Related papers: Lacunary recurrences for Eisenstein series

In this paper we propose a definition of a recurrence relation homomorphism and illustrate our definition with a few examples. We then define the period of a k-th order of linear recurrence relation and deduce certain preliminary results…

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue…

The authors learnt that similar results have been independently found by D.Zagier, V.Baranovsky and V.Balaji/A.King/P.Newstead. The corresponding references have been added (and some typos corrected).

In the former part of this paper, we summarize our previous results on infinite series involving the hyperbolic sine function, especially, with a focus on the hyperbolic sine analogue of Eisenstein series. Those are based on the classical…

Geometric Langlands predicts an isomorphism between Whittaker coefficients of Eisenstein series and functions on the moduli space of $\check{N}$-local systems. We prove this formula by interpreting Whittaker coefficients of Eisenstein…

Analogue of Springer's formula for the Poincar\'e series of the algebra invariants of ternary form is found.

We derive some q-analogs of Euler-Cassini-type identities and of recurrence formulas for powers of Fibonacci polynomials.

Eisenstein series play an important role in the theory of modular forms and have profound connections with $q$-series identities, partition theory, and special functions. Likewise, Ramanujan's mock theta functions, originally introduced in…

A generalization of Serre's $p$-adic Eisenstein series in the case of Siegel modular forms is studied and a coincidence between a $p$-adic Siegel Eisenstein series and a genus theta series associated with a quaternary quadratic form is…

We use the Jacquet-Langlands correspondence to generalize well-known congruence results of Mazur on Fourier coefficients and L-values of elliptic modular forms for prime level in weight 2 both to nonsquare level and to Hilbert modular…

By a classical principle of probability theory, sufficiently thin subsequences of general sequences of random variables behave like i.i.d.\ sequences. This observation not only explains the remarkable properties of lacunary trigonometric…

In this paper we present a simple method for deriving recurrence relations and we apply it to obtain two equations involving the Lerch Phi function and sums of Bernoulli and Euler polynomials. Connections between these results and those…

An interplay between the Lambert series and Euler's Pentagonal Number Theorem gives an Euler-type recurrence relation for any given arithmetical function. As consequences of this, we present Euler-type recurrence relations for some…

Using a structure theorem from [FG2010] we prove a version of multiple recurrence for sets of positive measure in a general stationary dynamical system.

A set of nonlinear differential equations associated with the Eisenstein series of the congruent subgroup $\Gamma_0(2)$ of the modular group $SL_2(\mathbb{Z})$ is constructed. These nonlinear equations are analogues of the well known…

Congruences of Fourier coefficients of modular forms have long been an object of central study. By comparison, the arithmetic of other expansions of modular forms, in particular Taylor expansions around points in the upper-half plane, has…

We develop an approach to establish $1/\pi$-series from bimodular forms. Utilizing this approach, we obtain new families of $2$-variable $1/\pi$-series associated to Zagier's sporadic Apery-like sequences.

We review some topics in the analytic theory of Eisenstein series, including meromorphic continuation, $L^2$-spectral expansion and Fourier coefficients. We also discuss some open problems.

The main objective of this paper is to give a summary of our recent work on recursion formulae for intersection numbers on moduli spaces of curves and their applications. We also present a conjectural relation between tautological rings and…

We extend the authors' previous work on Wiener-Wintner double recurrence theorem to the case of polynomials.