Hurwitz type results for sums of squares and sums of triangular numbers
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초록
Let rk(n) denote the number of representations of n as a sum of k squares and tk(n) denote the number of representations of n as a sum of k triangular numbers. Let r , (n) denote the number of representations of n as a sum of times a square and times another square and t , (n) denote the number of representations of n as a sum of times a triangular number and times another triangular number. We prove a number of results in which the generating function of r2(2k(an + b)), r4(2k(an + b)), r1,3(2k(an + b)), t2(2k(an + b)), t1,2(2k(an + b)), for various integer values of k, a and b, is a simple infinite product. We also obtain a relation between t1,3(3k+1n + 3k + 3k−1 · · · +3 + 1) and t1,3(n).
Abstract
Let rk(n) denote the number of representations of n as a sum of k squares and tk(n) denote the number of representations of n as a sum of k triangular numbers. Let r , (n) denote the number of representations of n as a sum of times a square and times another square and t , (n) denote the number of representations of n as a sum of times a triangular number and times another triangular number. We prove a number of results in which the generating function of r2(2k(an + b)), r4(2k(an + b)), r1,3(2k(an + b)), t2(2k(an + b)), t1,2(2k(an + b)), for various integer values of k, a and b, is a simple infinite product. We also obtain a relation between t1,3(3k+1n + 3k + 3k−1 · · · +3 + 1) and t1,3(n).
참고문헌 (8)
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