Related papers: New elementary formulas for any derivative of any …
A landmark result from rational approximation theory states that $x^{1/p}$ on $[0,1]$ can be approximated by a type-$(n,n)$ rational function with root-exponential accuracy. Motivated by the recursive optimality property of Zolotarev…
The main result of this paper shows a totally new necessary and sufficient condition to determine both real and complex zeros of derivative of all entire and meromorphic functions of one complex variable in the extended complex plane. By…
We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves…
In the present work, we determine explicitly the genus of any separable cubic extension of any global function field given the minimal polynomial of the extension. We give algorithms computing the ramification data and the genus of any…
We present a generalization of a formula of higher order derivatives and give a short proof.
Formula for the nth prime using elementary arithmetical functions based in a previous formula changing the characteristic function of prime numbers.
An explicit expression of the k-th derivative of the Bessel function $J_\nu(z)$, with respect to its order $\nu$, is given. Particularizations for the cases of positive or negative $\nu$ are considered.
Let $D(s)$ be a fractional derivation of order $s$. For a real $p\ne 0$, we construct an integral operator $A(p)$ in an appropriate functional space such that $A(p) D(s) A(p)^{-1}=D(p s)$ for all $s$. The kernel of the operator $A(p)$ is…
Here we present an algorithm to find elementary first integrals of rational second order ordinary differential equations (SOODEs). In \cite{PS2}, we have presented the first algorithmic way to deal with SOODEs, introducing the basis for the…
In this paper, we investigate new class of sequences related to fully degenerate Bernoulli numbers and polynomials. From those sequences, we derive some formulae for the degenerate Bernoulli and Euler polynomials.
A recursion formula for derivatives of Chebyshev polynomials is replaced by an explicit formula.
In multicentric calculus one takes a polynomial $p$ with distinct roots as a new variable and represents complex valued functions by $\mathbb C^d$-valued functions, where $d$ is the degree of $p$. An application is e.g. the possibility to…
In this note, we review some facts about polynomials representing functions modulo primes p. In addition we prove that the polynomial f(x) = x^{p-2} + x^{p-3} + ... + x^3 + x^2 + 2x + 1 represents the transposition (0 1) modulo p, that is,…
We prove explicit formulas for the $p$-adic $L$-functions of totally real number fields and show how these formulas can be used to compute values and representations of $p$-adic $L$-functions.
The aim of this paper is to provide sufficient conditions for when a polynomial or rational function over a field K is prime using its order of vanishing at infinity and the resultant.
For any positive integer n, a new family of periodic functions in power series form and of period n is used to solve in closed form a class of polynomial equation of order n. The n roots are the values of the appropriate function from that…
We give a formula for matrix exponentials and partial fraction decompositions.
Let f be an irreducible polynomial of degree d>=3 with no fixed prime divisor. We derive an asymptotic formula for the number of primes p<x such that f(p) is (d-1)-free.
The necessary and sufficient conditions for a function to be totally or partially separable are derived. It is shown that a function is totally separable if and only if each component of the gradient vector of depends only on the…
In this note we compare two formulas for the higher order derivatives of the function 1/(exp(x) -1). We also provide an integral representation for these derivatives and obtain a classical formula relating zeta values and Bernoulli numbers.