scholarly journals Eisenstein Series Identities Involving the Borweins' Cubic Theta Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Ernest X. W. Xia ◽  
Olivia X. M. Yao
Keyword(s):  
Eisenstein Series ◽  
Theta Functions ◽  
Elliptic Functions ◽  
The Other ◽  
Fundamental Theory ◽  
Convolution Sums

Based on the theories of Ramanujan's elliptic functions and the (p,k)-parametrization of theta functions due to Alaca et al. (2006, 2007, 2006) we derive certain Eisenstein series identities involving the Borweins' cubic theta functions with the help of the computer. Some of these identities were proved by Liu based on the fundamental theory of elliptic functions and some of them may be new. One side of each identity involves Eisenstein series, the other products of the Borweins' cubic theta functions. As applications, we evaluate some convolution sums. These evaluations are different from the formulas given by Alaca et al.

EVALUATION OF CONVOLUTION SUMS AND FOR k = a · b = 21, 33, AND 35

2021 ◽  
pp. 1-20
Author(s):  
K. PUSHPA ◽  
K. R. VASUKI
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Divisor Function ◽  
Convolution Sums

Abstract The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$ , for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).

scholarly journals CONVOLUTION SUMS AND THEIR RELATIONS TO EISENSTEIN SERIES

2013 ◽  
Vol 50 (4) ◽  
pp. 1389-1413 ◽  
Author(s):  
Daeyeoul Kim ◽  
Aeran Kim ◽  
Ayyadurai Sankaranarayanan
Keyword(s):  
Eisenstein Series ◽  
Convolution Sums

Theta functions and elliptic functions

2016 ◽  
pp. 384-403
Author(s):  
J. Hietarinta ◽  
N. Joshi ◽  
F. W. Nijhoff
Keyword(s):  
Theta Functions ◽  
Elliptic Functions

Elliptic Functions, Theta Functions, and Riemann Surfaces

1974 ◽  
Vol 28 (127) ◽  
pp. 875
Author(s):  
Y. L. L. ◽  
Harry E. Rauch ◽  
Aaron Lebowitz
Keyword(s):  
Riemann Surfaces ◽  
Theta Functions ◽  
Elliptic Functions

scholarly journals VII.—On the Motion of a Heavy Body along the Circumference of a Circle

1865 ◽  
Vol 24 (1) ◽  
pp. 59-71
Author(s):  
Edward Sang
Keyword(s):  
Royal Society ◽  
Elliptic Functions ◽  
The Other ◽  
Philosophical Magazine ◽  
Circular Arc ◽  
Heavy Body ◽  
Great Ease ◽  
Fourth Volume ◽  
The Common ◽  
The One

In the year 1861 I laid before the Royal Society of Edinburgh a theorem concerning the time of descent in a circular arc, by help of which that time can be computed with great ease and rapidity. A concise statement of it is printed in the fourth volume of the Society's Proceedings at page 419.The theorem in question was arrived at by the comparison of two formulæ, the one being the common series and the other an expression given in the “Edinburgh Philosophical Magazine” for November 1828, by a writer under the signature J. W. L. Each of these series is reached by a long train of transformations, developments, and integrations, which require great familiarity with the most advanced branches of the infinitesimal calculus; yet the theorem which results from their comparison has an aspect of extreme simplicity, and seems as if surely it might be attained to by a much shorter and less rugged road. For that reason I did not, at the time, give an account of the manner in which it was arrived at, intending to seek out a better proof. On comparing it with what is known in the theory of elliptic functions, its resemblance to the beautiful theorem of Halle became obvious; but then the coefficients in Halle's formulæ are necessarily less than unit, whereas for this theorem they are required to be greater than unit.

Determinantal representations of elliptic curves via Weierstrass elliptic functions

2018 ◽  
Vol 34 ◽  
pp. 125-136 ◽  
Author(s):  
Mao-Ting Chien ◽  
Hiroshi Nakazato
Keyword(s):  
Elliptic Curves ◽  
Algebraic Curve ◽  
Theta Functions ◽  
Elliptic Functions ◽  
Determinantal Representation ◽  
Ternary Form ◽  
Riemann Theta Functions

Helton and Vinnikov proved that every hyperbolic ternary form admits a symmetric derminantal representation via Riemann theta functions. In the case the algebraic curve of the hyperbolic ternary form is elliptic, the determinantal representation of the ternary form is formulated by using Weierstrass $\wp$-functions in place of Riemann theta functions. An example of this approach is given.

scholarly journals The Vibrations of a Particle about a Position of Equilibrium—Part III

1922 ◽  
Vol 41 ◽  
pp. 128-140
Author(s):  
Bevan B. Baker
Keyword(s):  
Dynamical System ◽  
Series Solution ◽  
Elliptic Functions ◽  
The Other ◽  
Real Solution ◽  
Present Part ◽  
Image Position ◽  
Region Of Convergence ◽  
Range Of Values ◽  
Made In

In the two parts of this investigation previously published it has been shown that the solution in terms of elliptic functions represents the motion of the particular dynamical system under consideration throughout the whole range of values of s and g for which a real solution exists, except for those values for which s = 2g and k = 1, but that, on the other hand, the series solution is convergent and represents the motion only so long asfor values of s and g for which the sign of this inequality is reversed the trigonometric series representing the solution are divergent. It is of importance to investigate what discontinuities, if any, of the system correspond to values of s and g which lie on the boundary of the region of convergence; the present part is concerned primarily with showing that under such circumstances no discontinuity of the system exists, thus confirming the suggestions made in Part I., § 12.

Weil Representation and Waldspurger Formula

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Integral Representation ◽  
Eisenstein Series ◽  
Local Field ◽  
Integral Formula ◽  
Theta Functions ◽  
Direct Consequence ◽  
Weil Representation ◽  
Computational Results ◽  
Weil Representations ◽  
Central Value

This chapter reviews some basic results on Weil representations, theta liftings and Eisenstein series. In particular, it introduces a proof of the Waldspurger formula. The theory of Weil representation is applied to an integral representation of the Rankin–Selberg L-function and to a proof of Waldspurger's central value formula. The chapter mostly follows Waldspurger's treatment with some modifications including Kudla's construction of incoherent Eisenstein series. It first describes the classical theory of Weil representation for an orthogonal space over a local field before discussing theta functions, the Siegel–Weil formula, and normalized local Shimizu lifting. The main result is an integral formula for the L-series using a kernel function. The Waldspurger formula is a direct consequence of the Siegel–Weil formula. After presenting the proof of Waldspurger formula, the chapter lists some computational results on three types of incoherent Eisenstein series.

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