Representations of squares by certain diagonal quadratic forms in odd number of variables

In this paper, we consider the following diagonal quadratic forms \begin{equation*} a_1x_1^2 + a_2x_2^2 + \cdots + a_{\ell}x_{\ell}^2, \end{equation*} where $\ell\ge 5$ is an odd integer and $a_i\ge 1$ are integers. By using the extended Shimura correspondence, we obtain explicit formulas for the number of representations of $|D|n^2$ by the above type of quadratic forms, where $D$ is either a square-free integer or a fundamental discriminant such that $(-1)^{(\ell-1)/2}D > 0$. We demonstrate our method with many examples, in particular, we obtain all the formulas (when $\ell =5$) obtained in the work of Cooper-Lam-Ye (Acta. Arith. 2013) and all the representation formulas for $n^2$ obtained by them in (Integers, 2013) when $n$ is even. The works of Cooper et. al make use of certain theta function identities combined with a method of Hurwitz to derive these formulas. It is to be noted that our method works in general with arbitrary coefficients $a_i$. As a consequence to some of our formulas, we obtain certain identities among the representation numbers and also some congruences involving Fourier coefficients of certain newforms of weights $6, 8$ and the divisor functions.