Amrik Singh Nimbran

The correspondence between the discrete and the continuous is a fascinating theme in mathematics. The Euler-Maclaurin sum formula, discovered independently and almost contemporaneously by Leonhard Euler (1707–1783) and Colin Maclaurin (1698–1746), in the early 1730s, relates the sum of the values of a function at the integers in the interval [ a ,...

The paper examines the structure of the periodic continued fraction for √ d and gives formulae for the central term as well as the repeated partial quotients occurring in its period. Motivated by discussion in Kraitchik's book [11, Ch.III], I examined the period of the continued fraction expansion of √ d with d a non-square positive integer. The ai...

The paper examines the structure of the periodic continued fraction for $\sqrt{d}$ and gives formulae for the central term as well as the repeated partial quotients occurring in its period.

We describe various methods to derive monotonic infinite series for fractions near π and obtain a variety of series for the special case of its convergents. These series immediately show that π is clearly different from these fractions, replicating with series the results in Dalzell [1, 2] and Lucas that used integrals with non-negative integrands...

In this paper we evaluate Euler-like sums involving harmonic/odd harmonic numbers and central binomial coefficients by using the expansion for powers of the inverse sine function. We also evaluate a number of integrals which in turn give rise to new central binomial coefficient and harmonic number identities.

We describe various methods to derive monotonic infinite series for fractions near π and obtain a variety of series for the special case of its convergents. These series immediately show that π is clearly different from these fractions, replicating with series the results in Dalzell and Lucas that used integrals with nonnegative integrands to repre...

It is a study of a chess opening with first move 1.Nh3.

We evaluate Euler-like sums involving harmonic numbers using expansions for powers of logarithmic, arctan and arctanh functions.

We present here some new and interesting Euler sums obtained by means of related integrals and elementary approach. We supplement Euler's general recurrence formula with two general formulas of the form n≥1 O (m) n 1 (2n−1) p + 1 (2n) p and n≥1 On (2n−1) p (2n+1) q , where O (m) n = n j=1 1 (2j−1) m. Two formulas for ζ (5) are also derived.

In this paper, we present some Euler-like sums involving partial sums of the harmonic and odd harmonic series. First, we give a brief historical account of Euler’s work on the subject followed by notations used in the body of the paper. After discussing some alternating Euler sums, we investigate the connection of integrals of inverse trigonometric...

This paper contains a number of series whose coefficients are products of central binomial coefficients & harmonic numbers. An elegant sum involving ζ(2) and two other nice sums appear in the last section.

We show how a couple of Ramanujan’s series for 1/π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/\pi $$\end{document} can be deduced directly from Forsyth’s series a...

This paper contains a number of series whose coefficients are products of central binomial coefficients & harmonic numbers. An elegant sum involving $\zeta(2)$ and two other nice sums appear in the last section.

We present here two classes of infinite series and the associated continued fractions involving $\pi$ and Catalan's constant $G$ based on the work of Euler and Ramanujan. A few sundry continued fractions are also given.

This note contains some asymptotic formulas for the sums of various residue classes of Euler’s φ-function.

This paper contains a presumably new representation for the square root of an integer as an infinite product.

We present here two classes of infinite series and the associated continued fractions involving π and Catalan’s constant based on the work of Euler and Ramanujan. A few sundry continued fractions are also given.

Infinite series are an important topic in mathematical analysis and convergence is the most crucial concept in the theory of infinite series. The speed at which the sequence of the partial sums of a series approaches its limiting sum has been a subject of investigation for many a mathematician. Euler [1], Kummer [2] and Markoff [3] all developed te...

We present here some series obtained by leaving out terms from the ζ(2) series and changing the sign scheme in the original series.

100.29 Some odd series for π - Volume 100 Issue 549 - Amrik Nimbran, Paul Levrie

In this note we present some sundry infinite products for √ 2 and six general formulas for some rational powers of 2.

This book discusses the three most fundamental concepts of Marxian thought and through them gives a concise exposition of his social philosophy.

We give here a neat Taylor series for the arctan function and use that to deduce a dozen BBP-type formulas for π. In addition, an alternative approach for deriving more formulas is also explained.

This paper contains a survey of interesting infinite products of rational functions based on Euler's Gamma function and infinite product expansions for sin x and cos x. Though the index in many products ranges, as usual, over the positive integers, we also treat products where the index ranges over the prime numbers. Historical information concerni...

In this note we present a decomposition of Catalan's infinite product for e leading to a new infinite product for e. Further, a generalization of Catalan's result is also obtained that yields products for e n .

we give here two series derived by us in 2014.

We describe here a method for deriving series for inverse pi using Legendre Polynomials and elementary method for this.

