Showing posts with label q-series. Show all posts
Showing posts with label q-series. Show all posts

### Telescoping continued fractions

Krishnan Rajkumar (JNU) and I have a new preprint -- on telescoping continued fractions. I have written previously on telescoping and several times on continued fractions, but this one is unique. I don't think anyone has tried to combine the two ideas. We apply it to find lower bounds for the remainder term for Stirling's formula. Ultimately, we discovered a nice new technique, proved several things required to make it work, but were unable to take it to its natural conclusion (so far!). The preprint (now available at arxiv) has several conjectures.

The pic is taken on June 9, 2022, in the Blue Door Cafe, Khan Market where we revised the paper. (That day I showed symptoms of COVID, my second time.)

Here is a link to the preprint.

Here is the abstract:

Title: Telescoping continued fractions for the error term in Stirling's formula
Authors: Gaurav Bhatnagar, Krishnan Rajkumar
Categories: math.CA math.NT

In this paper, we introduce telescoping continued fractions to find lower
bounds for the error term $r_n$ in Stirling's approximation $n! =\sqrt{2\pi}n^{n+1/2}e^{-n}e^{r_n}.$ This improves lower bounds given earlier by Cesàro (1922), Robbins (1955), Nanjundiah (1959), Maria (1965) and Popov (2017). The expression is in terms of a continued fraction, together with an algorithm to find successive terms of this continued fraction. The technique we introduce allows us to experimentally obtain upper and lower bounds for a sequence of convergents of a continued fraction in terms of a difference of two continued fractions.

Here is a talk introducing the method. I presented in the Ashoka Math Colloquium on November 2, 2021. It has an overview of the technique. The talk was made keeping undergraduate students in mind, so there is something here which is quite accessible. In particular I have outlined Robbins' approach at the beginning of the talk.

### Andrews' approach to conjecture the Rogers-Ramanujan identities

If something is worth doing, then I suppose it is worth doing again.

I had previously written an article on how to discover the Rogers-Ramanujan identities. That was based on ideas of Dick Askey. In this talk I presented an introduction to partitions  presenting many results of Euler and ending with George Andrews' approach to discover the Rogers-Ramanujan identities. This approach was given in his Number Theory book, and it seems that it is not as well known as it should be. A notation that my collaborator Hartosh Singh Bal and I use to gain intuition is also explained here.

Abstract

The two Rogers-Ramanujan identities were sent by Ramanujan to Hardy in a letter in 1913. As an example, here is the first Rogers-Ramanujan identity:

$$1+\frac{q}{(1-q)}+\frac{q^4}{(1-q)(1-q^2)}+\frac{q^9}{(1-q)(1-q^2)(1-q^3)}+\cdots$$

$$=\frac{1}{(1-q)(1-q^6)(1-q^{11})\dots}\times \frac{1}{(1-q^4)(1-q^{9})(1-q^{14})\dots}$$

They look less forbidding when interpreted in terms of partitions, which is how MacMahon considered them. A partition of a number  is a way of writing it as an unordered sum of other numbers. Unordered means that $2+3$ and $3+2$ are considered the same. For example,

$$5 = 4+1 = 3+2 = 3+1+1 = 2+1+1+1$$

are partitions  of $5$. (Two partitions of $5$ are missing in this list; can you find them?) The theory of partitions is an attractive area of mathematics, where many complicated formulas are rendered completely obvious by making the `right picture'. However, while each side of the Rogers--Ramanujan identities are represented naturally in terms of partitions, they are still far from obvious.

In this talk, we will introduce partitions, explain how to enumerate them systematically, represent them graphically, and write their generating functions. We present an experimental approach  to discover the Rogers-Ramanujan identities. This approach is due to Professor George Andrews of Penn State University.

### Thank you, Dick

 From R to L (facing camera): Ae Ja Yee, Bruce Berndt, Dick Askey, Shaun Cooper, Michael Schlosser, and me At Alladi 60 conference at a conference reception at the Alladi residence

Howard Cohl and Mourad Ismail created a Liber Amicorum (Friendship Book) to present to Richard Askey. Askey is not well, and he and his wife Liz have moved into a Hospice in Wisconsin. Askey is our leader, the leader of our field, and of the people in the field.

