Fourier coefficients of Siegel-Eisenstein series of odd genus

We give a new expression of the T-th Fourier coefficients of the Siegel-Eisenstein series of odd genus. Unlike even genus case, only modified divisor sums and no generalized class numbers enter into our formula.     

Convergence of series of Fourier coefficients for multiplicative convolutions

We study the convergence of series of the Fourier-Vilenkin coefficients of functions represented as multiplicative convolutions. In the trigonometric case similar results are obtained by C. Onneweer, M. Izumi, and S. Izumi. Moreover, we consider certain analogs of I. Hirshman and W. Rudin transforms of Fourier coefficients. Some results are proved to be unimprovable in a certain sense.     

Asymptotic relations among Fourier coefficients of real-analytic Eisenstein series

Following Wolpert, we find a set of asymptotic relations among the Fourier coefficients of real-analytic Eisenstein series. The relations are found by evaluating the integral of the product of an Eisenstein series with an exponential factor along a horocycle. We evaluate the integral in two ways by exploiting the automorphicity of ; the first of these evaluations immediately gives us one coefficient, while the other evaluation provides us with a sum of Fourier coefficients. The second evaluation of the integral is done using stationary phase asymptotics in the parameter is the eigenvalue of ) for a cubic phase.

As applications we find sets of asymptotic relations for divisor functions.

     

Asymptotic expansion of Fourier coefficients of reciprocals of Eisenstein series

The Ramanujan Journal - In this paper we give a classification of the asymptotic expansion of the q-expansion of reciprocals of Eisenstein series $$E_k$$ of weight k for the modular group $$mathop...     

On Fourier coefficients of Eisenstein series and Niebur Poincaré series of integral weight

We give a Katok-Sarnak type correspondence for Niebur type Poincaré series and Eisenstein series on the three-dimensional hyperbolic space.     

Sojourns and extremes of Fourier sums and series with random coefficients

Let X(t) be the trigonometric polynomial Σ^k_j=0a_j(U_tcosjt+V_jsinjt), –∞< t<∞, where the coefficients U_t and V_t are random variables and a_j is real. Suppose that these random variables have a joint distribution which is invariant under all orthogonal transformations of R^2k–2. Then X(t) is stationary but not necessarily Gaussian. Put L_t(u) = Lebesgue measure {s: 0?s?t, X(s) > u}, and M(t) = max{X(s): 0?s?t}. Limit theorems for L_t(u) and $\text{P}\text{(M(t) > u)}$ for u→∞ are obtained under the hypothesis that the distribution of the random norm (Σ^k_j=0(U²_j+V²_j))^{1 2} belongs to the domain of attraction of the extreme value distribution exp{ e^–2}. The results are also extended to the random Fourier series (k=∞).     

An explicit formula for the Fourier coefficients of Eisenstein series attached to lattices

We prove an explicit formula for local densities of inhomogeneous quadratic forms. The formula is derived by calculating explicitly the Fourier coefficients of various types of Eisenstein series.     

Convolutions of Dirichlet series with the Fourier coefficients of cusp forms of weight 0

By Rankin's method one constructs a meromorphic continuation and a functional equation for the convolution of two Dirichlet series corresponding to cusp forms of weight O. One investigates the summator function of the coefficients of this convolution.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 93, pp. 204–217, 1980.     

A note on certain divisibility properties of the Fourier coefficients of normalized Eisenstein series

In this note, we shall prove certain divisibility properties of the Fourier coefficients of a class of normalized Eisenstein series modulo certain prime powers.     

Radial limit of lacunary Fourier series with coefficients in non-commutative symmetric spaces

Let be a rearrangement invariant space, an arbitrary set and a von Neumann algebra with a semifinite normal faithful trace. It is proved that the associated symmetric space of measurable operators has -RNP if and only if has -RNP extending in this way some previous results by Q. Xu.

     

A simplified variant of the method of computation of the Fourier coefficients of a series

Making use of the results of Golubeva and Fomenko (Zapiski Nauchn. Sem. Leningr. Otd. Mat. Inst.,71, 85–95 (1977)), one computes the Fourier coefficients of the nonanalytic Eisenstein series relative to Sp₂() by a method which is simpler than that of Kaufhold (Math. Ann.,137, No. 5, 454–476 (1959)).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 76, pp. 60–64, 1978.     

Fourier coefficients of real analytic Eisenstein series at various cusps (II)

The Ramanujan Journal - The purpose of this paper is to consider the computation of the Fourier coefficients of real-analytic Eisenstein series associated to general cusps of Hecke congruence...     

Recurrence relations for the coefficients of the Fourier series expansions with respect to q-classical orthogonal polynomials

We propose an algorithm to construct recurrence relations for the coefficients of the Fourier series expansions with respect to the q-classical orthogonal polynomials p_k(x;q). Examples dealing with inversion problems, connection between any two sequences of q-classical polynomials, linearization of ϑ_m(x) p_n(x;q), where ϑ_m(x) is x^mor (x;q)_m, and the expansion of the Hahn-Exton q-Bessel function in the little q-Jacobi polynomials are discussed in detail. This revised version was published online in June 2006 with corrections to the Cover Date.