On a generalized function-to-sequence transform

Slobodan Trickovic,Miomir Stankovic

doi:10.2298/aadm180908005t

Abstract

By attaching a sequence {?n}n?N0 to the binomial transform, a new operator D? is obtained. We use the same sequence to define a new transform T? mapping derivatives to the powers of D?, and integrals to D-1?. The inverse transform B? of T? is introduced and its properties are studied. For ?n = (-1)n, B? reduces to the Borel transform. Applying T? to Bessel's differential operator d/dx x d/dx, we obtain Bessel's discrete operator D?nN?. Its eigenvectors correspond to eigenfunctions of Bessel's differential operator.

Highlights

AND PRELIMINARIESOne can pose a question regarding the character of the reality
Obvious example of duality is the Poisson distribution in the probability theory. It is discrete, expressing the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event
Complicated integrals of continuous functions are reduced in numerical analysis to finite sums, and to be solved, some difference equations have to be transformed into differential ones

Summary

INTRODUCTION

One can pose a question regarding the character of the reality. Is it discrete or continuous? The answer is simple: it is dual. Obvious example of duality is the Poisson distribution in the probability theory It is discrete, expressing the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event. Complicated integrals of continuous functions are reduced in numerical analysis to finite sums, and to be solved, some difference equations have to be transformed into differential ones. In this sense, we develop a specific transformation mapping the continuous into the discrete and vice versa.

A NEW OPERATOR

THE FUNCTION-TO-SEQUENCE TRANSFORM AND ITS INVERSE

Tα-TRANSFORM OF BASIC ELEMENTARY FUNCTIONS

THE INVERSE TRANSFORM

TRANSFORMS Tα AND Bα AND CONVOLUTION OF SEQUENCES