In 1929, G. N. Watson and B. M. Wilson agreed to begin the enormous task of editing Ramanujan’s notebooks. Apparently, Wilson was assigned the task of editing Chapters 2-12, while Watson was charged with Chapters 13-21 of the second notebook, which is a revised, enlarged edition of the first notebook. For six years, until his death in 1935, Wilson devoted all of his efforts to this undertaking. Watson worked tirelessly on primarily Chapters 16-21 until the late 1930’s when his interest in the project evidently waned. The main purpose of this paper is to prove and expound upon all of the results set forth by Ramanujan in Chapter 3. The paper draws partly upon the notes which Wilson left. These notes were transferred to Watson upon the death of Wilson. After Watson passed away in 1965, the manuscripts of both Watson and Wilson were donated to Trinity College Library, Cambridge. Chapter 3 is considerably more analytical in character than is Chapter 2 which is more elementary and which has also been edited 191. Although no combinatorial problems are mentioned in Chapter 3, the contents of this chapter belong under the umbrella of combinatorial analysis, as will be made plain in the sequel. Chapter 3 contains the statements of 86 theorems and formulas. As with Chapter 2, Ramanujan very briefly sketches the proofs of some of his findings in Chapter 3. It has frequently been declared that much of Ramanujan’s early work before he departed for England is the rediscovery of the work of others. Hardy [40, p. lo] estimated that two-thirds of Ramanujan’s best work is rediscovery. Indeed, some of the results in Chapter 3 can be traced back to Lambert, Lagrange, Euler, Rothe, Abel, and others. On the other hand, much of Ramanujan’s work in Chapter 3 has been rediscovered by others unaware of his work. For example, the single variable Bell polynomials were first thoroughly examined in print by Touchard (611 in 1933 and by Bell f 7 1 in

Chapter 7 of Ramanujan’s second notebook

In this paper we introduce a method to find the sum of powers on arithmetic progressions by using Cauchy’s equation and obtain a general formula. Then we apply our results to show how to determine some other sums of powers and sums of products. Our results are more general than those in [9]. Finally we discuss the sum of powers on arithmetic progressions in commmutative rings with characteristic 2 and find ‘full polynomials’.

Finding Sum of Powers on Arithmetic Progressions with Application of Cauchy’s Equation

This collection of sums and integrals has been harvested from the mathematical and physical literature in unstructured ways. Its main use is backtracking the original sources whenever an integral of the reader's application resembles one of the items in the collection.

Yet Another Table of Integrals.

General solution of a system of functional equations satisfied by sums of powers on arithmetic progressions

For integer $k \geq 0$, let $S_k$ denote the sum of the $k$th powers of the first $n$ positive integers $1^k + 2^k + \cdots + n^k$. For any given $k$, the power sum $S_k$ can in principle be determined by differentiating $k$ times (with respect to $x$) the associated exponential generating function $\sum_{k=0}^{\infty}S_k x^k/k!$, and then taking the limit of the resulting differentiated function as $x$ approaches $0$. In this paper, we exploit this method to establish a couple of seemingly novel recurrence relations, one of them involving the even-indexed power sums $S_2, S_4,\ldots, S_{2k}$, and the other the odd-indexed power sums $S_{1}, S_3, \ldots, S_{2k-1}$, with both recurrence relations depending explicitly on the parameter $N = n + \frac{1}{2}$. From this, we obtain a determinantal formula of order $k$ which yields $S_{2k}$ [$S_{2k-1}$] in the Faulhaber form, that is, as an odd [even] polynomial in $N$. As a byproduct, we discover a new determinantal formula for the Bernoulli number $B_{2k}$. Furthermore, we show that $S_{2k}$ and $S_{2k-1}$ can be obtained by taking the corresponding higher order derivatives of the Chebyshev polynomials of the second kind.