UPDATE (October 9, 2019): Alex Berkovich and Howard Cohl informed that Dick is no more.

My entry for his book is here: Thank you, Dick

The title is appropriate. When Dick autographed my copy of the book, Special Functions by Andrews, Askey and Roy, he wrote "Thank you for your work, early and now". (Here is my book review of this book.) He gave extensive comments on receiving a draft copy of my unpublished book "Experience Mathematics" and tried to help me get it published. My paper "How to discover the Rogers-Ramanujan Identities" is essentially an expansion of something that took Dick a couple of paragraphs.

When I returned to mathematics, I hung out with him in many conferences, and he was very supportive, coming for my talks, making remarks. In general, he was very welcoming. I have missed him the last couple of years.

### 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.

UPDATE (September 16, 2019). The paper has been accepted to appear in the Andrews 80 Special Issue in the Annals of Combinatorics.

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!

### 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 to 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.

### A bibasic Heine transformation formula

While studying chapter 1 of Andrews and Berndt's Lost Notebook, Part II, I stumbled upon a bibasic Heine's transformation. A special case is Heine's 1847 transformation. Other special cases include an identity of Ramanujan (c. 1919), and  a 1966 transformation formula of Andrews. Eventually, I realized that it follows from a Fundamental Lemma given by Andrews in 1966. Still, I'm happy to have rediscovered it. Using this formula one can find many identities proximal to Ramanujan's own $_2\phi_1$ transformations.

And of course, the multiple series extensions (some in this paper, and others appearing in another paper) are all new.

Here is a preprint.

Here is a video of a talk I presented at the Alladi 60 Conference. March 17-21, 2016.

Update (November 10, 2018). The multi-variable version has been accepted for publication in the Ramanujan Journal. This has been made open access. It is now available online, even though the volume and page number has not been decided yet. The title is: Heine's method and $A_n$ to $A_m$ transformation formulas.

Here is a reprint.
--

UPDATE (Feb 11, 2016). This has been published. Reference (perhaps to be modified later): A bibasic Heine transformation formula and Ramanujan's $_2\phi_1$ transformations, in Analytic Number Theory, Modular Forms and q-Hypergeometric Series, In honor of Krishna Alladi's 60th Birthday, University of Florida, Gainesville, Mar 2016,  G. E. Andrews and F. G. Garvan (eds.), 99-122 (2017)

The book is available here. The front matter from the Springer site.

--

UPDATE (June 16, 2016).  The paper has been accepted to appear in: Proceedings of the Alladi 60 conference held in Gainesville, FL. (Mar 2016), K. Alladi, G. E. Andrews and F. G. Garvan (eds.)

### WP Bailey Lemmas (Elliptic, multivariable)

After many many years, Michael Schlosser and I wrote another joint paper. We first collaborated in 1995-96 when both of us were Ph.D. students or shortly thereafter. Our joint work was part of his thesis, and published in Constructive Approximation. This time around, I was his post-doc in Vienna from Feb 1, 2016 to Feb 28, 2017.

The picture was taken in Strobl, a favorite place for small meetings and conferences for Krattenthaler's group in the University of Vienna.

In this paper, we give multivariable extensions (over root systems) of the elliptic well-poised (WP) Bailey Transform and Lemma. In the classical (i.e. dimension = 1) case, this work was done by Spiridonov, who in turn extended the work of Andrews and Bailey. It is Andrews' exposition which we found very useful while finding generalizations. We used  previous $q$-Dougall summations due to Rosengren, and Rosengren and Schlosser, and found a few of our own along with some new elliptic Bailey $_{10}\phi_9$ transformation formulas, extending some fundamental formulas given in the classical case by Frenkel and Turaev in 1997. Along the way, we discovered a nice trick to generalize the theorem of my advisor, Steve Milne, that  I had named "Fundamental Theorem of $U(n)$ series" in my thesis.

Hopefully, there will be many more collaborative ventures in the near future.

