## Sudler’s Products. Some New Observations

#### When/Where:

April 17, 2018, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

I will discuss my latest joint work with Ali Uncu (RISC JKU)

April 17, 2018, 3:00 — 3:50 pm at LIT 368.

I will discuss my latest joint work with Ali Uncu (RISC JKU)

March 20, 2018, 3:00 — 3:50 pm at LIT 368.

I consider the leading root \(x_0(q)\) of Ramanujan’s function (or \(q\)-Airy function) \(\sum\limits_{n=0}^\infty\frac{(-x)^nq^{n^2}}{(1-q)(1-q^2)\ldots(1-q^n)}\). I prove that its formal power series expansion

\(qx_0(-q)=1+q+q^2+2q^3+4q^4+8q^5+\ldots\)

has positive integer-valued coefficients, by giving an explicit combinatorial interpretation of these numbers in terms of trees whose vertices are decorated with polyominos.

Similar results are also obtained for the leading roots of the partial Theta function and the Painleve Airy function.

March 13, 2018, 3:00 — 3:50 pm at LIT 368.

We introduce a deformed version of Dyck paths (DDP), where additional to the steps allowed for Dyck paths, “jumps” orthogonal to the preferred direction of the path are permitted. We consider the generating function of DDP, weighted with respect to their half length, area and number of jumps. This represents the first example of a hierarchy of exact solvable two-dimensional lattice vesicle models showing higher-order multi-critical points with scaling functions expressible via generalized Airy functions, as conjectured bu John Cardy.

February 27, 2018, 3:00 — 3:50 pm at LIT 368.

We derive new expressions for the generating functions of q-tangent and q-secant numbers from enumerating path diagrams given by Dyck paths together with partial fillings below the paths. In doing so, we provide expressions for path diagrams with restricted height involving basic hypergeometric functions, obtained by solving recurrences arising from the continued fraction representation of the generating functions.

February 20, 2018, 3:00 — 3:50 pm at LIT 368.

Euler showed that the values of the Riemann zeta function at positive even integer arguments \(2k\) are rational multiples of \(\pi^{2k}\), these rationals being given in terms of Bernoulli numbers. Over the years, several proofs of this celebrated result of Euler have been given. We will discuss a new proof by simply starting with the determination of the Fourier coefficients of \(f(x)=x^k\), and using the Parseval identity. This leads to a pair of intertwining recurrences, which when investigated closely leads to a very different proof of Euler’s formula and a surprising new identity for Bernoulli numbers.

This is joint work with Colin Defant. Time permitting,

I will also discuss connections between Bernoulli polynomials and the analytic continuation of the Riemann zeta function.

February 13, 2018, 3:00 — 3:50 pm at LIT 368.

The Mock Theta Conjectures were identities stated by Ramanujan for his so called fifth order mock theta functions. Andrews and the speaker showed how two of these fifth order functions are related to rank differences mod 5. Hickerson was first to prove these identities and was also able to relate the three Ramanujan seventh order mock theta functions to rank differences mod 7. Based on work of Zwegers, Zagier observed that the two fifth order functions and the three seventh order functions are holomorphic parts of real analytic vector modular forms on \(SL_2(Z)\). Zagier gave an indication how these functions could be generalized. We give details of these generalizations and show how Zagier’s 11th order functions are related to rank differences mod 11.

February 6, 2018, 3:00 — 3:50 pm at LIT 368.

David Newman has number theory as his hobby. He and I have corresponded for more than 20 years. Recently we have written two joint papers. The first concerns special, non-standard infinite product representations of classical theta functions. The second concerns partition problems related to Fraenkel’s MEX function in partitions. MEX = MinimalEXcludant.

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

Let

\(\displaystyle (1-q)(1-q^2) … (1-q^m) = \sum_{n>= 0} c_m (n) q^n.\)

In this talk I discuss how to use \( q\)- binomial theorem together with the Euler pentagonal number theorem to show that \(\max( |c_m|) = 1\) iff \( m = 0,\ 1,\ 2,\ 3,\ 5\). There are many other similar results. For example, it can be proven that no positive integer m exists such that \(\max( |c_m|) = 9\).

This talk is based on my recent joint work with Ali Uncu (RISC, Linz).

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

In this talk, I will summarize forthcoming work with Griffin, Ono, and Zagier. In 1927 Pólya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann’s Xi-function. This hyperbolicity has been proved for degrees \(d\leq3\). We obtain an arbitrary precision asymptotic formula for the derivatives \(\Xi^{(2n)}(0)\), which allows us to prove the hyperbolicity of \(100\%\) of the Jensen polynomials of each degree. We obtain a general theorem which models such polynomials by Hermite polynomials. This general condition also confirms a conjecture of Chen, Jia, and Wang.

March 21, 2017, 3:00 — 3:50 pm at LIT 368.

An overview of application of the Fourier analysis applied to the study of multiplicative groups of rationals.