Related papers: On convolution powers of 1/x

In this paper, we present a new approach to the convolved Fibonacci numbers arising from the generating function of them and give some new and explicit identities for the convolved Fibonacci numbers.

We formulate and prove a general recurrence relation that applies to integrals involving orthogonal polynomials and similar functions. A special case are connection coefficients between two sets of orthonormal polynomials, another example…

For convex univalent functions we give instances where the sharp bound for various coefficient functionals are identical to those for the corresponding bound for the inverse function. We give instances where the sharp bounds differ and also…

This work presents a method of computing Voigt functions and their derivatives, to high accuracy, on a uniform grid. It is based on an adaptation of Fourier-transform based convolution. The relative error of the result decreases as the…

``Quasi-elliptic'' functions can be given a ring structure in two different ways, using either ordinary multiplication, or convolution. The map between the corresponding standard bases is calculated. A related structure has appeared…

If a function $f:\mathbb{R}\to\mathbb{R}$ can be represented as the sum of $n$ periodic functions as $f=f_1+\dots+f_n$ with $f(x+\alpha_j)=f(x)$ ($j=1,\dots,n$), then it also satisfies a corresponding $n$-order difference equation…

In this note we investigate the solutions of certain meta-Fibonacci recurrences of the form $f(n)=f(n-f(n-1))+f(n-2)$ for various sets of initial conditions. In the case when $f(n)=1$ for $n\leq 1$, we prove that the resulting integer…

For the Fibonacci numbers $F_n$, we have the self-convolution formula $5 \sum_{i=0}^n F_i F_{n-i} = (2n)F_{n+1} - (n+1)F_n$. We find the corresponding self-convolution formula for the Narayana numbers $R_n$ which satisfy $R_n = R_{n-1} +…

Different types of convolution operations involving large Vandermonde matrices are considered. The convolutions parallel those of large Gaussian matrices and additive and multiplicative free convolution. First additive and multiplicative…

We consider simple rational functions $R_{mn}(x)=P_m(x)/Q_n(x)$, with $P_m$ and $Q_n$ polynomials of degree $m$ and $n$ respectively. We look for "nice" functions, which we define to be ones where as many as possible of the roots, poles,…

We study a class of functional problems reducible to computing $f^{(n)}(x)$ for inputs $n$ and $x$, where $f$ is a polynomial-time bijection. As we prove, the definition is robust against variations in the type of reduction used in its…

Using simultaneously two operator identities, we consider the inversion of the convolution operators on a rectangular. The structure of the inverse operators and of some corresponding forms, which are important in signal processing, is…

A recurrence relations for sums of powers of complex functions can be written as a system of linear equation AX=B. Using properties of determinant and Cramer's rule for solving systems of linear equation, this paper presents an absolutely…

In this paper, we will extend the falling and rising factorial transforms \cite{ref. 1} which in this case every arbitrary function can be applied. Then, the properties of these transforms will be investigated and some corollaries will be…

Generalizations of some known results on the best, best linear and best one-sided approxima- tions by trigonometric polynomials of the classes of 2\pi - periodic functions presented in the form of convolutions to the case of set-valued…

Six families of generalized hypergeometric series in a variable $x$ and an arbitrary number of parameters are considered. Each of them is indexed by an integer $n$. Linear recurrence relations in $n$ relate these functions and their product…

While transformers have proven enormously successful in a range of tasks, their fundamental properties as models of computation are not well understood. This paper contributes to the study of the expressive capacity of transformers,…

We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some linear methods of this approximation for univariate functions in the class induced by the convolution…

We consider a self-convolutive recurrence whose solution is the sequence of coefficients in the asymptotic expansion of the logarithmic derivative of the confluent hypergeometic function $U(a,b,z)$. By application of the Hilbert transform…

Inversion of function sinc(x) is studied. New series and integral representations of branches of inverse function are obtained using Fourier analysis.