## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 34, Number 1 (2010), 13-30.

### A product formula defined by the Beta function and Gauss's hypergeometric function

#### Abstract

Let $c$ be a constant in ${\bf R}^t$. For a plane algebraic curve $r^{2m-n} = 2c^n \cos n\theta$, which depends on $m$ and $n$ in ${\bf N}$, we show that the whole length of the curve are given by a value of a product formula defined by the Beta function and Gauss's hypergeometric function depending $m$ and $n$ in ${\bf N}$. Besides, we point out the fact to be a similar model and an expansion for the complete elliptic integral of the second kind. Last, we give a background for the fact explaining the special case $m = n$.

#### Article information

**Source**

Tsukuba J. Math., Volume 34, Number 1 (2010), 13-30.

**Dates**

First available in Project Euclid: 8 September 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1283967405

**Digital Object Identifier**

doi:10.21099/tkbjm/1283967405

**Mathematical Reviews number (MathSciNet)**

MR2723721

**Zentralblatt MATH identifier**

1223.33029

**Subjects**

Primary: 11A67: Other representations 33C75: Elliptic integrals as hypergeometric functions

Secondary: 33B15: Gamma, beta and polygamma functions 33C05: Classical hypergeometric functions, $_2F_1$

**Keywords**

beta function hypergeometric function transcendental number and the complete elliptic integral of the second
kind

#### Citation

Ogawa, Takuma; Kamata, Yasuo. A product formula defined by the Beta function and Gauss's hypergeometric function. Tsukuba J. Math. 34 (2010), no. 1, 13--30. doi:10.21099/tkbjm/1283967405. https://projecteuclid.org/euclid.tkbjm/1283967405