Since it seems you added \prod
to the question, I take my answer at Different Sum signs, and using the \fonttable
(commented in the MWE), determine the slots for text and display style \prod
as 81 and 89, respectively.
\documentclass{article}
\usepackage{mathptmx,mathtools}
% =============================================
%Import symbols from font cmex without importing the whole package
% =============================================
\DeclareFontFamily{U} {cmex}{}
\DeclareFontShape{U}{cmex}{m}{n}{
<-8> cmex7
<8-9> cmex8
<9-10> cmex9
<10-> cmex10}{}
\DeclareSymbolFont{Xcmex} {U} {cmex}{m}{n}
\DeclareMathSymbol{\Xdsum}{\mathop}{Xcmex}{88}
\DeclareMathSymbol{\Xtsum}{\mathop}{Xcmex}{80}
\DeclareMathOperator*{\Xsum}{\mathchoice{\Xdsum}{\Xtsum}{\Xtsum}{\Xtsum}}
\DeclareMathSymbol{\Xdprod}{\mathop}{Xcmex}{89}
\DeclareMathSymbol{\Xtprod}{\mathop}{Xcmex}{81}
\DeclareMathOperator*{\Xprod}{\mathchoice{\Xdprod}{\Xtprod}{\Xtprod}{\Xtprod}}
% =============================================
%\usepackage{fonttable}
\begin{document}
\centering
sum and prod under mathptmx:\par
$\sum_{i=1}^2 x_i \ne \prod_{i=1}^2 x_i$
\[\sum_{i=1}^2 x_i \ne \prod_{i=1}^2 x_i \]
Defined Xsum and Xprod from cmex:\par
$\Xsum_{i=1}^2 x_i \ne \Xprod_{i=1}^2 x_i$
\[\Xsum_{i=1}^2 x_i \ne \Xprod_{i=1}^2 x_i\]
%\tiny\fonttable{cmex8}
\end{document}

\sum
is just a numeric reference to a character in the operators font for the current math settings, so\amssum
and\ptmxsum
have identical definitions. – David Carlisle Dec 21 '17 at 9:55