Some Results On The Riordan Arrays And Their Applications

This paper gives a generalization of Riordan functional array by using the theoryof Exponential Family. Some properties and applications of Riordan array andRiordan functional array are discussed. Generally speaking, the method relies onLagrange inversion formula, generating funcion and derivative technique and so on.Some identities containing special combinatorial numbers are obtained, in whichsome known identities are only special cases.As follows is the idiographic research:Chapter1mainly introduces the backgrounds and the current researches ofRiordan arrays and its applications. The following two chapters are the results of thepresent thesis.Chapter2does further study to the properties of Rordan arryas. By usingLagrange inversion formula and derivative technique, some identities containingspecial combinatorial sequences are displayed.Chapter3can be divided into two parts. In the first part, the row summationformula and the recursive relationship of exponential Riordan array are obtained. Inthe second part, three aspects research are done. First, two new identities related to theHermite polynomials is established. Then, the Pascal functional matrices aregeneralized. This section provides a simple new proof to the existing results by usingthe theory of Riordan array. The third section gives the exponential Riordan arrayform of Bell maticeBn, which elements are Bell polynomial. Further more, therecursive relationship of Bell matrices is obtained.