABSTRACT FROM AUTHOR]

In this paper, we construct extensions of the differential field obtained from Halphen's system of ordinary differential equations (ODEs) using quasi-modular forms of levels two, four, and six. In addition, we derive analogs of Jacobi's derivative formula and Jacobi's quartic identity using theta functions with rational characteristics. Moreover, we also treat representability as sums of squares, sums of triangular numbers and mixed sums of squares and triangular numbers. [ABSTRACT FROM AUTHOR]

Ramanujan presented four identities for third-order mock theta functions in his Lost Notebook. In 2005, with the aid of complex analysis, Yesilyurt first proved these four identities. Recently, Andrews et al. proved these identities by using q-series. In this paper, using some identities for the universal mock theta function g (x ; q) = x - 1 - 1 + ∑ n = 0 ∞ q n 2 (x ; q) n + 1 (q x - 1 ; q) n , we provide different proofs of these four identities. [ABSTRACT FROM AUTHOR]

Shimura defined a family of maps from the space of modular forms of half-integral weight to the space of modular forms of integral weight. Selberg in his unpublished work found explicitly this correspondence (the first Shimura map 𝒮 1 ) for the class of forms which are products of a Hecke eigenform of level one and a Jacobi theta function. Later, Cipra generalized the work of Selberg to the case where Jacobi theta functions are replaced by the theta functions associated to Dirichlet character of prime power moduli, and the level one Hecke eigenforms are replaced by newforms of arbitrary level. Hansen and Naqvi generalized Cipra's work (on the image of a class of modular forms under the first Shimura map 𝒮 1 ) to cover theta functions associated to Dirichlet characters of arbitrary moduli. In this paper, we show that the earlier results can be modified to get similar results for the t th Shimura lifts 𝒮 t , for any positive square-free integer t. [ABSTRACT FROM AUTHOR]

We give an algorithm to compute (ℓ , ℓ , ℓ) -isogenies from the Jacobians of genus three hyperelliptic curves to the Jacobians of non-hyperelliptic curves over a finite field of characteristic different from 2 in time O ~ (ℓ 3) , where ℓ is an odd prime which is coprime to the characteristic. An important application is to reduce the discrete logarithm problem in the Jacobian of a hyperelliptic curve to the corresponding problem in the Jacobian of a non-hyperelliptic curve. [ABSTRACT FROM AUTHOR]

The Davey–Stewartson equation has been instrumental in describing various physical phenomena, especially (2+1)-dimensional breathers and rogue waves. In this paper, we present a direct approach to studying the quasi-periodic breathers of the Davey–Stewartson equation. By employing Hirota's bilinear method and leveraging certain identities of theta functions, the problem is transformed into an over-determined nonlinear algebraic system, which can be formulated as a nonlinear least square problem and solved by classical numerical iterative algorithms. Through asymptotic analysis and numerical experiments, we categorize these solutions into three cases: quasi-periodic breathers, quasi-periodic stationary breathers, and quasi-periodic homoclinic orbits. The latter exhibit behavior reminiscent of quasi-periodic rogue waves, which are often observed in oceanic rogue wave phenomena. This work advances our understanding of the dynamics and properties of (2+1)-dimensional quasi-periodic breathers and rouge waves. • Direct investigation of quasi-periodic breathers to (2+1)-dimensional DS equation. • Systematic categorization of quasi-periodic breathers. • An efficient numerical method for the computation of quasi-periodic breathers. [ABSTRACT FROM AUTHOR]

Singular overpartition functions were defined by Andrews. Let Ck,i(n) denote the number of (k, i)-singular overpartitions of n, which counts the number of overpartitions of n in which no part is divisible by k and only parts ±i (mod k) may be overlined. A number of congruences modulo 3, 9 and congruences modulo powers of 2 for Ck,i(n) were discovered by Ahmed and Baruah, Andrews, Chen, Hirschhorn and Sellers, Naika and Gireesh, Shen and Yao for some pairs (k, i). In this paper, we prove some congruences modulo powers of 2 for C48, 6(n) and C48, 18(n).

