Theta Functions

Shaun Cooper
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Shaun Cooper: Massey University, Institute of Natural and Mathematical Science

Chapter Chapter 3 in Ramanujan's Theta Functions, 2017, pp 171-242 from Springer

Abstract: Abstract This chapter contains a detailed study of Ramanujan’s theta functions ϕ ( q ) = ∑ j = − ∞ ∞ q j 2 , and ψ ( q ) = ∑ j = 0 ∞ q j ( j + 1 ) ∕ 2 , $$\phi (q) =\sum _{ j=-\infty }^{\infty }q^{j^{2} },\quad \mbox{ and}\quad \psi (q) =\sum _{ j=0}^{\infty }q^{j(j+1)/2},$$ and the Borweins’ theta functions a ( q ) = ∑ j = − ∞ ∞ ∑ k = − ∞ ∞ q j 2 + j k + k 2 , b ( q ) = ∑ j = − ∞ ∞ ∑ k = − ∞ ∞ ω j − k q j 2 + j k + k 2 , ω = exp ( 2 π i ∕ 3 ) , and c ( q ) = ∑ j = − ∞ ∞ ∑ k = − ∞ ∞ q ( j + 1 3 ) 2 + ( j + 1 3 ) ( k + 1 3 ) + ( k + 1 3 ) 2 . $$\displaystyle\begin{array}{rcl} a(q)& =& \sum _{j=-\infty }^{\infty }\sum _{ k=-\infty }^{\infty }q^{j^{2}+jk+k^{2} }, {}\\ b(q)& =& \sum _{j=-\infty }^{\infty }\sum _{ k=-\infty }^{\infty }\omega ^{j-k}q^{j^{2}+jk+k^{2} },\quad \omega =\exp (2\pi i/3), {}\\ \mbox{ and}\quad c(q)& =& \sum _{j=-\infty }^{\infty }\sum _{ k=-\infty }^{\infty }q^{(j+\frac{1} {3} )^{2}+(j+\frac{1} {3} )(k+\frac{1} {3} )+(k+\frac{1} {3} )^{2} }. {}\\ \end{array}$$

Date: 2017
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DOI: 10.1007/978-3-319-56172-1_4

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