Update (Mar 22, 2018): The paper has been published. Here is the reference and Link:
G. Bhatnagar and  M.J. Schlosser, Elliptic well-poised Bailey transforms and lemmas on root systems, SIGMA, 14 (2018), 025, 44pp.

### Spiral Determinants

We consider Spiral Determinants of the kind
$$\text{det}\left( \begin{matrix} {16}&{15}&{14}&{13}\\ {5}&{4}&{3}&{12}\\ {6}&1&{2}&{11}\\ {7}&{8}&{9}&{10} \end{matrix} \right)$$
and
$$\text{det} \left( \begin{matrix} {17}&{16}&{15}&{14}&{13}\\ {18}&{5}&{4}&{3}&{12}\\ {19}&{6}&1&{2}&{11}\\ {20}&{7}&{8}&{9}&{10}\\ {21}&{22}&{23}&{24}&{25} \end{matrix} \right)$$
where the entries spiral out from the center. Christian Krattenthaler, who is one of the greatest experts on determinants, tells the story of how he came across such determinants and how he went about discovering the formulas for such determinants. The preprint is available on arxiv.

I have wanted to work with Christian ever since my Ph.D. days, when I tried to generalize a matrix inversion due to him. Finally, we have a joint paper. This also means that my Erdos number has come down from 4 to 3.

The picture above is from Christian's course on "Bijections" which I had an opportunity to attend in the University of Vienna during the period October 2016 to January 2017.

Update: April 26, 2017  The paper has been accepted and will appear in Linear Algebra and its Applications. Here is a preprint on arxiv.
Update: May 10, 2017. The paper is published online. The reference is:
G. Bhatnagar and C. Krattenthaler, Spiral Determinants, Linear Algebra Appl., 529 (2017) 374-390.
Here is a link to the publisher's site: https://www.sciencedirect.com/science/article/pii/S0024379517302719

### How to Discover the Rogers-Ramanujan Identities

Dec 22, 2012: It is Ramanujan's 125th birthday, but how many of his famous identities do you know? Here we examine a method to conjecture two very famous identities that were conjectured by Ramanujan, and later found to be known to Rogers.

Here is a link:  How to Discover the Rogers-Ramanujan Identities.

This was presented to a some high school math teachers in a conference. I tried to write it in a way that it could be understood by a motivated high school student.

Update (May 26, 2015): The article has been published. Here is a reference. Resonance, 20 (no. 5), 416-430, (May 2015).

Update (January 18, 2014): This article has been accepted for publication in Resonance, a popular science magazine aimed at the undergraduate level.

### How to prove Ramanujan's q-Continued Fractions

 The photograph of a page in Ramanujan's Lost Notebook where he expanded a ratio of two series in terms of three continued fractions. These three are among the continued fractions explained in this paper.

Its the 125th year of Ramanujan's birth, but how many of his formulas do you know? Here is an opportunity to get familiar with 9 of Ramanujan's continued fraction formulas. These include the three continued fractions that appear in the Lost Notebook in the above photograph.

Abstract:
By using Euler's approach of using Euclid's algorithm to expand a power series into a continued fraction, we show how to derive Ramanujan's $q$-continued fractions in a systematic manner.

A Preprint of this expository paper is now available from arXiv. The latest version fixes a typo. The final version appears in this book. You may wish to buy/access the entire volume from the AMS, its really an amazing piece of work.

Update (Sept 7, 2018): I presented this topic in IISER, Mohali, after adding a few ideas from the recent joint work with Mourad Ismail. Here is the presentation.

Update (December 20, 2014): Published in Contemporary Mathematics: Ramanujan 125, K. Alladi, F. Garvan, A. J. Yee (eds.)  627, 49-68 (2014)

Update (July 17, 2013): Accepted for publication in the proceedings of Ramanujan 125, where I presented this paper.

### The q-analog of the Gamma Function

I have begun reading Bruce Berndt's "Ramanujan's Notebooks", Part III. Here is a small morsel from Ramanujan's table: Entry 1(ii) of Chapter 16 of his Notebooks. Its a discovery proof of the limit of the $q$-Gamma function, as $q$ goes to 1. In my humble opinion, this is easier than the usual proof (due to Gosper) which appears in Gasper and Rahman.

