• Honam Mathematical Journal
• /
• v.43 no.2
• /
• pp.221-237
• /
• 2021

B. C. Berndt has found modular transformation formulae for a large class of functions coming from generalized Eisenstein series. Using those formulae, he established a lot of infinite series identities, some of which explain many infinite series identities given by Ramanujan. Continuing his work, the author proved a lot of new infinite series identities. Moreover, recently the author found transformation formulae for a class of functions coming from generalized non-holomorphic Eisenstein series. In this paper, using those formulae, we evaluate a few new infinite series identities which generalize the author's previous results.

• Honam Mathematical Journal
• /
• v.38 no.3
• /
• pp.479-494
• /
• 2016

In this paper, using a transformation formula for a certain series which comes from the generalized non-analytic Eisenstein series, we obtained some infinite series identities which contain Ramanujan's formula and author's previous results.

• Korean Journal of Mathematics
• /
• v.20 no.4
• /
• pp.451-461
• /
• 2012

B.C. Berndt has established many relations between various infinite series using a transformation formula for a large class of functions, which comes from a more general class of Eisenstein series. In this paper, continuing his study, we find some infinite series identities.

• Journal of the Chungcheong Mathematical Society
• /
• v.34 no.2
• /
• pp.139-155
• /
• 2021

Using a modular transformation formula for a class of functions related to generalized non-holomorphic Eisenstein series, we find a new class of infinite series about identities, some of which include generalized formulae of several Berndt's results.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.6
• /
• pp.1671-1683
• /
• 2016

Abstract. In this paper, for $a{\in}C$, we investigate functions $g_a$ and ${\psi}_a$ associated with three term relations. $g_a$ is defined by means of function ${\psi}_a$. By using these functions, we obtain some functional equations related to the Eisenstein series and the Riemann zeta function. Also we find a generalized difference formula of function $g_a$.

• Journal of the Chungcheong Mathematical Society
• /
• v.25 no.2
• /
• pp.299-312
• /
• 2012

In 1970s, B. C. Berndt proved a transformation formula for a large class of functions that includes the classical Dedekind eta function. From this formula, he evaluated several classes of infinite series and found a lot of interesting infinite series identities. In this paper, using his formula, we find new infinite series identities.

• Bulletin of the Korean Mathematical Society
• /
• v.39 no.2
• /
• pp.299-307
• /
• 2002

As a by-product of [5], we produce algebraic integers of certain values of quotients of Eisenstein series. And we consider the relation of $\Theta_3(0,\tau)$ and $\Theta_3(0,\tau^n)$. That is,we show that $$\mid$\Theta_3(0,\tau^n)$\mid$=$\mid$\Theta_3(0,\tau)$\mid$,\bigtriangleup(0,\tau)=\bigtriangleup(0,\tau^n)$ and $J(\tau)=J(\tau^n)$ for some $\tau\in\eta$.

• Bulletin of the Korean Mathematical Society
• /
• v.50 no.4
• /
• pp.1389-1413
• /
• 2013

In this paper, we consider several convolution sums, namely, $\mathcal{A}_i(m,n;N)$ ($i=1,2,3,4$), $\mathcal{B}_j(m,n;N)$ ($j=1,2,3$), and $\mathcal{C}_k(m,n;N)$ ($k=1,2,3,{\cdots},12$), and establish certain identities involving their finite products. Then we extend these types of product convolution identities to products involving Faulhaber sums. As an application, an identity involving the Weierstrass ${\wp}$-function, its derivative and certain linear combination of Eisenstein series is established.

• Korean Journal of Mathematics
• /
• v.30 no.1
• /
• pp.25-42
• /
• 2022

In this paper, we derive analogues of a couple of classes of infinite series identities with the confluent hypergeometric functions involving Dirichlet characters.

• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.3
• /
• pp.481-502
• /
• 2011

B. C. Berndt [4, 5] evaluated several classes of infinite series and established many relations between various infinite series. In this paper, continuing his work, we derive new relations between infinite series.