Studies On The Properties Of Riordan Arrays And Related Polynimial Sequences

The theory of Riordan arrays is an important method to study the properties of com-binatorial sequences,and to find and prove combinatorial summations and identities.In this thesis,we present a new characterization of the generalized Riordan arrays.Combining this characterization with the factorizations of Riordan arrays,the umbral composition of polynomial sequences and the Hadamard product of series,we further study the properties of the generalized Riordan arrays and the related(generalized Sheffer)polynomial sequences.Morever,we establish some relations between the polynomial sequences related to the Ri-ordan arrays.Thus,we give a unified approach to some well-known polynomial sequences appeared in combinatorics and the theory of special functions.The main contents of this thesis are as follows:(1)It is proved that the generalized Riordan arrays on(cn)can be characterized by two invertable series ?(t)and ?(t).Using the T(?(t)|?(t))characterization of the generalized Riordan arrays,the recurrences of the elements of the generalized Riordan arrays and the recurrence of the related(generalized Sheffer)polynomial sequences are established,some basic properties of the generalized Riordan arrays and Sheffer sequences are restated,and the two cases of cn=1 and cn=n! are discussed.(2)It is proved that the generalized Lucas-u sequence and Lucas-v sequence are all related to the classical Riordan arrays.By using the factorizations of Riordan arrays and the umbral composition of polynomial sequences,the expressions of the generalized Lucas-u sequence and Lucas-v sequence are established,and the relations between two special Lucas-u sequences or two special Lucas-v sequences are studied.In addition,the relation between the polynomial sequence related to the Riordan array(?)and the Lucas-u sequence is also derived.(3)It is proved that the generalized Humbert polynomials are related to the generalized Riordan arrays.The expressions,recurrence relations and differential recurrence relations of the generalized Humbert polynomials are established.Additionally,by the Hadamard product of series,the relation between the generalized Humbert polynomial sequence and the Lucas-u sequence are obtained.