In this study, we first give the definitions of (s,t)-Jacobsthal and (s,t)-Jacobsthal Lucas sequence. By using these formulas we define (s,t)-Jacobsthal and (s,t)-Jacobsthal Lucas matrix sequences. After that we estab...In this study, we first give the definitions of (s,t)-Jacobsthal and (s,t)-Jacobsthal Lucas sequence. By using these formulas we define (s,t)-Jacobsthal and (s,t)-Jacobsthal Lucas matrix sequences. After that we establish some sum formulas for these matrix sequences.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
In this paper we present combinatorial interpretations and polynomials generalizations for sequences including the Fibonacci numbers, the Pell numbers and the Jacobsthal numbers in terms of partitions. It is important...In this paper we present combinatorial interpretations and polynomials generalizations for sequences including the Fibonacci numbers, the Pell numbers and the Jacobsthal numbers in terms of partitions. It is important to mention that results of this nature were given by Santos and Ivkovic in two papers published on the Fibonacci Quarterly, Polynomial generalizations of the Pell sequence and the Fibonacci sequence [1] and Fibonacci Numbers and Partitions [2] , and one, by Santos, on Discrete Mathematics, On the Combinatorics of Polynomial generalizations of Rogers-Ramanujan Type Identities [3]. By these results one can see that from the q-series identities important combinatorial information can be obtained by a careful study of the two variable function introduced by Andrews in Combinatorics and Ramanujan's lost notebook [4].展开更多
文摘In this study, we first give the definitions of (s,t)-Jacobsthal and (s,t)-Jacobsthal Lucas sequence. By using these formulas we define (s,t)-Jacobsthal and (s,t)-Jacobsthal Lucas matrix sequences. After that we establish some sum formulas for these matrix sequences.
文摘有关组合和式的Dwork类型超同余式的研究,一直是组合学与数论学中的一个有意义的课题。本文主要研究孙智伟提出的关于g n(-1)和式的Dwork类型超同余式的特殊情况:设p为奇素数且r 2为正整数,有∑p r-1 k=0 g k(-1)≡(-1 p)∑p r-1-1 k=0 g k(-1)(mod p r+1),这里Legendre符号为(.)。证明的关键是利用分拆方法,Jacobsthal二项式同余式和有关调和数的同余性质。
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.
基金Partially supported by FAPESP(Fundacao de Amparo a Pesquisa do Estado de Sao Paulo).
文摘In this paper we present combinatorial interpretations and polynomials generalizations for sequences including the Fibonacci numbers, the Pell numbers and the Jacobsthal numbers in terms of partitions. It is important to mention that results of this nature were given by Santos and Ivkovic in two papers published on the Fibonacci Quarterly, Polynomial generalizations of the Pell sequence and the Fibonacci sequence [1] and Fibonacci Numbers and Partitions [2] , and one, by Santos, on Discrete Mathematics, On the Combinatorics of Polynomial generalizations of Rogers-Ramanujan Type Identities [3]. By these results one can see that from the q-series identities important combinatorial information can be obtained by a careful study of the two variable function introduced by Andrews in Combinatorics and Ramanujan's lost notebook [4].
