## Sunday, March 24, 2019

### Prime number conjectures from the Shapiro class structure

The first of hopefully many joint projects with my childhood friend Hartosh Singh Bal. For many reasons this has been a most exciting collaboration. For one thing, Hartosh and I have been discussing mathematical ideas since Class 11 in Modern School. So it was good to work on something which will lead to something new. For another, Shapiro was Hartosh's number theory Professor at NYU. And for three more reasons, you will have to look at the last section of this paper.

Here is the abstract:
The height $H(n)$ of  $n$, introduced by  Pillai in 1929,  is the  smallest positive integer $i$ such that the  $i$th iterate of Euler's totient function at $n$ is $1$.  H.  N. Shapiro (1943) studied the structure of the set of all numbers at a height. We provide a formula for the height function thereby extending a result of Shapiro. We list steps to generate numbers of any height which turns out to be a useful way to think of this construct. In particular, we extend some results of Shapiro regarding the largest odd numbers at a height. We present some theoretical and computational evidence to show that $H$ and its relatives are closely related to the important functions of number theory, namely $\pi(n)$ and the $n$th prime $p_n$. We conjecture formulas for $\pi(n)$ and $p_n$ in terms of the height function.
Here is a link to a preprint of the paper.

Prime number conjectures from the Shapiro class structure (with Hartosh Singh Bal), 17 pp.

From left to right: Sonit, Hartosh, me, Punya in 1983 or so

## Tuesday, February 12, 2019

### An infinite family of Borwein-type + - - conjectures

Another collaboration with Michael Schlosser written to celebrate the 80th birthday of Professor George Andrews. The paper's opening paragraph talks about a very interesting event that took place  in a conference celebrating George's birthday in June 2018.

The so-called Borwein conjectures, due to Peter Borwein (circa 1990), were  popularized by Andrews. The first of these concerns the expansion of finite products of the form
$$(1-q)(1-q^2)(1-q^4)(1-q^5)(1-q^7)(1-q^8)\cdots$$
into a power series in $q$ and the sign pattern displayed by the coefficients.  In June 2018, in a conference at Penn State celebrating Andrews' 80th birthday,  Chen Wang, a young Ph.D. student studying at the University of Vienna, announced that he has vanquished the first of the Borwein conjectures. In this paper, we  propose another set of Borwein-type conjectures.  The conjectures here are consistent with the first two Borwein conjectures as well as what is known about their refinement proposed by Andrews. At the same time, they do not appear to be very far from these conjectures in form and content.
Our first conjecture considers products of the form
$$\prod_{i=0}^{n-1} (1-q^{3i+1}) (1-q^{3i+2}) \prod_{j=1}^m \prod_{i=-n}^{n-1} (1-p^jq^{3i+1})(1- p^jq^{3i+2}) .$$
These are motivated by theta products.

Here is a link to a preprint of the paper.
A partial theta function Borwein conjecture, by Gaurav Bhatnagar and Michael Schlosser.

Here is a picture from a trip to Hong Kong for an OPSF meeting in June 2017.  From left to right: Heng Huat Chan (Singapore), Michael Schlosser (Vienna), Hjalmar Rosengren (Sweden), Shaun Cooper (New Zealand), me. A special team of Special Functions people from around the world!

## Thursday, January 31, 2019

### Orthogonal polynomials associated with continued fractions

My first joint paper with Professor Mourad Ismail. This has been a most interesting collaboration. Mourad taught me what to do on the sidelines of a series of meetings around the world. Most of these meetings were organized by the OPSF activity group of SIAM, one of the most interesting and diverse group of mathematicians and physicists. It began with a couple of meetings in Maryland in July 2016. The next one was in Hong Kong in July 2017, followed by a week long visit of Mourad to Austria (Oct 2017), where he gave me an exclusive, one-on-one, tutorial. Next we met in a summer school on $q$-series in Tianjin university in July-Aug 2018. Finally, we finished up things when I met him in Baltimore at the joint AMS meetings (Jan 2019), followed by a weekend trip Orlando, right after visiting UF in Gainesville.  This work was presented in Baltimore in a special session on continued fractions.

I hope it is the first in a series on Orthogonal Polynomials. There is much to learn and much to do.

Here is a link to the preprint on ArXiv.

***

Orthogonal polynomials associated with a continued fraction of Hirschhorn

Gaurav Bhatnagar and Mourad E. H. Ismail

Abstract

We study orthogonal polynomials associated with a continued fraction due to Hirschhorn.
Hirschhorn's continued fraction contains as special cases the famous Rogers--Ramanujan continued fraction and two of Ramanujan's generalizations. The orthogonality measure of the set of
polynomials obtained has an absolutely continuous component. We find generating functions, asymptotic formulas, orthogonality relations, and the Stieltjes transform of the measure. Using standard generating function techniques, we show how to obtain formulas for the convergents of Ramanujan's continued fractions, including a formula that Ramanujan recorded himself as Entry 16 in Chapter 16 of his second notebook.

***
Here is a picture of Mourad with me in Tianjin (July-Aug 2018).

The picture below is the conference group photo from Hong Kong (July 2017).

Mourad is seated in the front row second from the left. Many of the leading lights of the OPSF world are in this picture.