Let $e$ and $q$ be fixed co-prime integers satisfying $1\lt e\lt q$. Let $\mathscr {C}$ be a certain family of deformations of the curve $y^e=x^q$. That family is called the $(e,q)$-curve and is one of the types of curves called plane telescopic curves. Let $\varDelta$ be the discriminant of $\mathscr {C}$. Following pioneering work by Buchstaber and Leykin (BL), we determine the canonical basis $\{ L_j \}$ of the space of derivations tangent to the variety $\varDelta =0$ and describe their specific properties. Such a set $\{ L_j \}$ gives rise to a system of linear partial differential equations (heat equations) satisfied by the function $\sigma (u)$ associated with $\mathscr {C}$, and eventually gives its explicit power series expansion. This is a natural generalisation of Weierstrass’ result on his sigma function. We attempt to give an accessible description of various aspects of the BL theory. Especially, the text contains detailed proofs for several useful formulae and known facts since we know of no works which include their proofs.

H. H. Chan, K. S. Chua and P. Solé [‘Quadratic iterations to $\pi $ associated to elliptic functions to the cubic and septic base’, Trans. Amer. Math. Soc. 355(4) (2002), 1505–1520] found that, for each positive integer d, there are theta series $A_d, B_d$ and $C_d$ of weight one that satisfy the Pythagoras-like relationship $A_d^2=B_d^2+C_d^2$. In this article, we show that there are two collections of theta series $A_{b,d}, B_{b,d}$ and $C_{b,d}$ of weight one that satisfy $A_{b,d}^2=B_{b,d}^2+C_{b,d}^2,$ where b and d are certain integers.

Let ${\mathcal {E}}$ be a complex elliptic curve and S be a non-empty finite subset of ${\mathcal {E}}$. We show that the functions $\tilde {\Gamma }$ introduced in [BDDT] out of string theory motivations give rise to a basis (as a vector space) of the minimal algebra $A_{{\mathcal {E}}{\smallsetminus } S}$ of holomorphic multivalued functions on ${\mathcal {E}}{\smallsetminus } S$ which is stable under integration, introduced in [EZ]; this basis is alternative to the basis of $A_{{\mathcal {E}}{\smallsetminus } S}$ constructed in loc. cit. using elliptic analogs of the hyperlogarithm functions.

For $r\in(0,1)$, let $\mu \left( r\right) $ be the modulus of the plane Grötzsch ring $\mathbb{B}^2\setminus[0,r]$, where $\mathbb{B}^2$ is the unit disk. In this paper, we prove that

\begin{equation*}\mu \left( r\right) =\ln \frac{4}{r}-\sum_{n=1}^{\infty }\frac{\theta _{n}}{2n}r^{2n},\end{equation*}

with $\theta _{n}\in \left( 0,1\right)$. Employing this series expansion, we obtain several absolutely monotonic and (logarithmically) completely monotonic functions involving $\mu \left( r\right) $, which yields some new results and extend certain known ones. Moreover, we give an affirmative answer to the conjecture proposed by Alzer and Richards in H. Alzer and K. Richards, On the modulus of the Grötzsch ring, J. Math. Anal. Appl. 432(1): (2015), 134–141, DOI 10.1016/j.jmaa.2015.06.057. As applications, several new sharp bounds and functional inequalities for $\mu \left( r\right) $ are established.

The spectrum and orthogonal eigenbasis are computed of a tridiagonal matrix encoding a finite-dimensional reduction of the difference Lamé equation at the single-gap integral value of the coupling parameter. This entails the exact solution, in terms of single-gap difference Lamé wave functions, for the spectral problem of a corresponding open inhomogeneous isotropic $XY$ chain with coupling constants built from elliptic integers.

Let $\Omega $ be a complex lattice which does not have complex multiplication and $\wp =\wp _\Omega $ the Weierstrass $\wp $-function associated with it. Let $D\subseteq \mathbb {C}$ be a disc and $I\subseteq \mathbb {R}$ be a bounded closed interval such that $I\cap \Omega =\varnothing $. Let $f:D\rightarrow \mathbb {C}$ be a function definable in $(\overline {\mathbb {R}},\wp |_I)$. We show that if f is holomorphic on D then f is definable in $\overline {\mathbb {R}}$. The proof of this result is an adaptation of the proof of Bianconi for the $\mathbb {R}_{\exp }$ case. We also give a characterization of lattices with complex multiplication in terms of definability and a nondefinability result for the modular j-function using similar methods.

We prove general Dwork-type congruences for constant terms attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and p-adic analytic properties of functions originating from polynomial solutions modulo $p^s$ of hypergeometric and Knizhnik–Zamolodchikov (KZ) equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application, we show that the simplest example of a p-adic KZ connection has an invariant line subbundle while its complex analog has no nontrivial subbundles due to the irreducibility of its monodromy representation.

The signature four elliptic theory of Ramanujan is provided with a counterpart to the Jacobian modular sine; this counterpart yields natural direct proofs of several hypergeometric identities recorded by Ramanujan, bypassing the signature four transfer principle of Berndt et al. [‘Ramanujan’s theories of elliptic functions to alternative bases’, Trans. Amer. Math. Soc. 347 (1995), 4163–4244].