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 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. [ABSTRACT FROM AUTHOR]

We construct a class of companion elliptic functions associated with even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function ℘ (z | τ) as a blueprint, we will derive their representations in terms of q-series and partial fractions. We also explore the significance of the coefficients of their power series expansions and establish the modular properties under the actions of the arithmetic groups Γ 0 (N) and Γ 1 (N) . [ABSTRACT FROM AUTHOR]

Recently, Romik determined (in: Ramanujan J, 2019, https://doi.org/10.1007/s11139-018-0109-5) the Taylor expansion of the Jacobi theta constant θ 3 , around the point x = 1 . He discovered a new integer sequence, (d (n)) n = 0 ∞ = 1 , 1 , - 1 , 51 , 849 , - 26199 , ... , from which the Taylor coefficients are built, and conjectured that the numbers d(n) satisfy certain congruences. In this paper, we prove some of these conjectures, for example that d (n) ≡ (- 1) n + 1 (mod 5) for all n ≥ 1 , and that for any prime p ≡ 3 (mod 4), d(n) vanishes modulo p for all large enough n. [ABSTRACT FROM AUTHOR]

Let pod(n) denote the number of partitions of an integer n wherein the odd parts are distinct. Recently, a number of congruences for pod(n) have been established. In this paper, we establish the generating function of pod(5n + 2) and then prove new infinite families of congruences modulo 5 and 9 for pod(n) by using the formulas for t3(n) and t5(n), where tk (n) is the number of representations of n as a sum of k triangular numbers. In particular, we generalize a congruence for pod(n) due to Radu and Sellers. [ABSTRACT FROM AUTHOR]

We assign to a non-compact Riemannian symmetric space X a theta function and a zeta function. We compute all of the Minakshisundaram–Pleijel coefficients in a short-time asymptotic expansion of the theta function especially when X is of complex type, which we use to compute the one-loop effective potential—whose relevance for quantum field theory, for example, is briefly commented on. These coefficients, for X of general type, are also shown to be specifically related to residues and special values of the zeta function. The results presented extend previous ones in the literature where the rank of X is assumed to be one. [ABSTRACT FROM AUTHOR]

In this paper, we focus on applications of high-level versions of Jacobi's derivative formula to number theory, such as quaternary quadratic forms and convolution sums of some arithmetical functions. [ABSTRACT FROM AUTHOR]

We consider a deformation $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ of the Dedekind eta function depending on two $d$ -dimensional simple lattices $(L,\unicode[STIX]{x1D6EC})$ and two parameters $(m,t)\in (0,\infty)$ , initially proposed by Terry Gannon. We show that the minimisers of the lattice theta function are the maximisers of $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ in the space of lattices with fixed density. The proof is based on the study of a lattice generalisation of the logarithm, called the lattice logarithm, also defined by Terry Gannon. We also prove that the natural logarithm is characterised by a variational problem over a class of one-dimensional lattice logarithms. [ABSTRACT FROM AUTHOR]

Let $\mathbb{Z}$ and $\mathbb{Z}^{+}$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in \mathbb{Z}^{+}$ , let $t(a,b,c,d;n)$ be the number of representations of $n$ by $\frac{1}{2}ax(x+1)+\frac{1}{2}by(y+1)+\frac{1}{2}cz(z+1)+\frac{1}{2}dw(w+1)$ with $x,y,z,w\in \mathbb{Z}$. Using theta function identities we prove 13 transformation formulas for $t(a,b,c,d;n)$ and evaluate $t(2,3,3,8;n)$ , $t(1,1,6,24;n)$ and $t(1,1,6,8;n)$. [ABSTRACT FROM AUTHOR]

We investigate the arithmetic properties of the second-order mock theta function $B(q)$ and establish two identities for the coefficients of this function along arithmetic progressions. As applications, we prove several congruences for these coefficients. [ABSTRACT FROM AUTHOR]

In the present paper, we establish an elliptic interpolation formula with the help of the (f,g)-inversion formula. As applications, some basic theta function identities are presented, including an equivalent algebraic form of Weierstrass' theta identity. [ABSTRACT FROM AUTHOR]

We study the Taylor expansion around the point x = 1 of a classical modular form, the Jacobi theta constant θ 3 . This leads naturally to a new sequence (d (n)) n = 0 ∞ = 1 , 1 , - 1 , 51 , 849 , - 26199 , ... of integers, which arise as the Taylor coefficients in the expansion of a related "centered" version of θ 3 . We prove several results about the numbers d(n) and conjecture that they satisfy the congruence d (n) ≡ (- 1) n - 1 (mod 5) and other similar congruence relations. [ABSTRACT FROM AUTHOR]

In this paper, we evaluate two parameters hk,n and h ' k,n of some P-Q type Theta functions ϕ(q) for any positive real numbers k and n. During this process, we also evaluate Ramanujan's cubic continued fraction. [ABSTRACT FROM AUTHOR]