Related papers: Regularization of certain divergent series of poly…
Let $\theta >2$ be real and non-integral with integer part $n = \lfloor \theta \rfloor$ and let $ \phi (x)$ be a generalised polynomial with leading term $x^\theta.$ We establish a mean value estimate for the exponential sum…
The twisted $T$-adic exponential sum associated to a polynomial in one variable is studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the…
In the work we have considered p-adic functional series with binomial coefficients and discussed its p-adic convergence. Then we have derived a recurrence relation following with a summation formula which is invariant for rational argument.…
We explore differential operators, $T$, that diagonalize on a simple basis, $\{B_n(x)\}_{n=0}^\infty$, with respect to some sequence of real numbers, $\{a_n\}_{n=0}^\infty$, and sequence of polynomials, $\{Q_k(x)\}_{k=0}^\infty$, as in $…
We describe mean value estimates for exponential sums of degree exceeding 2 that approach those conjectured to be best possible. The vehicle for this recent progress is the efficient congruencing method, which iteratively exploits the…
First we generalize a famous lemma of Gallagher on the mean square estimate for exponential sums by plugging a weight in the right hand side of Gallagher's original inequality. Then we apply it in the special case of the Cesaro weight, in…
This is the translation of Leonhard Euler's paper "De Seriebus divergentibus" written in Latin into English. Leonhard Euler defines and discusses divergent series. He is especially interested in the example $1!-2!+3!-\text{etc.}$ and uses…
In this paper we study the convergence of multiple Dirichlet L-series. In particular we give an integral representation of the series in the region of convergence by using Abel's summation formula. A certain generalized result is also…
We investigate the zeros of polynomial solutions to the differential-difference equation \[ P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x), n=0,1,... \] where $A_{n}$ and $B_{n}$ are polynomials of degree at most 2 and 1…
The purpose of this paper consists to study the sums of the type $P(n) + P(n - d) + P(n - 2 d) + \dots$, where $P$ is a real polynomial, $d$ is a positive integer and the sum stops at the value of $P$ at the smallest natural number of the…
Several problems on Fourier series and trigonometric approximation on a hexagon and a triangle are studied. The results include Abel and Ces\`aro summability of Fourier series, degree of approximation and best approximation by trigonometric…
In this note we associate a sequence of non-negative integers to any convergent series of positive real numbers and study this sequence for the series $\sum_{n \geq 1} n^{-k}$ where $k$ is an integer $\geq 2$.
We discuss certain special cases of algebraic approximants that are given as zeroes of so-called "effective characteristic polynomials" and their generalization to a multiseries setting. These approximants are useful for the convergence…
We analyze the sequence of polynomials defined by the differential-difference equation $P_{n+1}(x)=P_{n}^{\prime}(x)+x(n+1)P_{n}(x)$ asymptotically as $n\to\infty$. The polynomials $P_{n}(x)$ arise in the computation of higher derivatives…
An inverse polynomial has a Chebyshev series expansion 1/\sum(j=0..k)b_j*T_j(x)=\sum'(n=0..oo) a_n*T_n(x) if the polynomial has no roots in [-1,1]. If the inverse polynomial is decomposed into partial fractions, the a_n are linear…
We investigate an infinite sequence of polynomials of the form: \[a_0T_{n}(x)+a_{1}T_{n-1}(x)+\cdots+a_{m}T_{n-m}(x)\] where $(a_0,a_1,\ldots,a_m)$ is a fixed m-tuple of real numbers, $a_0,a_m\ne0$, $T_i(x)$ are Chebyshev polynomials of the…
We generalize the classical Olivier's theorem which says that for any convergent series $\sum_n a_n$ with positive nonincreasing real terms the sequence $(n a_n)$ tends to zero. Our results encompass many known generalizations of Olivier's…
Uniform asymptotic expansions are derived for the zeros of the reverse generalized Bessel polynomials of large degree $n$ and real parameter $a$. It is assumed that $-\Delta_{1} n+\frac{3}{2} \leq a \leq \Delta_{2} n$ for fixed arbitrary…
Explicit determinations of several classes of trigonometric sums are given. These sums can be viewed as analogues or generalizations of Gauss sums. In a previous paper, two of the present authors considered primarily sine sums associated…
We study a generalization of the classical Pentagonal Number Theorem and its applications. We derive new identities for certain infinite series, recurrence relations and convolution sums for certain restricted partitions and divisor sums.…