Proofs of some conjectures of Sun on the relations between t(a, b, c; n) and N(a, b, c; n)

摘要

Abstract Let Zdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathbb {Z}}$$end{document} and Z+documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathbb {Z}}^{+}$$end{document} be the set of integers and the set of positive integers, respectively. For a,b,c,n∈Z+documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$a, b, c, nin {mathbb {Z}}^{+}$$end{document} and x,y,z∈Zdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$x, y, zin {mathbb {Z}}$$end{document}, let N(a, b, c; n) be the number of representations of n as ax2+by2+cz2documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ax^2+by^2+cz^2$$end{document} and t(a, b, c; n) be the number of representations of n as ax(x+1)2+by(y+1)2+cz(z+1)2documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$afrac{x(x+1)}{2}+bfrac{y(y+1)}{2}+cfrac{z(z+1)}{2}$$end{document}. Recently, Sun established many relations between t(a, b, c; n) and N(a,b,c;8n+a+b+c)documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$N(a,b,c;8n+a+b+c)$$end{document} and listed 43 relations need to be confirmed. More recently, Xia and Zhang proved 19 relations conjectured by Sun. In this paper, by employing Ramanujan’s theta function identities, we prove the remaining 24 relations.