# Research

## Analysis

### Friday, November 22, 2019

Title: Zagier-type formula for special multiple Hurwitz zeta values
Speaker: Cezar Lupu, Texas Tech University
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

In this talk, we provide a Zagier-type formula for the multiple $$t$$-values (special Hurwitz zeta values), \begin{gather*} t\left(k_{1},k_{2},\dotsc,k_{r}\right)=2^{-\left(k_{1}+k_{2}+\dotsb+k_{r}\right)}\zeta\left(k_{1},k_{2},\dotsc,k_{r};-\frac{1}{2},-\frac{1}{2},\dotsc,-\frac{1}{2}\right) \\ =\sum_{1\le n_{1}< n_{2}<\dotsb< n_{r}}\frac{1}{\left(2n_{1}-1\right)^{k_{1}}\left(2n_{2}-1\right)^{k_{2}}\dotsm\left(2n_{r}-1\right)^{k_{r}}}. \end{gather*}

Our formula is similar with Zagier's formulas for MZVs $$\zeta(2,\dotsc,2,3)$$ and will involve $$\mathbb{Q}$$-linear combinations of powers of $$\pi$$ and odd zeta values. The derivation of the formula for $$t(2,\dotsc,2,3)$$ relies on a rational zeta series approach via a Gauss hypergeometric argument.

### Friday, November 17, 2019

Title: Convergence behavior on the circle of optimal approximants, II
Speaker: Catherine Bénéteau
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, November 8, 2019

Title: Convergence behavior on the circle of optimal approximants
Speaker: Catherine Bénéteau
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

The objects of interest of my talk will be polynomials, called optimal approximants, that indirectly approximate inverses of functions in certain weighted analytic function spaces of the open unit disk. If the function being approximated is itself a polynomial, I will discuss an efficient method for computing the coefficients of the optimal approximants via projections using reproducing kernels, and will talk about how to use that information to examine the convergence properties of the optimal approximants on the unit circle.

### Friday, October 25, 2019

Title: The minimal embedding dimension for deformations of commutant pairs, II
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, October 18, 2019

Title: The minimal embedding dimension for deformations of commutant pairs
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

Let $$R$$ be an irreducible polynomial in two variables, and $$X$$, $$Y$$ two differential operators acting on functions of one complex variable. We call $$\{X,Y\}$$ a commutant pair if, for any polynomial $$P(X,Y)$$, we have $$\left[P(X,Y),[X,Y]\right]=0$$. Given $$R$$, a deformation of the commutant pair $$\{X,Y\}$$ is a family of pairs $$\left\{X_t,Y_t\right\}$$, where $$t$$ belongs to the interval $$[0,1]$$, such that $$R\left(X_t,Y_t\right)=0$$ for all $$t$$, and $$\left\{X_0,Y_0\right\}=\{X,Y\}$$. We discuss the problem of finding the minimal dimension, $$n$$, of the representation of $$\left\{X_t,Y_t\right\}$$ into the space of linear operators acting on entire functions taking values in $$C^n$$. The problem has a remarkable pedigree and surprisingly diverse connections to various mathematical fields.

### Friday, October 11, 2019

Title: The Mittag-Leffler Reproducing Kernel Hilbert Space, III
Speaker: Joel Rosenfeld
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, October 4, 2019

Title: The Mittag-Leffler Reproducing Kernel Hilbert Space, II
Speaker: Joel Rosenfeld
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, September 27, 2019

Title: The Mittag-Leffler Reproducing Kernel Hilbert Space
Speaker: Joel Rosenfeld
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

This talk will introduce the concept of fractional calculus, and presents a complex function space developed by me and my colleagues. The Mittag-Leffler space of entire functions is a one parameter generalization of the Fock space, and as such, many familiar properties of functions in the Fock space have direct generalizations in the Mittag-Leffler space. This talk will present an integral form of the norm of functions in the Mittag-Leffler space, demonstrate a modification of the Mittag-Leffler space that allows for the intertwining of fractional differentiation and multiplication by $$z^q$$, and time permitting, we will discuss some results concerning weighted composition operators over this space.

### Friday, September 20, 2019

Title: Stahl's theorem on a Riemann surface, Part III
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, September 13, 2019

Title: Stahl's theorem on a Riemann surface, Part II
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, September 6, 2019

Title: Stahl's theorem on a Riemann surface
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

H. Stahl's theorem on convergence of Pade approximats for analytic functions with branch points is one of the fundamental results in the theory of rational approximations of analytic functions. It is also one of the basic facts in the theory of orthogonal polynomials. Denominators of Pade approximats are complex (non-hermitian) orthogonal polynomials and Stahl's created an original method of investigating asymptotics of complex orthogonal polynomials based directly on complex orthogonality.