Sums of powers of integers via differentiation

This paper presents a direct and simple approach to obtaining the formulas forS
 k(n)= 1
 k
 + 2
 k
 + ... +n
 k
 wheren andk are nonnegative integers. A functional equation is written based on the functional properties ofS
 k
 (n) and several methods of solution are presented. These lead to several recurrence relations for the functions and a simple one-step differential-recurrence relation from which the polynomials can easily be computed successively. Arbitrary constants which arise are (almost) the Bernoulli numbers when evaluated and identities for these modified Bernoulli numbers are obtained. The functional equation for the formulas leads to another functional equation for the generating function for these formulas and this is used to obtain the generating functions for theS
 k
 's and for the modified Bernoulli numbers. This leads to an explicit representation, not a recurrence relation, for the modified Bernoulli numbers which then yields an explicit formula for eachS
 k
 not depending on the earlier ones. This functional equation approach has been and can be applied to more general types of arithmetic sequences and many other types of combinatorial functions, sequences, and problems.

Formulas for sums of powers of integers by functional equations

A sufficiency technique in calculus of variations using Caratheodory's equivalent problems approach

Rayleigh’s Principle provides an inequality which gives upper bounds to an eigenvalue of a differential equation by evaluating a ratio of quadratic functionals with any function from a prescribed class. It also shows that the value of the functional evaluated with the eigenfunction is exactly the eigenvalue. This paper shows how minimizing the functional 

$$ J\left[ u \right] = \iint {\left[ {p\nabla u \cdot \nabla u + \left( {q - \lambda r} \right){u^2}} \right]} $$

by a modification of Carath€odory’s equivalent-problems method yields Rayleigh’s Principle for the partial-differential-equation (PDE) eigenvalue problem ▽ · (p▽u) – (q – λr)u = 0 on D, u = 0 on ∂D. The approach leads to a Hamilton-Jacobi equation for a vector variable, and seeking special forms of solution to this leads to a scalar PDE $ \nabla \cdot \sigma + \frac{1}{p}\sigma \cdot \sigma = q - \lambda r $ for the vector variable σ(x,y). This PDE is an obvious generalization of a Riccati ordinary differential equation, and it can be linearized by the transformation $ \sigma = p\frac{{\nabla w}}{w} $ . The resulting linear PDE is the original eigenvalue PDE, which has a nonvanishing solution by hypothesis. This guarantees the existence of a “nice” equivalent problem which immediately yields Rayleigh’s Principle, both the upper-bound and equality parts. The proof given here is for the two-dimensional case, but the specialization to the one-dimensional case and the generalization to the more-than-two-dimensional cases are immediate.

Rayleigh’s Principle by Equivalent Problems

In this paper, we discuss the functional inequality $ p(n + m){\kern 1pt} \le {\kern 1pt} (_{n}^{{n + m}}){\kern 1pt} p(n){\kern 1pt} p(m) $, which arises in tournament theory and other parts of combinatorics. A simple transformation removes the binomial coefficient, and then the solution set divides naturally into three classes of functions. One class consists of all the nonpositive functions since this inequality puts no restriction on such functions. The counting-function solutions, i.e., the nonnegative solutions, all lie in the other two classes and satisfy easily obtainable exponential growth bounds. This set of solutions also possesses a structure in the sense that various combinations of these solutions, e.g., sums and products, are again in the set. Various solution functions and properties of solutions are obtained by introducing a slack function to convert the functional inequality to a functional equation. The general solution to this functional equation is obtained by transforming it to another functional equation whose general solution is known. Solution functions found in this manner occur in pairs and are sometimes even from different solution classes. This slack-function concept has modifications, so it can be applied in other ways to the functional inequality and to other inequalities.

Donald R. Snow

Papers

Formulas for sums of powers of integers by functional equations

A sufficiency technique in calculus of variations using Caratheodory's equivalent problems approach

Rayleigh’s Principle by Equivalent Problems

A Functional Inequality Arising in Combinatorics