This paper evaluates two De Doelder like Euler sums and two sums of the type found by Bailey, Borwein and Girgensohn based on their results by a simple method. We also derive classes of series for ζ(3).

We give here a class of series for Pi.

We give here a simple and neat series for Pi.

We give here a recurrence relation and a theorem which generalizes Leibniz Pi series.

We give here a general theorem which generates an infinite class of pi series including Leibniz series as a special case.

We discuss in this paper Gausss formula for evaluating the Psi function at rational arguments through a generalization of the Euler’s constant.

This note contains a method for computing the values of Psi function and is related to Gauss's formulas.

This paper slightly modifies Lagrange’s theorem proving Wilson’s theorem.

This note offers a simple method for deriving an infinite number of efficient series for π^2. We illustrate here a method by deriving some progressively faster series basing on a series of Euler.

We have discovered a beautiful theorem which generates an infinite class of series for Pi with Leibniz series as a particular case.

This paper reviews Glaisher’s series for 1 π and contains many series derived by the author through an elementary algebraic approach.

Wallis discovered in 1655 a marvelous infinite product for Pi. Around eighty years later, Euler developed the product representation for the sine function which generalized Wallis formula. Basing on these, Sondow and Huang derived few such products recently. Their paper motivated the author to discover three general formulae which generate infinite...

It is a collection of infinite products for pi.

The Gamma function Γ(introduced by Euler (in his two letters dated 13 October, 1729 and January 8, 1730 addressed to Goldbach) as a generalization of the Factorial function, the Psi function as the derivative of ln Γ(Euler's constant gamma that appeared in 1734 as integration constant C, and the Zeta function defined originally as the sum of recipr...

The article contains some series for inverse pi derived from the formula for length of the perimeter of ellipse.

This paper explains methods for deriving a general formula for ∑ . These formulae involve one of the three: (1) (2) (3) The author employs the concept of ℎ in the second and that of in the third method. He also gives some results of his own relating to these numbers. of the second kind. The sums of like powers of consecutive natural numbers have fa...

The prime number theorem states that the number of primes up to a given number is approximated by the logarithmic integral function. To compute the value of this function, the author offers two formulae deduced from truncated series – one convergent and the other divergent. He also gives a table of values computed by him for this function. Introduc...

This paper discusses Stirling numbers. The author gives a few results of his own and presents some new formulae relating to Stirling numbers of the second kind derived through an unorthodox method. His method enables one to derive rest of the closed formulae. AMS Subject Classification: 11B73. Key words and phrases: Bell numbers, Stirling numbers o...

This paper introduces general harmonic series for log extends Euler's constant and evaluates sums of the series ∑ {(− 1) , k integer >1. The author also derives an elegant infinite series for π.

The paper describes a method to solve the Diophantine equation + = and gives its parametric solutions. It also discusses the nature of these solutions and puts forward a conjectural theorem regarding them. We know that while the equation + = has infinitely many solutions in integers, the equation + = has none.[3] What about the equation + = ? The o...

This paper studies a problem in the theory of figurate num-bers: identifying and investigating those numbers which are polygonal in two ways -triangular and square. The author has discovered many identities relating to these and other associated numbers. The ancient Greeks, particularly the Pythagoreans, linked arithmetic with geometry thus initiat...

The computation of π, the ratio of a circle's circumference to its diameter, has attracted mathematicians since antiquity. The discovery of the Madhava-Gregory series and that of John Machin's celebrated identity (1706): π = 4 − , made it easier and quicker. Electronic computers are now being used for computational work as well as for derivation of...

The object of the paper is to prove that the equation x, + y, = z, is insolvable for non_ zero integers. Around t637 Fermat stated that it was impossible to decompose a cube in to two cubes. rn 1770, Euler gave an inlellus il i.i;;;;;"o"iiy nr.,nut,, method of infinitc descent. He stated without i l3uert+:u,oi,i;;; j,HT,ll1if 1,J;':i,1x,'liHf J*,H,...

The well-known Wilson's theorem states that if p is a prime number, then (− 1)!≡–1 (mod). (1) Let 1≤ ≤ , then −≡ − (mod). By replacing − (− 1), − (− 2), − (− 3), . . . , − 1 by −(− 1), −(− 2), −(− 3), . . . , −1, we obtain from (1) (−)! (− 1)!≡(−1) (mod). (2) Also (−)! (− 1)!≡(−1) (+ 1) (mod). Putting = in (2), we get (− 1)!≡(−1) (mod). If is an od...

In this paper I propose to investigate into the problem of the repetition of digits and numbers. I shall introduce the concept of repeating numbers – numbers which if raised to some power are repeated at the end of the product. Those (repeating) numbers which are repeated on every power will be called self-generating or replicating numbers. We know...