On sums of an even number of squares, and an even number of triangular numbers: An elementary approach based on Ramanujan's \(_1\psi_1\) summation formula (Q2782410)

Let NEWLINE\[NEWLINEF(a,t) := \sum_{j=-\infty}^\infty \frac{t^j}{1-aq^{2j}}.NEWLINE\]NEWLINE K. Venkatachaliengar's fundamental multiplicative identity states that NEWLINE\[NEWLINEF(a,t)F(b,t)=t\frac{\partial }{\partial t}F(ab,t)+F(ab,t)(\rho(a)+\rho(b)),NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\rho(z) = \frac{1}{2}+\mathop{{\sum}'}_{j} \frac{z^j}{1-q^{2j}}.NEWLINE\]NEWLINE Here, the prime denotes the summation is over all integers \(j\) excluding 0. Using the above identity, the author shows that the derivatives of \(f_i(\theta), i=1,2,3,\) where NEWLINE\[NEWLINEf_1(\theta)=\frac{1}{i}F(e^{i\pi},e^{i\theta}),\quad f_2(\theta) = \frac{e^{i\theta/2}}{i}F(e^{i\pi\tau},e^{i\theta}), \quad\text{and}\quad f_3(\theta)=\frac{e^{i\theta/2}}{i}F(e^{i\pi+i\pi\tau},e^{i\theta}),NEWLINE\]NEWLINE can be expressed in terms of the \(f_i(\theta)\)'s. These observations then give rise to a representation of NEWLINE\[NEWLINE2^{2m}[(f_1^2)^{(2m-2)}(\pi)+(-1)^m(f_1^2)^{(2m-2)}(\pi\tau)]NEWLINE\]NEWLINE in terms of the \(f_i\)'s. Using Ramanujan's \(_1\psi_1\) summation formula, the author writes \(f_i\)'s in terms of infinite products and establishes their relations with \(\varphi^{4m}(q)\), where NEWLINE\[NEWLINE\varphi(q)=\sum_{j=-\infty}^\infty q^{j^2}.NEWLINE\]NEWLINE Next, by computing NEWLINE\[NEWLINE2^{2m}[(f_1^2)^{(2m-2)}(\pi)+(-1)^m(f_1^2)^{(2m-2)}(\pi\tau)]NEWLINE\]NEWLINE using the Lambert type series expansion provided by \(F(a,t)\), the author proves that \(\varphi^{4m}(q)\) is a sum of an Eisenstein series (this is the Lambert series present in Ramanujan's formula) and a cusp form (this appears as a sum of infinite products). A formula for \(r_{4m}(n)\), which is the number of representations of \(n\) as sums of \(4m\) squares, is an immediate consequence since its generating function is \(\varphi^{4m}(q).\) In a similar way, the author derives a formula for \(r_{4m+2}(q).\)NEWLINENEWLINEFor the entire collection see [Zbl 0980.00024].