HURWITZ TYPE RESULTS FOR CERTAIN REPRESENTATIONS OF INTEGERS AS SUMS OF SQUARES
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초록
Let rα,β(n) denote the number of representations of n as a sum of α times a square and β times another square. In the recent past, a number of authors have obtained Hurwitz type results for representations of integers as sums of squares. Motivated by their works, in this paper we prove many results in which the generating function of r2,3(λk(an + b)), r2,4(an + b) and r1,5(λk(an + b)) for various non-negative integer values of λ, k, a and b are infinite products. To obtain our main results we use the theta function identities of Ramanujan found in chapter 16 of his second notebook.
Abstract
Let rα,β(n) denote the number of representations of n as a sum of α times a square and β times another square. In the recent past, a number of authors have obtained Hurwitz type results for representations of integers as sums of squares. Motivated by their works, in this paper we prove many results in which the generating function of r2,3(λk(an + b)), r2,4(an + b) and r1,5(λk(an + b)) for various non-negative integer values of λ, k, a and b are infinite products. To obtain our main results we use the theta function identities of Ramanujan found in chapter 16 of his second notebook.
참고문헌 (9)
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- S. Ramanujan, "Notebooks, 2 Volumes" (1957), Tata Institute of Fundamental Research
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