Relations between B-spline curves and eulerian type polynomials and their applications

In this thesis, some properties of families of spline curve including B-spline curves have been investigated. Families of special spline curve with which B-spline curves are related, such as cardinal spline, exponential spline, exponential Euler spline and Bernoulli monospline families have also been studied and some basic properties have been given. The construction of the families of B-spline curve with the help of cardinal spline families have been studied. Relations among families of cardinal spline and B-spline curve and Eulerian type polynomials and their generating functions have been investigated in detail. The definitions and some properties of the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Genocchi numbers and polynomials, the Eulerian type numbers and polynomials and other some families of special numbers and polynomials have been given. In addition, these families of special numbers and polynomials and families of novel cardinal type spline curve have also been defined. By using Mathematica package program in the Wolfram, graphs of these families have been drawn. Moreover, the families of Bézier curve constructed with Bernstein basis functions, which are closely related to families of B-spline curve, have also been studied. Finally, results including the curve families studied in this thesis and families of special polynomials are given and new formulas, relations and identities including some well-known special numbers and polynomials such as the Bernoulli numbers and polynomials, Eulerian type numbers and polynomials, the Genocchi numbers and polynomials, the Frobenius-Euler numbers and polynomials, the Stirling numbers of the second kind, Bernoulli numbers of higher order, the Catalan numbers are obtained.