Allied Series Research Articles

Parameterizations for quintic Eisenstein series

In his Lost Notebook, Ramanujan gave product expansions for a pair of weight two Eisenstein series of level five. We show that Ramanujanʼs formulas are special cases of more general parameterizations for quintic Eisenstein series. In particular, we prove that the Eisenstein series for the Hecke subgroup of level five are expressible as homogeneous polynomials in two parameters closely connected with the Rogers–Ramanujan functions. Moreover, the coefficients of each polynomial are symmetric in absolute value about the middle terms. Corresponding polynomial expansions for allied series, including Eisenstein series on the full modular group, are also derived.

Absolute matrix summability of the allied series of a Fourier series

In this paper we have proved a theorem for infinite triangular matrix using new techniques.

On the behaviour of a series associated with the allied series of a Fourier series

On the behaviour of a series associated with the allied series of a Fourier series

On the absolute logarithmic summability of the allied series of a Fourier series

On the absolute logarithmic summability of the allied series of a Fourier series

On the absolute logarithmic summability of the allied series of a Fourier series

On the absolute logarithmic summability of the allied series of a Fourier series

On the absolute convergence of a series associated with a Fourier series

(1. 7) 4'(t) = {f(x + t) f(x t) }/2. The ordinary Cesaro summability of the series (1.3) was first studied by Hardy and Littlewood [5] who observed that the relation of (1.3) to the integral fo(4(t)/t)dt is very similar to that between the allied series and the integral fo(#(t)/t)dt. Zygmund [10] gave a necessary and sufficient condition for the convergence of the same series (1.3). The object of the present note is to study the absolute convergence and absolute Riesz summability of (1.3). We prove the following THEOREM 1. If (1) 4)1(t) log k/t is of bounded variation in (0, 7r), Received by the editors November 28, 1955.

The Absolute Cesaro Summability of the Successively Derived Allied Series of a Fourier Series

We suppose that f(t) is integrable in the Lebesgue sense m (π, π) and is periodic with period 2π. We denote its Fourier series byThen the allied series is

The absolute convergence of the allied series of a Fourier series

The absolute convergence of the allied series of a Fourier series

Note on the sum of an oscillating series, II

This note is a sequel to a former one, a knowledge of which will be assumed. We first prove a theorem which is analogous to O Theorem I, and is indeed a simple consequence of it. As an easy inference from this theorem we obtain a necessary condition that the series allied with a Fourier series may be summable (C, k) for all k >p, where p is any number greater than 1. We then show that the condition thus obtained is sufficient. We finally show that the same method can be applied to Fourier series. This note thus contains a solution of the Cesàro summability problem for Fourier series and for the allied series, which is the most precise one yet obtained.

On certain sufficient conditions for the convergence and Cesàro summability of the allied series of a double Fourier series

On certain sufficient conditions for the convergence and Cesàro summability of the allied series of a double Fourier series

On Fourier series and functions of bounded variation

1. In a previous communication to the Society I have pointed out that the succession of constants obtained by multiplying together two successions of Fourier constants in the manner which naturally suggests itself is a succession of Fourier constants, and I have discussed the summability of the function with new constants are associated. We may express the matter in another way by saying that I have shown that the use of the Fourier constants of an even function g(x) as convergence factors in the Fourier series of a function f(x) changes the latter series into a series which is associated with the new series is increased. The use of the Fourier constants of an odd function as convergence factors, on the other hand, has the effect of changing the allied series of the Fourier series of f (x) into a Fourier series, even when the allied series is not itself a Fourier series. It at once suggests itself that the former of the two statements in this form of the result is not the most that can be said. Indeed, the series, whose general term is cos nx , and whose coefficients are accordingly unity, may clearly take the place of the Fourier series of g(x) , although it is not a Fourier series. On the other hand, it is the derived series of the Fourier serious of a function of bounded variation, which is, moreover, odd. We are thus led to ask ourselves whether this is not the trivial case of a general theorem. In the present communication I propose to show, among other things, that the answer to this questions is in affirmative. The following theorems are, in fact, true:—

Societies and Academies

Societies and Academies

On a mode of generating fourier series

§ 1. In a recent paper published in the 'Proceedings' of this Society I obtained formulæ for the sum of the series formed by the Fourier constants: of a function, when each of these is divided by a power of the index giving its place in the series. As I there remarked, the scope of these results is really wider than at first appears, and I undertook to develop later their further consequences. The present communication is the outcome of this undertaking. I have, however, already in part indicated in a second still more recent paper the general nature of the argument which I propose to employ. I start with a Fourier series f ( x ) ∼ Ʃ ∞ n = 1 ( a n cos nx + b n sin nx ), and its allied series Ʃ ∞ n = 1 (- a n sin nx + b n cos nx ).