George E. Andrews

The combinatorics of the mock theta functions $$\nu(q)$$

When/Where:

March 14, 2017, 3:00 — 3:50 pm at LIT 368.

Abstract:

The third order mock theta function $$\nu(q)$$ has a variety of interesting combinatorial aspects.  We shall explore several and shall introduce a new class of partitions complete with a new crank and new congruence properties.

Alexander Berkovich

Non-negative Thoughts

When/Where:

February 28, 2017, 3:00 — 3:50 pm at LIT 368.

Abstract:

I discuss some new  inequalities involving certain  partitions with simple restrictions on largest and smallest parts. I will employ  these inequalities to show that

$$\displaystyle \sum_{n>0} \frac{q^{n(n+1)/2}}{(-q;q)_n(1-q^{L+n})(q;q)_L}$$

has non-negative  q-series coefficients for all positive integers $$L$$.

This talk is based on my current joint work with Ali K.  Uncu.

Colin Defant

Unitary Cayley Graphs

When/Where:

February 14, 2017, 3:00 — 3:50 pm at LIT 368.

Abstract:

If $$R$$ is a commutative ring with unity, then the unitary Cayley graph of $$R$$, denoted $$G_R$$ , is the graph with vertex set $$V(G_R)=R$$ and edge set $$E(G_R)=\{\{x,y\} : x-y\text{ is a unit in }R\}$$. We will focus specifically on the unitary Cayley graph of $$\mathbb{Z}/n\mathbb{Z}$$, which we may view as the graph with vertices $$0,1,…,n-1$$ in which two vertices are adjacent if and only if their difference is relatively prime to $$n$$. We provide the values of many graph parameters of these unitary Cayley graphs and find that they are intimately related to some interesting arithmetic functions. We also discuss an open problem concerning the domination numbers of these graphs.

Ali K. Uncu

Some new observations on partitions and divisors

When/Where:

January 31, 2017, 3:00 — 3:50 pm at LIT 368.

Abstract:

We will look through the history and many proofs from many academics of the result best known due to Fokkink, Fokkink and Wang. We will add more proofs to the list of proofs of this identity. Later we will move onto refinements of these results.

This discussion is about my recent joint work with Alexander Berkovich.

George E. Andrews

Topics in Partitions

When/Where:

January 24, 2017, 3:00 — 3:50 pm at LIT 368.

Abstract:

I will begin with the Bhargava-Adiga summation and how it relates Gauss’s Eureka Theorem to partitions.  Assuming that time permits, I will conclude by discussing joint work with David Newman on bizarre factorizations of the classical theta functions.

Atul Dixit

Transformations involving $$r_k(n)$$ and Bessel functions

When/Where:

January 10, 2017, 3:00 — 3:50 pm at LIT 368.

Abstract:

Let $$r_k(n)$$ denote the number of representations of the positive integer $$n$$ as the sum of $$k$$ squares, where $$k\geq 2$$. In 1934, the Russian mathematician Alexander Ivanovich Popov, who is more popularly known as one of the world experts in Finno-Ugric Linguistics, obtained a beautiful transformation between two series involving $$r_k(n)$$ and Bessel functions. Unfortunately, Popov’s proof appears to be defective since there are subtleties involved in extending the available results on $$r_2(n)$$ to those involving $$r_k(n)$$, $$k>2$$, and since the usual techniques do not carry over. In this work, we give a rigorous proof of Popov’s result by observing that N. S. Koshliakov’s ingenious proof of the Voronoi summation formula for coefficients of a Dirichlet series satisfying a functional equation with one gamma factor circumvents these difficulties. We then obtain an analogue of a double Bessel series identity on page 335 of Ramanujan’s Lost Notebook in the spirit of Popov’s identity.
In the second part of our talk, we will obtain a proof of a more general summation formula for $$r_k(n)$$ due to A. P. Guinand, which is claimed in his work without proof and without any conditions, and apply it to obtain a new transformation of a series involving $$r_k(n)$$ and a product of two Bessel functions. This transformation can be considered as a massive generalization of many well-known results in the literature, for example, those of A.L. Dixon and W.L. Ferrar, of G.H. Hardy, and of a classical result of Popov. This is joint work with Bruce C. Berndt, Sun Kim and Alexandru Zaharescu.

Frank Garvan

New Mock Theta Function Identities II

When/Where:

December 1, 2016, 3:00 — 3:50 pm at LIT 368.

Abstract:

This is a continuation of my previous talk.
In his last letter to Hardy, Ramanujan defined there mock theta functions of order 7 and stated that these three functions are not related. We find that there are actually some surprising relationships between these functions.

Frank Garvan

New Mock Theta Function Identities

When/Where:

November 10, 2016, 3:00 — 3:50 pm at LIT 368.

Abstract:

We give simple proofs of new Hecke-double sum identities for fifth and seventh order mock functions. The method involves various Bailey and conjugate Bailey pairs and an identity that Alex Berkovich mentioned in his seminar talk earlier this semester.

Alexander Berkovich

Partitions with non-repeating even parts. New Observations

When/Where:

October 27, 2016, 3:00 — 3:50 pm at LIT 368.

Abstract:

I explain how to  use the q-binomial theorem and the Gauss sum to prove a number of new theorems for partitions $$\pi$$  with non-repeating even parts, counted with weight $$(-1)^{\#\pi}$$.

This talk is based on my joint work with Ali Uncu.

Alexander Berkovich

New weighted partition identities involving partitions with distinct odd parts

When/Where:

October 20, 2016, 3:00 — 3:50 pm at LIT 368.

Abstract:

I  will show  how to use the Franklin involution to derive the Ramanujan-Fine identity. I will discuss  new combinatorial interpretation of this identity.This interpretation involves partitions with distinct odd parts such that the smallest  positive integer that is not a part of partition   is odd.

This talk is based on my joint work with Ali Uncu.