# Functions and Analysis

## 1003 Submissions

[2] **viXra:1003.0166 [pdf]**
*replaced on 20 Mar 2010*

### A Self-Recurrence Method for Generalizing Known Scientific Results

**Authors:** Florentin Smarandache

**Comments:** 7 pages

A great number of articles widen known scientific results (theorems, inequalities,
math/physics/chemical etc. propositions, formulas), and this is due to a simple procedure,
of which it is good to say a few words:
Let suppose that we want to generalizes a known mathematical proposition P(a) ,
where a is a constant, to the proposition P(n) , where n is a variable which belongs to
subset of N .
To prove that P is true for n by recurrence means the following: the first step is
trivial, since it is about the known result P(a) (and thus it was already verified before by
other mathematicians!). To pass from P(n) to P(n + 1) , one uses too P(a) : therefore one
widens a proposition by using the proposition itself, in other words the found
generalization will be paradoxically proved with the help of the particular case from
which one started!
We present below the generalizations of Hölder, Minkovski, and respectively
Tchebychev inequalities.

**Category:** Functions and Analysis

[1] **viXra:1003.0105 [pdf]**
*submitted on 10 Mar 2010*

### Orthogonal Polynomials, Moment Problem and the Riemann XI-Function ξ(1/2 + Iz)

**Authors:** Jose Javier Garcia Moreta

**Comments:** 12 Pages.

In this paper we study a set of orthogonal Polynomials with respect a certain given measure related to
the Taylor series expansion of the Xi-function , this paper is based on a previous conjecture by Carlon and
Gaston related to the fact that Riemann Hypothesis (with simple zeros) is equivalent to the limit for a
certain set of orthogonal Polynomials, we study the 'Hamburger moment problem' for even 'n' and 0 for n
odd here the moments are related to the power series expansion of Xi-function , we also give the
integral representation for the generating function , in terms of the Laplace transform of , and
in the end of the paper we study the connection of our orthogonal polynomial set with the Kernel ,
through all the paper we will use the simplified notation (see paper for abstract with equations)

**Category:** Functions and Analysis