On power series with positive coefficients

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On the power series coefficients of certain quotients of Eisenstein series

In their last joint paper, Hardy and Ramanujan examined the coefficients of modular forms with a simple pole in a fundamental region. In particular, they focused on the reciprocal of the Eisenstein series . In letters written to Hardy from nursing homes, Ramanujan stated without proof several more results of this sort. The purpose of this paper is to prove most of these claims.

     

On Laurent series with multiply positive coefficients

We consider the class of doubly infinite sequences {a _k } _k ^∞ = −∞ whose truncated sequences {a _k } _n ^k = −n are 3-times positive in the sense of Pólya and Fekete for all n = 1, 2, ..., and a ₀ ≠ 0. We obtain a characterization of this class in terms of independent parameters. We also find an estimate of the growth order of the corresponding Laurent series ∑ _{k= −∞} ^∞ akz ^k .     

On random power series with nonidentically distributed coefficients

The distribution of the radius of convergence r(ω) of a random power series with nonidentically distributed coefficients is considered. The results obtained here extend the well-known work in the identically distributed case.     

On the coefficients of Neumann series of Bessel functions

Recently Pogány and Süli (Proc. Amer. Math. Soc. 137 (7) (2009) 2363-2368) derived a closed-form integral expression for Neumann series of Bessel functions. In this note we precisely characterize the class of functions α that generate the integral representation of a Neumann series of Bessel functions in the sense that the restriction αN_|=(αn) of a function α to the set N of all positive integers is the sequence of coefficients of the initial Neumann series.     

On convergence of multiple trigonometric series with monotone coefficients

We study the Pringsheim pointwise convergence of multiple trigonometric series. We obtain a condition on the coefficients of the series that guarantees its Pringsheim convergence and prove the unimprovability of this condition.     

On the Laurent coefficients of a class of Dirichlet series

The generalized Euler constants for an arithmetic progression have been considered by several authors including D. H. Lehmer, W. E. Briggs, K. Dilcher and S. Kanemitsu as an important generalization of the ordinary Euler-Stieltjes constants. In this paper, answering a problem posed by Kanemitsu, we shall adopt the partial zeta-function, a special case of the Hurwitz zeta-function, as a genuine generating function for the generalized Euler constants and make extensive use of the Hurwitz zeta-function to derive all the preceding results of Dilcher and Kanemitsu in a unified and elucidated manner.     

On the sign changes of coefficients of general Dirichlet series

Under what conditions do the (possibly complex) coefficients of a general Dirichlet series exhibit oscillatory behavior? In this work we invoke Laguerre's Rule of Signs and Landau's Theorem to provide a rather simple answer to this question. Furthermore, we explain how our result easily applies to a multitude of functions.

     

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.     

On some properties of trigonometric series with monotone decreasing coefficients

В статье рассматрива ются одномерные и дву мерные тригонометрические ряды с моно-тонными коэффициентами. Дает ся пример двойного тригонометрическог о ряда (1) $$\mathop \sum \limits_{n,k = 1}^\infty a_{nk} \sin nx\sin ky,$$ , коэффициенты которо го монотонны поk и поп, любая последовательность \(\{ S_{n_k m_k } (x,y)\} _{k = 1}^\infty\) прямоугольных части чных сумм ряда (1), где min(n _k ,m _k )→∞ приk→∞, расходится по чти всюду на (0,n)². Кроме того, изучается мера множеств нулей ф ункций (2) $$f(x) = \frac{{a_0 }}{2} + \mathop \sum \limits_{n = 1}^{a_0 } a_n \cos nx\tilde f(x) = \mathop \sum \limits_{n = 1}^\infty a_n \sin nx,$$ , гдеа _n ↓ приn→ ∞, и доказ ьшается несколько те орем о скорости убывания ко эффициентовa _n рядов (2), если все част ичные суммыS _n (f,x) или \(S_n (\tilde f,x)\) дляn=1,2,... неотрицате ль-ны на (0,n).