# On total positivity of Riordan arrays

## Submission Type

Event

## Faculty Advisor

Tian-Xiao He

## Expected Graduation Date

2020

## Location

Center for Natural Sciences

## Start Date

4-4-2020 2:00 PM

## End Date

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.

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.