I find the following equation in a quantum theory book where multivariate Taylor expansion was discussed. The equation uses Kronecker delta to change the subscripts which looks bizarre to me, and it goes
$$\phi_{\alpha \beta}\left(\boldsymbol{R}_{i}+\boldsymbol{d}_{\kappa}-\boldsymbol{R}_{j}-\boldsymbol{d}_{\nu}\right)\left.u_{i \kappa, \alpha} u_{i \kappa, \beta}\right.=\left[\sum_{j^{\prime} \nu^{\prime}} \phi_{\alpha \beta}\left(\boldsymbol{R}_{i}+\boldsymbol{d}_{\kappa}-\boldsymbol{R}_{j^{\prime}}-\boldsymbol{d}_{\nu^{\prime}}\right)\right] \delta_{i j} \delta_{\kappa \nu}u_{i \kappa, \alpha} u_{j \nu, \beta},\tag{1}$$
where $\phi$ is a function of the difference of two coordinates, $\boldsymbol{R}_i+\boldsymbol{d}_{\kappa}$ and $\boldsymbol{R}_j+\boldsymbol{d}_{\nu}$. And $\phi_{\alpha\beta}$ is $$\phi_{\alpha \beta}\left(\boldsymbol{R_{i}+d_{\kappa}-R_{j}-d_{\nu}}\right)=\frac{\partial^{2} \phi\left(\boldsymbol{R_{i}+d_{\kappa}-R_{j}-d_{\nu}}\right)}{\partial R_{i \alpha} \partial R_{i \beta}}.\tag{2}$$
with $R_{i\alpha}$ being the $\alpha$-element of coordinate $\boldsymbol{R}_i$. If $u_{i\kappa,\alpha}$ is the displacement of the coordinate $\boldsymbol{R}_i+\boldsymbol{d}_{\kappa}$ at $\alpha-$direction, then where does the summation in (1) come from? I'm trying to derive (1) using the properties of Kronecker delta but so far not much progress has been made.
The RHS of eq.(1) should be first written as:
$$\sum_{j^{\prime}\nu^{\prime}}\phi_{\alpha\beta}(\boldsymbol{R_i+d_{\kappa}-R_{j^{\prime}}-d_{\nu^{\prime}}})\delta_{jj^{\prime}}\delta_{\nu\nu^{\prime}}u_{i\kappa,\alpha}u_{i\kappa,\beta}$$
and then
$$\sum_{j^{\prime}\nu^{\prime}}\phi_{\alpha\beta}(\boldsymbol{R_i+d_{\kappa}-R_{j^{\prime}}-d_{\nu^{\prime}}})\sum_{j\nu}\delta_{jj^{\prime}}\delta_{\nu\nu^{\prime}}u_{i\kappa,\alpha}\delta_{ji}\delta_{\nu\kappa}u_{j\nu,\beta}$$
which gives
$$\sum_{j^{\prime}\nu^{\prime}}\phi_{\alpha\beta}(\boldsymbol{R_i+d_{\kappa}-R_{j^{\prime}}-d_{\nu^{\prime}}})\delta_{ij^{\prime}}\delta_{\kappa\nu^{\prime}}u_{i\kappa,\alpha}u_{j\nu,\beta}$$