The $q$-analog of the Gamma Function

The objective of this note is to show how to arrive at the definition of the $q$-analog of the Gamma function. To do so, we "discover" the limit:
\label{entry1ii} \newcommand{\qrfac}[2]{{\left({#1}; q\right)_{#2}}} \lim_{q\to 1} \frac{\qrfac{q}{\infty}}{(1-q)^x \qrfac{q^{x+1}}{\infty}}= \Gamma (x+1).

Recall the limit definition of the Gamma function (from, for example Rainville [5, p. 11]):
$$\Gamma(x+1):=\lim_{n\to \infty} \frac{n! n^x}{(x+1)(x+2)\cdots (x+n)}.$$
To derive \eqref{entry1ii}, we find a $q$-analog of this limit. To that end, we use:
1.  $\displaystyle \lim_{q\to 1} \frac{\qrfac{q}{n}}{(1-q)^n} = n!$
2. $\displaystyle \lim_{q\to 1} \left(\frac{1-q^n}{1-q}\right)^x =n^x$
3. $\displaystyle \lim_{q\to 1} \frac{\qrfac{q^{x+1}}{n}}{(1-q)^n}=(x+1)(x+2)\cdots (x+n)$
Thus, we have
\begin{align*}\require{cancel} \Gamma(x+1)&= \lim_{n\to \infty} \frac{n! n^x}{(x+1)(x+2)\cdots (x+n)}\cr
& = \lim_{n\to \infty} \lim_{q\to 1} \frac{(1-q)^n}{\qrfac{q^{x+1}}{n}}\cdot \frac{\qrfac{q}{n}}{(1-q)^n} \cdot \left(\frac{1-q^n}{1-q}\right)^x\cr &= \lim_{q\to 1}\lim_{n\to \infty} \frac{\cancel{(1-q)^n}}{\qrfac{q^{x+1}}{n}}\cdot \frac{\qrfac{q}{n}}{\cancel{(1-q)^n}} \cdot \left(\frac{1-q^n}{1-q}\right)^x\cr &= \lim_{q\to 1} \frac{\qrfac{q}{\infty}}{\qrfac{q^{x+1}}{\infty}} \frac{1}{(1-q)^x}. \end{align*}
Here, we assume that the limits can be interchanged, and $|q|<1$. This completes the derivation of \eqref{entry1ii}.

Given the relation \eqref{entry1ii}, we can define the $q$-Gamma function, for $|q|<1$, as \label{qgammadef} \Gamma_q (x)= \frac{\qrfac{q}{\infty}}{(1-q)^{x-1} \qrfac{q^{x}}{\infty}}.

Remarks. The proof by Gosper, reported by Andrews [1] and reproduced in Gasper and Rahman [4] uses Euler's Product definition of the Gamma Function. Equation \eqref{entry1ii} is Entry 1(ii) in Berndt [2, ch.16]. The limit definition is entry 2293 in Carr's book [3], so Ramanujan had access to it.

References
1. G. E. Andrews, $q$-Series: Their development and application in analysis, number theory, combinatorics, physics and computer algebra, NSF CBMS Regional Conference Series, 66 1986.
2. B. C. Berndt, Ramanujan's Notebooks, Part III, Springer Verlag, New York, 1991.
3. G. S. Carr, Formulas and Theorems of Pure Mathematics, 2nd ed., Chelsea, NY, 1970.
4. G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics And Its Applications 35, Cambridge University Press, Cambridge, 1990; Second Ed. (2004).
5. E. D. Rainville, Special Functions, Chelsea, NY (1960).

### The q-disease

Special Functions,
Here's a belated review,
and a thank you.

~*~*~*~*~*~

Beauty in mathematics,
said Polya,
is seeing the truth
without effort.

Everything
in The Book
is as elegant,
as could be.

Everything
as simple,
as effortless,
as should be.

Everything
as beautiful,
as it is.

~*~*~*~*~*~