IJPAM: Volume 36, No. 2 (2007)

ON THE HYPERFACTORIAL FUNCTION,
HYPERTRIANGULAR FUNCTION, AND
THE DISCRIMINANTS OF CERTAIN POLYNOMIALS

Mohammad K. Azarian
Department of Mathematics
University of Evansville
1800 Lincoln Avenue, Evansville, IN 47722, USA
e-mail: azarian@evansville.edu


Abstract.For any natural number $n$, let $H_{f}%
(n)=1^{1}.2^{2}.3^{3}.4^{4}...n^{n}$ be the hyperfactorial function of $n$, and let $H_{t}(n)=1^{1}+2^{2}+3^{3}+4^{4}+...+n^{n}$, be the hypertriangular function of $n.$ Also, let $r_{1},r_{2},r_{3},...,r_{k},$ represent the roots of the polynomial equation $p_{k}(z)=z^{k}-1=0$ (for a fixed $k\geq2$). First, we show that $k^{k}$ (for any integer $k\geq2$) can be written in terms of the discriminant of $p_{k}(z).$ Then, we use this result to show that $H_{f}(n)$ and $H_{t}(n)$ can be written as a product of discriminants of certain polynomials, and as a sum of discriminants of these polynomials, respectively. Also, we use a known result to present an upper and a lower bound for $H_{t}(n).$ Finally, we pose two questions for the reader.

Received: February 9, 2007

AMS Subject Classification: 11A

Key Words and Phrases: $n$-th roots of unity, hyperfactorial function, hypertriangular function, $K-$function, the discriminant of a polynomial, Barnes $G-$function, Euler gamma function, Sylvester matrix, resultant of two polynomials

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 36
Issue: 2