# On total positivity of Riordan arrays

Event

Tian-Xiao He

2020

## Location

Center for Natural Sciences

4-4-2020 2:00 PM

4-4-2020 3:00 PM

## Disciplines

Education | Mathematics

## Abstract

A Riordan array �� = (��(��), ��(��)) is defined as an infinite lower triangular matrix whose generating function of the kth column is ��(��)��(��)��, where �� and �� are formal power series with ��(0)=1, ��(0)=0, and ��′(0) ≠ 0. The set of all Riordan arrays forms a group called the Riordan group. The total positivity of R can be characterized by using the generating functions of its A- and Z- sequences. A finite sequence of nonnegative numbers is a Pólya frequency sequence (PF for short) if and only if its generating function only has real zeros. In particular, the set of all Bell-type Riordan arrays is an important subgroup of the Riordan group. Pascal triangle, for example, is one of the well-known Bell-type Riordan arrays. A Riordan array is total positive if the A-sequence is a PF sequence. We will study the total positivity of Bell-type Riordan arrays and construct Bell-type Riordan arrays with total positivity by using their A-sequences. We will also give the combinatorial interpretations of those Riordan arrays by using lattice paths. As one of the results, we find new sequences that are not included in OEIS.

## Share

COinS

Apr 4th, 2:00 PM Apr 4th, 3:00 PM

On total positivity of Riordan arrays

Center for Natural Sciences

A Riordan array �� = (��(��), ��(��)) is defined as an infinite lower triangular matrix whose generating function of the kth column is ��(��)��(��)��, where �� and �� are formal power series with ��(0)=1, ��(0)=0, and ��′(0) ≠ 0. The set of all Riordan arrays forms a group called the Riordan group. The total positivity of R can be characterized by using the generating functions of its A- and Z- sequences. A finite sequence of nonnegative numbers is a Pólya frequency sequence (PF for short) if and only if its generating function only has real zeros. In particular, the set of all Bell-type Riordan arrays is an important subgroup of the Riordan group. Pascal triangle, for example, is one of the well-known Bell-type Riordan arrays. A Riordan array is total positive if the A-sequence is a PF sequence. We will study the total positivity of Bell-type Riordan arrays and construct Bell-type Riordan arrays with total positivity by using their A-sequences. We will also give the combinatorial interpretations of those Riordan arrays by using lattice paths. As one of the results, we find new sequences that are not included in OEIS.