For an integer p≥2 we construct vertical and horizontal one-pth Riordan arrays from a Riordan array.When p=2 one-pth Riordan arrays are reduced to well known half Riordan arrays.The generating functions of the A-sequ...For an integer p≥2 we construct vertical and horizontal one-pth Riordan arrays from a Riordan array.When p=2 one-pth Riordan arrays are reduced to well known half Riordan arrays.The generating functions of the A-sequences of vertical and horizontal one-pth Riordan arrays are found.The vertical and horizontal one-pth Riordan arrays provide an approach to construct many identities.They can also be used to verify some well known identities readily.展开更多
Given a Riordan array,its vertical half and horizontal half are studied separately before.In the present paper,we introduce the(m,r,s)-halves of a Riordan array.This allows us to discuss the vertical half and horizont...Given a Riordan array,its vertical half and horizontal half are studied separately before.In the present paper,we introduce the(m,r,s)-halves of a Riordan array.This allows us to discuss the vertical half and horizontal half in a uniform context.As applications,we find several new identities involving Fibonacci,Pell and Jacobsthal sequences by applying the(m,r,s)-halves of Pascal and Delannoy matrices.展开更多
We discuss two different procedures to study the half Riordan arrays and their inverses.One of the procedures shows that every Riordan array is the half Riordan array of a unique Riordan array.It is well known that ev...We discuss two different procedures to study the half Riordan arrays and their inverses.One of the procedures shows that every Riordan array is the half Riordan array of a unique Riordan array.It is well known that every Riordan array has its half Riordan array.Therefore,this paper answers the converse question:Is every Riordan array the half Riordan array of some Riordan arrays?In addition,this paper shows that the vertical recurrence relation of the column entries of the half Riordan array is equivalent to the horizontal recurrence relation of the original Riordan array’s row entries.展开更多
In this paper,we consider the difference properties of Riordan arrays.As their applications,we provide some difference identities involving several classical combinatorial sequences such as the generalized Stirling nu...In this paper,we consider the difference properties of Riordan arrays.As their applications,we provide some difference identities involving several classical combinatorial sequences such as the generalized Stirling numbers,including Stirling numbers of the first and second kinds and Stirling numbers of type B of the first and second kinds,and Gegenbauer-Humbert-type polynomials.展开更多
This paper gives a unified approach to Hsu's two classes of extended GSN pairs in the setting of Hsu-Riordan partial monoid which is a generalization of Shapiro's Riordan group, and moreover Hsu-Wang transfer ...This paper gives a unified approach to Hsu's two classes of extended GSN pairs in the setting of Hsu-Riordan partial monoid which is a generalization of Shapiro's Riordan group, and moreover Hsu-Wang transfer theorem, Drown-Sprugnoli transfer formula and generalized Brown transfer lemma which display some transfer methods of different kinds of Hsu-Riordau arrays and identities respectively.展开更多
In this paper we consider the enumeration of subsets of the set, say Dm, of those Dyck paths of arbitrary length with maximum peak height equal to m and having a strictly increasing sequence of peak height （as one go...In this paper we consider the enumeration of subsets of the set, say Dm, of those Dyck paths of arbitrary length with maximum peak height equal to m and having a strictly increasing sequence of peak height （as one goes along the path）. Bijections and the methods of generating trees together with those of Riordan arrays are used to enumerate these subsets, resulting in many combinatorial structures counted by such well-known sequences as the Catalan nos., Narayana nos., Motzkin nos., Fibonacci nos., Schroeder nos., and the unsigned Stirling numbers of the first kind. In particular, we give two configurations which do not appear in Stanley＇s well-known list of Catalan structures.展开更多
In this paper, using exponential Riordan arrays, we investigate the Bessel numbers and Bessel matrices. By exploring links between the Bessel matrices, the Stirling matrices and the degenerate Stirling matrices, we sh...In this paper, using exponential Riordan arrays, we investigate the Bessel numbers and Bessel matrices. By exploring links between the Bessel matrices, the Stirling matrices and the degenerate Stirling matrices, we show that the Bessel numbers are special case of the degenerate Stirling numbers, and derive explicit formulas for the Bessel numbers in terms of the Stirling numbers and binomial coefficients.展开更多
文摘For an integer p≥2 we construct vertical and horizontal one-pth Riordan arrays from a Riordan array.When p=2 one-pth Riordan arrays are reduced to well known half Riordan arrays.The generating functions of the A-sequences of vertical and horizontal one-pth Riordan arrays are found.The vertical and horizontal one-pth Riordan arrays provide an approach to construct many identities.They can also be used to verify some well known identities readily.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1210128011861045)the Science Foundation for Youths of Gansu Province(Grant No.20JR10RA187)。
文摘Given a Riordan array,its vertical half and horizontal half are studied separately before.In the present paper,we introduce the(m,r,s)-halves of a Riordan array.This allows us to discuss the vertical half and horizontal half in a uniform context.As applications,we find several new identities involving Fibonacci,Pell and Jacobsthal sequences by applying the(m,r,s)-halves of Pascal and Delannoy matrices.
文摘We discuss two different procedures to study the half Riordan arrays and their inverses.One of the procedures shows that every Riordan array is the half Riordan array of a unique Riordan array.It is well known that every Riordan array has its half Riordan array.Therefore,this paper answers the converse question:Is every Riordan array the half Riordan array of some Riordan arrays?In addition,this paper shows that the vertical recurrence relation of the column entries of the half Riordan array is equivalent to the horizontal recurrence relation of the original Riordan array’s row entries.
文摘In this paper,we consider the difference properties of Riordan arrays.As their applications,we provide some difference identities involving several classical combinatorial sequences such as the generalized Stirling numbers,including Stirling numbers of the first and second kinds and Stirling numbers of type B of the first and second kinds,and Gegenbauer-Humbert-type polynomials.
文摘This paper gives a unified approach to Hsu's two classes of extended GSN pairs in the setting of Hsu-Riordan partial monoid which is a generalization of Shapiro's Riordan group, and moreover Hsu-Wang transfer theorem, Drown-Sprugnoli transfer formula and generalized Brown transfer lemma which display some transfer methods of different kinds of Hsu-Riordau arrays and identities respectively.
文摘In this paper we consider the enumeration of subsets of the set, say Dm, of those Dyck paths of arbitrary length with maximum peak height equal to m and having a strictly increasing sequence of peak height （as one goes along the path）. Bijections and the methods of generating trees together with those of Riordan arrays are used to enumerate these subsets, resulting in many combinatorial structures counted by such well-known sequences as the Catalan nos., Narayana nos., Motzkin nos., Fibonacci nos., Schroeder nos., and the unsigned Stirling numbers of the first kind. In particular, we give two configurations which do not appear in Stanley＇s well-known list of Catalan structures.
基金Supported by the Natural Science Foundation of Gansu Province (Grant No.1010RJZA049)
文摘In this paper, using exponential Riordan arrays, we investigate the Bessel numbers and Bessel matrices. By exploring links between the Bessel matrices, the Stirling matrices and the degenerate Stirling matrices, we show that the Bessel numbers are special case of the degenerate Stirling numbers, and derive explicit formulas for the Bessel numbers in terms of the Stirling numbers and binomial coefficients.
