## Sunday, February 20, 2022

### Krishnan Rajkumar (JNU) - Thursday Feb 24, 2022 - 3:55 PM (IST)

The next talk is by Krishnan Rajkumar. This talk will be recorded but not available online immediately.

Talk Announcement:

Title: The Binet function and telescoping continued fractions

Speaker: Krishnan Rajkumar (Jawaharlal Nehru University (JNU))

When: Thursday, Feb 24, 2022 - 4:00 PM - 5:00 PM (IST)

Abstract:

The Binet function $J(z)$ defined by the equation $\Gamma(z) = \sqrt{2 \pi} z^{z-\frac{1}{2}}e^{-z} e^{J(z)}$ is a well-studied function. The Stirling approximation comes from the property $J(z) \rightarrow 0$ as $z\rightarrow \infty$, $|arg z|<\pi$. In fact, an asymptotic expansion $J(z) \sim z^{-1} \sum_{k=0}^{\infty} c_k z^{-2k}$ holds in this region, with closed form expressions for $c_k$ and explicit integrals for the error term for any finite truncation of this asymptotic series.

In this talk, we will discuss two different classical directions of research. The first is exemplified by the work of Robbins (1955) and Cesaro (1922), and carried forward by several authors, the latest being Popov (2018), where elementary means are used to find rational lower and upper bounds for $J(n)$ which hold for all positive integers $n$. All of these establish inequalities of the form $J(n)-J(n+1) > F(n)-F(n+1)$ for an appropriate rational function $F$ to derive the corresponding lower bounds by telescoping.

The second direction is to use moment theory to derive continued fractions of specified forms for $J(x)$. For instance, a modified S-fraction of the form $\frac{a_1}{x \ +} \frac{a_2}{x \ +}\frac{a_3}{x \ +} \cdots$ can be formally derived from the above asymptotic expansion using a method called the qd-algorithm. The resulting continued fraction can then be shown to converge to $J(x)$ by the asymptotic properties of $c_k$ and powerful results from moment theory. There are no known closed-form expressions for the $a_k$.

We will then outline what we call the method of telescoping continued fractions to extend the elementary methods of the first approach to derive the modified S-fraction for $J(x)$ obtained in the second by a new algorithm. We will describe several results that we can prove and some conjectures that together enhance our understanding of the numbers $a_k$ as well as provide upper and lower bounds for $J(x)$ that improve all known results.

This is joint work with Gaurav Bhatnagar.
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Gaurav Bhatnagar (Ashoka), Atul Dixit (IIT, Gandhinagar) and Krishnan Rajkumar (JNU)

## Saturday, February 5, 2022

### Surbhi Rai (IIT, Delhi) - Thursday, Feb 10, 2022 - 3:55 PM IST

Dear all,

The next talk is by Surbhi Rai, a graduate student in IIT, Delhi. The announcement is as follows.

Talk Announcement:

Title: Expansion Formulas for Multiple Basic  Hypergeometric Series Over Root Systems

Speaker: Surbhi Rai (IIT, Delhi)

When: Thursday, Feb 10, 2022 - 4:00 PM - 5:00 PM (IST)

Abstract:  In a series of works, Zhi-Guo Liu extended some of the central summation and transformation formulas of basic hypergeometric series. In particular, Liu extended Rogers' non-terminating very-well-poised  $_{6}\phi_{5}$  summation formula, Watson's transformation
This talk will present several infinite families of extensions of Liu's fundamental formulas to multiple basic hypergeometric series over root systems. We will also discuss results that extend Wang and Ma's generalizations of Liu's work which they obtained using $q$-Lagrange inversion. Subsequently, we will look at an application based on the expansions of infinite products. These extensions have been obtained using the $A_n$ and $C_n$ Bailey transformation and other summation theorems due to Gustafson, Milne, Milne and Lilly, and others, from $A_n$, $C_n$ and $D_n$ basic hypergeometric series theory. We will observe how this approach brings Liu's expansion formulas within the Bailey transform methodology.