Event Title

On total positivity of Riordan arrays

Faculty Advisor

Tian-Xiao He

Graduation Year

2020

Location

Center for Natural Sciences

Start Date

4-4-2020 2:00 PM

End Date

4-4-2020 3:00 PM

Description

A Riordan array 𝑅 = (𝑔(π‘₯), 𝑓(π‘₯)) is defined as an infinite lower triangular matrix whose generating function of the kth column is 𝑔(π‘₯)𝑓(π‘₯)π‘˜, where 𝑔 and 𝑓 are formal power series with 𝑔(0)=1, 𝑓(0)=0, and 𝑓′(0) β‰  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 PΓ³lya 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.

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

On total positivity of Riordan arrays

Center for Natural Sciences

A Riordan array 𝑅 = (𝑔(π‘₯), 𝑓(π‘₯)) is defined as an infinite lower triangular matrix whose generating function of the kth column is 𝑔(π‘₯)𝑓(π‘₯)π‘˜, where 𝑔 and 𝑓 are formal power series with 𝑔(0)=1, 𝑓(0)=0, and 𝑓′(0) β‰  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 PΓ³lya 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.