Date of Award

Spring 1959

Degree Type

Thesis - Restricted

Degree Name

Master of Science (MS)



First Advisor

Talacko, Joseph V.

Second Advisor

Heider, J. L.

Third Advisor

Hanneken, C.B.


A substantial part of every text on the calculus of finite differences is devoted to the derivation of various finite difference formulas. The aim of this thesis is to present an alternative and more powerful way, different from the algebraic approach, which not only accomplishes the same task, but also simplifies formulas which are otherwise very complicated. This method is based on the employing of certain symbolic linear operators. Although symbolic operators were introduced by George Boole in 1860, it has been only in recent years that their usefulness and importance have been recognized. Among the authors of various texts certain operators are in common use and for some there is an accepted notation. For others agreement is not general, and in any case, there appear to be gaps in the scheme. Since this paper is concerned primarily with t he use of symbolic operators, certain sections of the calculus of finite differences that do not lend themselves to this method will not be treated. Of these sections I may mention the determination of the error term and the more involved difference equations. The application of the operational technique in these cases would be more complicated than the algebraic method and in some instances cannot be done. Those sections in which symbolic operators are put to use are not, by any means, exhaustive and all-inclusive. It was necessary to choose carefully from among the numerous formulas so as to present a general notion of the application of this method to the various problems of interpolation, differentiation, summation, and integration. The historical notes, although not dealing directly with the problem in question, are interesting and, therefore, have been included.