Let $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ be a lattice in $\mathbb {C}$ with invariants $g_2,g_3$ and $\sigma _{\Omega }(z)$ the associated Weierstrass $\sigma $-function. Let $\eta _1$ and $\eta _2$ be the quasi-periods associated to $\omega _1$ and $\omega _2$, respectively. Assuming $\eta _2/\eta _1$ is a nonzero real number, we give an upper bound for the number of algebraic points on the graph of $\sigma _{\Omega }(z)$ of bounded degrees and bounded absolute Weil heights in some unbounded region of $\mathbb {C}$ in the following three cases: (i) $\omega _1$ and $\omega _2$ algebraic; (ii) $g_2$ and $g_3$ algebraic; (iii) the algebraic points are far from the lattice points.

We develop tools for constructing rigid analytic trivializations for Drinfeld modules as infinite products of Frobenius twists of matrices, from which we recover the rigid analytic trivialization given by Pellarin in terms of Anderson generating functions. One advantage is that these infinite products can be obtained from only a finite amount of initial calculation, and consequently we obtain new formulas for periods and quasi-periods, similar to the product expansion of the Carlitz period. We further link to results of Gekeler and Maurischat on the $\infty $ -adic field generated by the period lattice.

In addition to the features of the two-parameter Chinese restaurant process (CRP), the restaurant under consideration has a cocktail bar and hence allows for a wider range of (bar and table) occupancy mechanisms. The model depends on three real parameters, $\alpha$ , $\theta_1$ , and $\theta_2$ , fulfilling certain conditions. Results known for the two-parameter CRP are carried over to this model. We study the number of customers at the cocktail bar, the number of customers at each table, and the number of occupied tables after n customers have entered the restaurant. For $\alpha>0$ the number of occupied tables, properly scaled, is asymptotically three-parameter Mittag–Leffler distributed as n tends to infinity. We provide representations for the two- and three-parameter Mittag–Leffler distribution leading to efficient random number generators for these distributions. The proofs draw heavily from methods known for exchangeable random partitions, martingale methods known for generalized Pólya urns, and results known for the two-parameter CRP.

In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn–Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function $\Xi (z)={\textrm {e}}^{|z|^{p}}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution is a global-in-time solution or it shall blow up in a finite time if the initial value is regular. In the case of singular initial data, the local-in-time/global-in-time existence and uniqueness are derived. Also, the regularity of the mild solution is investigated. For the IVP associated with the second one, some modifications to the generalized formula are made to deal with the nonlinear term. We also establish some important estimates for the derivatives of resolvent operators, they are the basis for using the Picard sequence to prove the local-in-time existence of the solution.

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.

We introduce a generalization ${\rm{\pounds}}_d^{(\alpha)}(X)$ of the finite polylogarithms ${\rm{\pounds}}_d^{(0)}(X) = {{\rm{\pounds}}_d}(X) = \sum\nolimits_{k = 1}^{p - 1} {X^k}/{k^d}$ , in characteristic p, which depends on a parameter α. The special case ${\rm{\pounds}}_1^{(\alpha)}(X)$ was previously investigated by the authors as the inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential which is instrumental in a grading switching technique for nonassociative algebras. Here, we extend such generalization to ${\rm{\pounds}}_d^{(\alpha)}(X)$ in a natural manner and study some properties satisfied by those polynomials. In particular, we find how the polynomials ${\rm{\pounds}}_d^{(\alpha)}(X)$ are related to the powers of ${\rm{\pounds}}_1^{(\alpha)}(X)$ and derive some consequences.

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 $t(1,1,6,8;n)$.

In a recent study of how the output voltage of a Hall plate is affected by the shape of the plate and the size of its contacts, U. Ausserlechner has come up with a remarkable double integral that can be viewed as a generalisation of the classical elliptic ‘arithmetic–geometric mean (AGM)’ integral. Here we discuss transformation properties of the integral, which were experimentally observed by Ausserlechner, as well as its analytical and arithmetic features including connections with modular forms.

We consider an extension of the Ramanujan series with a variable $x$. If we let $x=x_{0}$, we call the resulting series ‘Ramanujan series with the shift $x_{0}$’. Then we relate these shifted series to some $q$-series and solve the case of level $4$ with the shift $x_{0}=1/2$. Finally, we indicate a possible way towards proving some patterns observed by the author corresponding to the levels $\ell =1,2,3$ and the shift $x_{0}=1/2$.

We obtain explicit expressions for genus 2 degenerate sigma-function in terms of genus 1 sigma-function and elementary functions as solutions of a system of linear partial differential equations satisfied by the sigma-function. By way of application, we derive a solution for a class of generalized Jacobi inversion problems on elliptic curves, a family of Schrödinger-type operators on a line with common spectrum consisting of a point and two segments, explicit construction of a field of three-periodic meromorphic functions. Generators of rank 3 lattice in ℂ2 are given explicitly.

We prove model completeness for the expansion of the real field by the Weierstrass ℘ function as a function of the variable z and the parameter (or period) τ. We need to existentially define the partial derivatives of the ℘ function with respect to the variable z and the parameter τ. To obtain this result, it is necessary to include in the structure function symbols for the unrestricted exponential function and restricted sine function, the Weierstrass ζ function and the quasi-modular form E2 (we conjecture that these functions are not existentially definable from the functions ℘ alone or even if we use the exponential and restricted sine functions). We prove some auxiliary model-completeness results with the same functions composed with appropriate change of variables. In the conclusion, we make some remarks about the non-effectiveness of our proof and the difficulties to be overcome to obtain an effective model-completeness result, and how to extend these results to appropriate expansion of the real field by automorphic forms.