Circular summations of theta functions in Ramanujan's lost notebook
Let \[ f(a,b) = \sum_{k\in \mathbb Z} a^{k(k+1)/2}b^{k(k-1)/2},\quad | ab| <1. \] In this article, the author proves the following result found on page 54 of Ramanujan's Lost Notebook: \noindent Let \(n\) be a positive integer. Let \[ U_j=(a^{1/n})^{j(j+1)/2}(b^{1/n})^{j(j-1)/2} \text{\quad and \quad} V_j=(a^{1/n})^{j(j-1)/2}(b^{1/n})^{j(j+1)/2} \] for each integer \(j\). Then \[ \sum_{-\frac{n}{2}<r\leq \frac{n}{2}} U_r^n f^n\left(\frac{U_{n+r}}{U_r},\frac{V_{n-r}}{U_r}\right) =f(a,b)F_n(a,b), \] where \[ F_n(x) = 1+2nx^{(n-1)/2}+\cdots, \quad n\geq 3. \] The author then establishes Ramanujan's expressions for \(F_n(x)\) when \(n=2,3,4,5\) and \(7\). These results are first proved by \textit{S. S. Rangachari} using theory of theta functions and weight polynomials in coding theory [J. Number Theory 48, No. 3, 364--372 (1994; Zbl 0813.11030)]. The author presents proofs of the above using various identities found by Ramanujan. Readers who are interested in the above results should also consult papers by \textit{K. Ono} [J. Number Theory 76, No. 1, 62--65 (1999; Zbl 0927.11025)], \textit{S. Ahlgren} [Proc. Am. Math. Soc. 128, No. 5, 1333--1338 (2000; Zbl 0953.11006)] and \textit{K. S. Chua} [Proc. Am. Math. Soc. 130, No. 1, 1--8 (2002; Zbl 1061.11019), Ramanujan J. 5, No. 4, 353--354 (2001; Zbl 0990.11027)].
scientific article; zbMATH DE number 2153729