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Gaurav Bhatnagar

Publications and source records attributed to Gaurav Bhatnagar.

At least 19 recordsLinked to original sources
arXiv2026

The method of telescoping continued fractions

We give an approach to discover continued fractions for series of the form $$\sum_{k=0}^\infty \frac{\epsilon^k}{(x+k)^s},$$ where $\epsilon = \pm 1$. We find a continued fraction of the form \begin{equation*} \frac{a_1}{b_1(x)} \fplus \frac{a_2}{b_2(x)} \fplus \fdots \end{equation*} where $a_k$ are constants and $b_k(x)$ are polynomials. Our technique involves telescoping continued fractions. This provides a discovery approach to continued fractions given by Ramanujan for $$2\sum_{k=1}^\infty \frac{(-1)^{k+1}}{x+2k-1}, 2\sum_{k=0}^\infty \frac{1}{(x+2k+1)^2}, 2\sum_{k=0}^\infty \frac{(-1)^k}{(x+2k+1)^2}, \sum_{k=1}^\infty \frac{1}{(x+k)^3}, $$ and the like. We display the first few terms of several continued fractions obtained in this manner for larger values of $s$, including $s=5, 7, 9, 11$. They do not follow as simple a pattern as Ramanujan's continued fractions.

math.NT↗
arXiv2024

Expansion formulas for elliptic hypergeometric series

We provide an alternate approach to obtaining expansion formulas on the lines of the well-poised Bailey lemma. We recover results due to Spiridonov and Warnaar and one new formula of this type. These formulas contain an arbitrary sequence as an argument, and are thus flexible in the number of parameters they contain. As a result, we are able to derive $19$ new transformation formulas for elliptic hypergeometric series. These transformation formulas appear to be new even in the basic hypergeometric case, when $p=0$.

math.NT↗
arXiv2023

An esoteric identity with many parameters and other elliptic extensions of elementary identities

We provide elliptic extensions of elementary identities such as the sum of the first $n$ odd or even numbers, the geometric sum and the sum of the first $n$ cubes. Many such identities, and their $q$-analogues, are indefinite sums, and can be obtained from telescoping. So we used telescoping in our study to find elliptic extensions of these identities. In the course of our study, we obtained an identity with many parameters, which appears to be new even in the $q$-case. In addition, we recover some $q$-identities due to Warnaar.

math.NT↗
arXiv2023

Telescoping continued fractions for the error term in Stirling's formula

In this paper, we introduce telescoping continued fractions to find lower bounds for the error term $r_n$ in Stirling's approximation $\displaystyle n! = \sqrt{2π}n^{n+1/2}e^{-n}e^{r_n}.$ 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.

math.CA↗
arXiv2023

A weighted extension of Fibonacci numbers

We extend Fibonacci numbers with arbitrary weights and generalize a dozen Fibonacci identities. As a special case, we propose an elliptic extension which extends the $q$-Fibonacci polynomials appearing in Schur's work. The proofs of most of the identities are combinatorial, extending the proofs given by Benjamin and Quinn, and in the $q$ case, by Garrett. Some identities are proved by telescoping.

math.CO↗
arXiv2022

Ramanujan's $q$-continued fractions

Ramanujan's $q$-continued fractions are a central part of Ramanujan's development of basic hypergeometric series. They appear in Chapter 16 of Part III and Chapter 32 of Part V of {\em Ramanujan's Notebooks} edited by Berndt, and in Volume I of Andrews and Berndt's {\em Ramanujan's Lost Notebook}. In these references the continued fractions as presented in the order in which they appear in Ramanujan's original notebooks. We summarize the work of several authors on this topic and re-organize Ramanujan's $q$-continued fractions.

math.CA↗
arXiv2022

Glaisher's divisors and infinite products

Ramanujan gave a recurrence relation for the partition function in terms of the sum of the divisor function $σ(n)$. In 1885, J.W. Glaisher considered seven divisor sums closely related to the sum of the divisors function. We develop a calculus to associate a generating function with each of these divisor sums. This yields analogues of Ramanujan's recurrence relation for several partition-theoretic functions as well as $r_k(n)$ and $t_k(n)$, functions counting the number of ways of writing a number as a sum of squares (respectively, triangular) numbers. As by-products of this association, we obtain several convolutions, recurrences and congruences for divisor functions. We give alternate proofs of two classical theorems, one due to Legendre and the other -- Ramanujan's congruence $p(5n+4) \equiv 0 \pmod 5$.

math.NT↗
arXiv2022

Orthogonal polynomials associated with a continued fraction of Hirschhorn

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.

math.CA↗
arXiv2022

Expansion formulas for multiple basic hypergeometric series over root systems

We extend expansion formulas of Liu given in 2013 to the context of multiple series over root systems. Liu and others have shown the usefulness of these formulas in Special Functions and number-theoretic contexts. We extend Wang and Ma's generalizations of Liu's work which they obtained using $q$-Lagrange inversion. We use the $A_n$ and $C_n$ Bailey transformation and other summation theorems due to Gustafson, Milne, Milne and Lilly, and others, from the theory of $A_n$, $C_n$ and $D_n$ basic hypergeometric series.

