Related papers: Opposite power series

Let $S= \{ p_1, \ldots, p_s\}$ be a finite, non-empty set of distinct prime numbers and $(U_{n})_{n \geq 0}$ be a linear recurrence sequence of integers of order $r$. For any positive integer $k,$ we define $(U_j^{(k)})_{j\geq 1}$ an…

We compute the number of irreducible linear representations of self-similar branch groups, by expressing these numbers as the co\"efficients a_n of a Dirichlet series sum a_n n^{-s}. We show that this Dirichlet series has a positive…

Let m be a probability measure supported on some infinite and compact set K in the complex plane and let p_n(z) be the corresponding degree n orthonormal polynomial with positive leading coefficient. Let v_n be the normalized zero counting…

The most simple and famous divergent power series coming from ODE may be the so-called Euler series $\sum_{n\ge 0}(-1)^n\,n!\,x^{n+1}$, that, as well as all its positive powers, is Borel-summable in any direction excepted the negative real…

Assume that $\{a_{n};\,n\geq0\}$ is a sequence of positive numbers and $\sum a_{n}^{\,-1}<\infty$. Let $\alpha_{n}=ka_{n}$, $\beta_{n}=a_{n}+k^{2}a_{n-1}$ where $k\in(0,1)$ is a parameter, and let $\{P_{n}(x)\}$ be an orthonormal polynomial…

In this paper we prove the following: let $\omega(t)$ be a continuous function, increasing in $[0,\infty)$ and $\omega(+0)=0$. Then there exists a series of the form$\sum_{k=-\infty}^\infty C_ke^{ikx}$ with $\sum_{k=-\infty}^\infty C^2_k…

In this note we present some results on the convergence of Nekrasov partition functions as power series in the instanton counting parameter. We focus on $U(N)$ ${\mathcal N}=2$ gauge theories in four dimensions with matter in the adjoint…

In this work we study the solutions of the equation $z^pR(z^k)=\alpha$ with nonzero complex $\alpha$, integer $p,k$ and $R(z)$ generating a (possibly doubly infinite) totally positive sequence. It is shown that the zeros of…

The power spectrum (PS) of mass density fluctuations, in the range $0.05 \leq k \leq 0.2 \ihmpc$, is derived from the Mark III catalog of peculiar velocities of galaxies, independent of ``biasing". It is computed from the density field as…

This paper focuses on the study of the order of power series that are linear combinations of a given finite set of power series. The order of a formal power series, known as $\textrm{ord}(f)$, is defined as the minimum exponent of $x$ that…

For a power series which converges in some neighborhood of the origin in the complex plane, it turns out that the zeros of its partial sums---its sections---often behave in a controlled manner, producing intricate patterns as they converge…

We show that the generating function $\sum_{n\ge0}M_n\,z^n$ for Motzkin numbers $M_n$, when coefficients are reduced modulo a given power of $2$, can be expressed as a polynomial in the basic series $\sum _{e\ge0} ^{} {z^{4^e}}/(…

In this article, we consider the reciprocal antisymmetric polynomial \[P(z) = \sum_{j = 0}^{s}(-1)^j\gamma_j\left(z^j - z^{N + s + 1 - j}\right), \ \gamma_0 = 1.\] It is shown that if all the zeros of $P(z)$ are located on the unit circle,…

This is an English translation of the Latin original "De summa seriei ex numeris primis formatae ${1/3}-{1/5}+{1/7}+{1/11}-{1/13}-{1/17}+{1/19}+{1/23}-{1/29}+{1/31}-$ etc. ubi numeri primi formae $4n-1$ habent signum positivum formae autem…

Let T : Lp --> Lp be a positive contraction, with p strictly between 1 and infinity. Assume that T is analytic, that is, there exists a constant K such that \norm{T^n-T^{n-1}} < K/n for any positive integer n. Let q strictly betweeen 2 and…

We show that it is possible for a square integrable function on the circle, which is a sum of an almost everywhere convergent series of exponentials with positive frequencies, to not belong to the Hardy space. A consequence in the…

We study the Kapteyn series $% %TCIMACRO{\dsum \limits_{n=1}^{\infty}}% %BeginExpansion {\displaystyle\sum\limits_{n=1}^{\infty}} %EndExpansion t^{n}\mathrm{J}_{n}(nz) $. We find a series representation in powers of $z$ and analyze its…

In this paper, we study a Dirichlet series generated by powers of harmonic numbers. As an application of these functions, we derive certain series involving harmonic numbers. We also study the analytic properties of these Dirichlet series…

Let $w$ be a weight on the unit disk $\mathbb{D}$ having the form \[w(z)=|v(z)|^2\prod_{k=1}^s\left|\frac{z-a_k}{1-z\overline{a}_k}\right|^{m_k}\,,\quad m_k>-2,\ |a_k|<1,\] where $v$ is analytic and free of zeros in $\overline{\mathbb{D}}$,…

We present several congruences modulo a power of prime $p$ concerning sums of the following type $\sum_{k=1}^{p-1}{m^k\over k^r}{2k\choose k}^{-1}$ which reveal some interesting connections with the analogous infinite series.