Relations among representations of integers by certain quadratic forms

Nayandeep Deka Baruah () and Hirakjyoti Das ()
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Nayandeep Deka Baruah: Tezpur University
Hirakjyoti Das: Tezpur University

Indian Journal of Pure and Applied Mathematics, 2022, vol. 53, issue 3, 672-682

Abstract: Abstract Let $$N(c_1,c_2,\ldots , c_k;n),$$ N ( c 1 , c 2 , … , c k ; n ) , $$r(c_1,c_2,\ldots , c_k;n),$$ r ( c 1 , c 2 , … , c k ; n ) , and $$t(c_1,c_2,\ldots , c_k;n)$$ t ( c 1 , c 2 , … , c k ; n ) count the representations of a positive integer n in the quadratic forms $$\begin{aligned}&c_1\left( x_1^2+x_1 x_2+x_2^2\right) +c_2\left( x_3^2+x_3 x_4+x_4^2\right) +\cdots +c_k\big (x_{2k-1}^2+x_{2k-1}x_{2k}+x_{2k}^2\big ), \\&\quad c_1x_1^2+c_2x_2^2+\cdots +c_kx_k^2, \\&\quad c_1x_1(x_1+1)/2+c_2x_2(x_2+1)/2+\cdots +c_kx_k(x_k+1)/2, \end{aligned}$$ c 1 x 1 2 + x 1 x 2 + x 2 2 + c 2 x 3 2 + x 3 x 4 + x 4 2 + ⋯ + c k ( x 2 k - 1 2 + x 2 k - 1 x 2 k + x 2 k 2 ) , c 1 x 1 2 + c 2 x 2 2 + ⋯ + c k x k 2 , c 1 x 1 ( x 1 + 1 ) / 2 + c 2 x 2 ( x 2 + 1 ) / 2 + ⋯ + c k x k ( x k + 1 ) / 2 , respectively, where $$c_i$$ c i ’s and $$x_i$$ x i ’s are integers. Using Ramanujan’s theta functions $$\varphi (q)$$ φ ( q ) , $$\psi (q)$$ ψ ( q ) , Borweins’ cubic theta function a(q), and simple dissecting techniques, we find relations satisfied by $$N(c_1,c_2,\ldots , c_k;n)$$ N ( c 1 , c 2 , … , c k ; n ) , $$r(c_1,c_2,\ldots , c_{2k};n)$$ r ( c 1 , c 2 , … , c 2 k ; n ) , and $$t(c_1,c_2,\ldots , c_{2k};n)$$ t ( c 1 , c 2 , … , c 2 k ; n ) when $$1

Keywords: Theta function; Cubic theta function; Quadratic form; 11E25; 11E20 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s13226-021-00158-w

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