math.CA↗
arXiv2021

Knowledge driven Description Synthesis for Floor Plan Interpretation

Image captioning is a widely known problem in the area of AI. Caption generation from floor plan images has applications in indoor path planning, real estate, and providing architectural solutions. Several methods have been explored in literature for generating captions or semi-structured descriptions from floor plan images. Since only the caption is insufficient to capture fine-grained details, researchers also proposed descriptive paragraphs from images. However, these descriptions have a rigid structure and lack flexibility, making it difficult to use them in real-time scenarios. This paper offers two models, Description Synthesis from Image Cue (DSIC) and Transformer Based Description Generation (TBDG), for the floor plan image to text generation to fill the gaps in existing methods. These two models take advantage of modern deep neural networks for visual feature extraction and text generation. The difference between both models is in the way they take input from the floor plan image. The DSIC model takes only visual features automatically extracted by a deep neural network, while the TBDG model learns textual captions extracted from input floor plan images with paragraphs. The specific keywords generated in TBDG and understanding them with paragraphs make it more robust in a general floor plan image. Experiments were carried out on a large-scale publicly available dataset and compared with state-of-the-art techniques to show the proposed model's superiority.

cs.CV↗
arXiv2021

GRIHA: Synthesizing 2-Dimensional Building Layouts from Images Captured using a Smart Phone

Reconstructing an indoor scene and generating a layout/floor plan in 3D or 2D is a widely known problem. Quite a few algorithms have been proposed in the literature recently. However, most existing methods either use RGB-D images, thus requiring a depth camera, or depending on panoramic photos, assuming that there is little to no occlusion in the rooms. In this work, we proposed GRIHA (Generating Room Interior of a House using ARCore), a framework for generating a layout using an RGB image captured using a simple mobile phone camera. We take advantage of Simultaneous Localization and Mapping (SLAM) to assess the 3D transformations required for layout generation. SLAM technology is built-in in recent mobile libraries such as ARCore by Google. Hence, the proposed method is fast and efficient. It gives the user freedom to generate layout by merely taking a few conventional photos, rather than relying on specialized depth hardware or occlusion-free panoramic images. We have compared GRIHA with other existing methods and obtained superior results. Also, the system is tested on multiple hardware platforms to test the dependency and efficiency.

cs.CV↗
arXiv2021

Stanley--Elder--Fine theorems for colored partitions

We give a new proof of a partition theorem popularly known as Elder's theorem, but which is also credited to Stanley and Fine. We extend the theorem to the context of colored partitions (or prefabs). More specifically, we give analogous results for $b$-colored partitions, where each part occurs in $b$ colors; for $b$-colored partitions with odd parts (or distinct parts); for partitions where the part $k$ comes in $k$ colors; and, overpartitions.

math.CO↗
arXiv2021

The Partition-Frequency Enumeration Matrix

We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite upper-triangular number-theoretic matrix. We call this matrix the Partition-Frequency Enumeration (PFE) matrix. This matrix unifies a large number of results connecting number-theoretic functions to partition-type functions. The calculus is extended to arbitrary generating functions, and functions with Weierstrass products. As a by-product, we recover (and extend) some well-known recurrence relations for many number-theoretic functions, including the sum of divisors function, Ramanujan's $\tau$ function, sums of squares and triangular numbers, and for $\zeta(2n)$, where $n$ is a positive integer. These include classical results due to Euler, Ewell, Ramanujan, Lehmer and others. As one application, we embed Ramanujan's famous congruences $p(5n+4)\equiv 0$ (mod $5)$ and $\tau(5n+5)\equiv 0$ (mod $5)$ into an infinite family of such congruences.

math.NT↗
arXiv2020

Student Mixture Model Based Visual Servoing

Classical Image-Based Visual Servoing (IBVS) makes use of geometric image features like point, straight line and image moments to control a robotic system. Robust extraction and real-time tracking of these features are crucial to the performance of the IBVS. Moreover, such features can be unsuitable for real world applications where it might not be easy to distinguish a target from the rest of the environment. Alternatively, an approach based on complete photometric data can avoid the requirement of feature extraction, tracking and object detection. In this work, we propose one such probabilistic model based approach which uses entire photometric data for the purpose of visual servoing. A novel image modelling method has been proposed using Student Mixture Model (SMM), which is based on Multivariate Student's t-Distribution. Consequently, a vision-based control law is formulated as a least squares minimisation problem. Efficacy of the proposed framework is demonstrated for 2D and 3D positioning tasks showing favourable error convergence and acceptable camera trajectories. Numerical experiments are also carried out to show robustness to distinct image scenes and partial occlusion.

cs.RO↗
arXiv2020

Prime number conjectures from the Shapiro class structure

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 state a formula for the height function due to Shapiro and use it to list steps to generate numbers at any height. This 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 $π(n)$ and the $n$th prime $p_n$. We conjecture formulas for $π(n)$ and $p_n$ in terms of the height function.

math.NT↗