Studies On Complementary Sheffer Sequences And Symmetric Sheffer Sequences

The Sheffer sequences are a topic of interest in combinatorics,and play an important role in statistics,theory of special functions,etc.In this thesis,based on the relation between(generalized)Sheffer sequences and(generalized)Riordan arrays,we study the[m]-complementary arrays of generalized Riordan arrays,the[m]-complementaty sequences of generalized Sheffer sequences and the[m]-complementary transformation of Sheffer group,and study systematically the properties of the[m]-complementary sequences of the Lucas u,v sequences and the dual arrays,complementary arrays and inverse arrays of the u,v-Riordan arrays.Moreover,we also give the generating function characterization of the exponential symmetric Sheffer sequences with the weight(wn).The main contents of this thesis are as follows:(1)We establish the expression of the generic element of the[m]-complementary arrays of generalized Riordan arrays.By using the[m]-complementary arrays of Riordan arrays,we give the definitions of the[m]-complementary sequences of(generalized)Sheffer sequences and[m]-complementary transformation of Sheffer group,study the elementary properties,and discuss the relations between the Sheffer sequences and their[m]-complementary sequences.The results are applied to some special Sheffer sequences,such as the exponential polynomial sequence,the falling factorial polynomial sequence,the Laguerre polynomial sequence and the polynomial sequences related to a class of self-dual Riordan arrays proposed by Luzon et al.(2)We study the[m]-complementary sequences of the Lucas u,v sequences and the dual arrays,complementary arrays and inverse arrays of the u,v-Riordan arrays.We show some properties of related Riordan arrays,including U-1=aV?,V-1=aU?.We also obtain the expressions of the[m]-complementary sequences of the Lucas u,v sequences,and establish the related umbral compositions.Moreover,we use the weighted Motzkin paths to give the interpretations of entries in the inverse arrays and two related identities,and show in what conditions the sequence in the 0th column of the inverse array can be a log-convex sequence.The above results are applied to some special Lucas u,v sequences and related Riordan arrays.(3)We give the definition of the exponential Sheffer sequences with the weight(wn),and establish the generating function characterization of the exponential symmetric Sheffer sequences with the weight(wn).We show that all the symmetric Sheffer sequences with weight(?n),(<?>n)or((?)n)are in fact constant multiples of the Charlier polynomial sequence,the Meixner polynomial sequence,or the Krawtchouk polynomial sequence,respectively.Using the Sheffer sequences and the theory of Riordan arrays,the basic properties of the above three polynomial sequences are given.