Here presented are the definitions of （c）-Riordan arrays and （c）-Bell polynomials which are extensions of the classical Riordan arrays and Bell polynomials. The characterization of （c）-Riordan arrays by means of...Here presented are the definitions of （c）-Riordan arrays and （c）-Bell polynomials which are extensions of the classical Riordan arrays and Bell polynomials. The characterization of （c）-Riordan arrays by means of the A- and Z-sequences is given, which corresponds to a horizontal construction of a （c）-Riordan array rather than its definition approach through column generating functions. There exists a one-to-one correspondence between GegenbauerHumbert-type polynomial sequences and the set of （c）-Riordan arrays, which generates the sequence characterization of Gegenbauer-Humbert-type polynomial sequences. The sequence characterization is applied to construct readily a （c）-Riordan array. In addition, subgrouping of （c）-Riordan arrays by using the characterizations is discussed. The （c）-Bell polynomials and its identities by means of convolution families are also studied. Finally, the characterization of （c）-Pdordan arrays in terms of the convolution families and （c）-Bell polynomials is presented.展开更多
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
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
Let ,4 = {1 ≤ a1 〈 a2 〈 ...} be a sequence of integers. ,4 is called a sum-free sequence if no ai is the sum of two or more distinct earlier terms. Let A be the supremum of reciprocal sums of sum-free sequences. In...Let ,4 = {1 ≤ a1 〈 a2 〈 ...} be a sequence of integers. ,4 is called a sum-free sequence if no ai is the sum of two or more distinct earlier terms. Let A be the supremum of reciprocal sums of sum-free sequences. In 1962, ErdSs proved that A 〈 103. A sum-free sequence must satisfy an ≥ （k ~ 1）（n - ak） for all k, n ≥ 1. A sequence satisfying this inequality is called a x-sequence. In 1977, Levine and O＇Sullivan proved that a x-sequence A with a large reciprocal sum must have al = 1, a2 = 2, and a3 = 4. This can be used to prove that λ 〈 4. In this paper, it is proved that a x-sequence A with a large reciprocal sum must have its initial 16 terms： 1, 2, 4, 6, 9, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, and 50. This together with some new techniques can be used to prove that λ 〈 3.0752. Three conjectures are posed.展开更多
文摘Here presented are the definitions of （c）-Riordan arrays and （c）-Bell polynomials which are extensions of the classical Riordan arrays and Bell polynomials. The characterization of （c）-Riordan arrays by means of the A- and Z-sequences is given, which corresponds to a horizontal construction of a （c）-Riordan array rather than its definition approach through column generating functions. There exists a one-to-one correspondence between GegenbauerHumbert-type polynomial sequences and the set of （c）-Riordan arrays, which generates the sequence characterization of Gegenbauer-Humbert-type polynomial sequences. The sequence characterization is applied to construct readily a （c）-Riordan array. In addition, subgrouping of （c）-Riordan arrays by using the characterizations is discussed. The （c）-Bell polynomials and its identities by means of convolution families are also studied. Finally, the characterization of （c）-Pdordan arrays in terms of the convolution families and （c）-Bell polynomials is presented.
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.
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.
基金supported by National Natural Science Foundation of China(Grant No.11071121)
文摘Let ,4 = {1 ≤ a1 〈 a2 〈 ...} be a sequence of integers. ,4 is called a sum-free sequence if no ai is the sum of two or more distinct earlier terms. Let A be the supremum of reciprocal sums of sum-free sequences. In 1962, ErdSs proved that A 〈 103. A sum-free sequence must satisfy an ≥ （k ~ 1）（n - ak） for all k, n ≥ 1. A sequence satisfying this inequality is called a x-sequence. In 1977, Levine and O＇Sullivan proved that a x-sequence A with a large reciprocal sum must have al = 1, a2 = 2, and a3 = 4. This can be used to prove that λ 〈 4. In this paper, it is proved that a x-sequence A with a large reciprocal sum must have its initial 16 terms： 1, 2, 4, 6, 9, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, and 50. This together with some new techniques can be used to prove that λ 〈 3.0752. Three conjectures are posed.
