### A Sequence of Modular Forms Associated with Higher-Order Derivatives of Weierstrass-Type Functions

#### Abstract

In this article, we first determine a sequence $\{f_{n}(\tau )\}_{n\in

\mathbb{N}}$ of modular forms with weight

\begin{equation*}

2^{n}k+4(2^{n-1}-1)\qquad (n\in \mathbb{N};\;k\in \mathbb{N}\setminus

\{1\};\;\mathbb{N}:=\{1,2,3,\cdots \}).

\end{equation*}

We then present some applications of this sequence which are related to the

Eisenstein series and the cusp forms. We also prove that higher-order

derivatives of the Weierstrass type $\wp _{2n}$-functions are related to the

above-mentioned sequence $\{f_{n}(\tau )\}_{n\in \mathbb{N}}$ of modular

forms.

\mathbb{N}}$ of modular forms with weight

\begin{equation*}

2^{n}k+4(2^{n-1}-1)\qquad (n\in \mathbb{N};\;k\in \mathbb{N}\setminus

\{1\};\;\mathbb{N}:=\{1,2,3,\cdots \}).

\end{equation*}

We then present some applications of this sequence which are related to the

Eisenstein series and the cusp forms. We also prove that higher-order

derivatives of the Weierstrass type $\wp _{2n}$-functions are related to the

above-mentioned sequence $\{f_{n}(\tau )\}_{n\in \mathbb{N}}$ of modular

forms.

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