Representation numbers of five sextenary quadratic forms

Ernest X. W. Xia¹ ; Olivia X. M. Yao² ; A. F. Y. Zhao³

¹ Department of Mathematics Jiangsu University Zhenjiang, Jiangsu 212013, P. R. China
² Department of Mathematics Jiangsu University Jiangsu, Zhenjiang 212013, P. R. China
³ School of Mathematical Sciences and Institute of Mathematics Nanjing Normal University Nanjing 210023, P. R. China

Colloquium Mathematicum, Tome 138 (2015) no. 2, pp. 247-254

Résumé

For nonnegative integers $a, b, c$ and positive integer $n$, let $N(a,b,c;n)$ denote the number of representations of $n$ by the form $$ \sum_{i=1}^a (x_i^2+x_iy_i+y_i^2)+2\sum_{j=1}^b(u_j^2+u_jv_j+v_j^2) +4\sum_{k=1}^c(r_k^2+r_ks_k+s_k^2). $$ Explicit formulas for $N(a,b,c;n)$ for some small values were determined by Alaca, Alaca and Williams, by Chan and Cooper, by Köklüce, and by Lomadze. We establish formulas for $N(2,1,0;n)$, $N(2,0,1;n)$, $N(1,2,0;n)$, $N(1,0,2;n)$ and $N(1,1,1;n)$ by employing the $(p, k)$-parametrization of three 2-dimensional theta functions due to Alaca, Alaca and Williams.

Détail

DOI : 10.4064/cm138-2-9

Keywords: nonnegative integers positive integer c denote number representations form sum sum sum explicit formulas c small values determined alaca alaca williams chan cooper lomadze establish formulas employing parametrization three dimensional theta functions due alaca alaca williams