Generalizations of some conjectures of Sun on the relations between N(a, b, c, d; n) and T(a, b, c, d; n)

Let T ( a ,  b ,  c ,  d ;  n ) denote the number of representations of n as a x ( x + 1 ) 2 + b y ( y + 1 ) 2 + c z ( z + 1 ) 2 + d w ( w + 1 ) 2 , where a , b , c , d are positive integers and n , x , y , z , w are arbitrary non-negative integers, and let N ( a ,  b ,  c ,  d ;  n ) denote the num...

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Bibliographic Details
Published inThe Ramanujan journal Vol. 48; no. 3; pp. 639 - 654
Main Author Yao, Olivia X. M.
Format Journal Article
LanguageEnglish
Published New York Springer US 15.04.2019
Springer Nature B.V
Subjects
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Integers
Mathematics
Mathematics and Statistics
Number Theory
Representations
Sun
11D85
11E25
Theta function identities
Sum of squares
Sum of triangular numbers
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Summary:Let T ( a ,  b ,  c ,  d ;  n ) denote the number of representations of n as a x ( x + 1 ) 2 + b y ( y + 1 ) 2 + c z ( z + 1 ) 2 + d w ( w + 1 ) 2 , where a , b , c , d are positive integers and n , x , y , z , w are arbitrary non-negative integers, and let N ( a ,  b ,  c ,  d ;  n ) denote the number of representations of n as a x 2 + b y 2 + c z 2 + d w 2 , where this time x , y , z and w are integers. Recently, Sun proved relations between N ( a ,  b ,  c ,  d ;  n ) and T ( a ,  b ,  c ,  d ;  n ) and posed 23 conjectures. In this paper, we not only confirm five conjectures due to Sun, but also generalize four of them.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-018-0067-y