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For |q|<1 >, define fi=∏∞n=1(1−qin) , and let (A(q),B(q)) be any of the pairs (f41,f81f22), (f41,f101f23), (f61,f42f21), (f61,f141f42), (f101,f62f21), (f141,f53f1),(f141,f82f21). For any such pair (A(q),B(q)) , define the sequences {a(n)} and {b(n)} to be the coefficients of qn of A(q) and B(q) , respectively. Then for each pair it is shown that a(n) vanishes if and only if b(n) vanishes. In each case, a criterion is given which states precisely when a(n)=b(n)=0 . Moreover, for the pairs (f261,f93f1) , (f261,f162f61) it is shown that a(n)=b(n)=0 if 12n+13 satisfies a criteria of Serre for a(n)=0 .


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International Journal of Number Theory



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Mathematics Commons



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