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Abstract. It is well-known that the fundamental solution of ut(n, t) = u(n + 1, t) − 2u(n, t) + u(n − 1, t), n ∈ Z, with u(n, 0) = δnm for every fixed m ∈ Z, is given by u(n, t) = e −2t In−m(2t), where I k (t) is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal seriesBy using semigroup theory, this formula allows us to analyze some operators associated with the discrete Laplacian. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted ℓ p (Z)-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. Interestingly, it is shown that the Riesz transforms coincide essentially with the so called discrete Hilbert transform defined by D. Hilbert at the beginning of the XX century. We also see that these Riesz transforms are limits of the conjugate harmonic functions.The results rely on a careful use of several properties of Bessel functions.

The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size h > 0for u, f : Z h → R, 0 < s < 1, is performed. The pointwise nonlocal formula for (−∆ h ) s u and the nonlocal discrete mean value property for discrete s-harmonic functions are obtained. We observe that a characterization of (−∆ h ) s as the Dirichlet-to-Neumann operator for a semidiscrete degenerate elliptic local extension problem is valid. Regularity properties and Schauder estimates in discrete Hölder spaces as well as existence and uniqueness of solutions to the nonlocal Dirichlet problem are shown. For the latter, the fractional discrete Sobolev embedding and the fractional discrete Poincaré inequality are proved, which are of independent interest. We introduce the negative power (fundamental solution)which can be seen as the Neumann-to-Dirichlet map for the semidiscrete extension problem. We then prove the discrete Hardy-Littlewood-Sobolev inequality for (−∆ h ) −s .As applications, the convergence of our fractional discrete Laplacian to the (continuous) fractional Laplacian as h → 0 in Hölder spaces is analyzed. Indeed, uniform estimates for the error of the approximation in terms of h under minimal regularity assumptions are obtained. We finally prove that solutions to the Poisson problem for the fractional Laplacian (−∆) s U = F, in R, can be approximated by solutions to the Dirichlet problem for our fractional discrete Laplacian, with explicit uniform error estimates in terms of h.2010 Mathematics Subject Classification. Primary: 35R11, 49M25. Secondary: 35K05, 65N15. Key words and phrases. Nonlocal discrete diffusion equations, fractional discrete Laplacian, regularity and extension problem, Sobolev and Poincaré inequalities, error of approximation, semidiscrete heat equation.

Abstract. We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials B n (x; λ) in detail. The starting point is their Fourier series on [0, 1] which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze oscillatory phenomena which arise in certain cases.These results are transferred to the Apostol-Euler polynomials E n (x; λ) via a simple relation linking them to the Apostol-Bernoulli polynomials.

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