Changing sum and product symbols

Question

I've been searching for answers, found a couple but could not make them to work in my particular situation. If there is a general procedure to do the change, I'd be glad to read it. Is there a way to change a couple of symbols from one symbol set to another? More specifically, I'd like to change the sum and product operators from mtpro2 (lite) to the ones in mathptmx. It'd be great to have a new set of commands, like e.g. \newsum or \newprod for the newly-defined symbols. I'm using XeLaTeX. A basic TeX file is the one I'm posting below. Thank you.

\documentclass[14pt, a4paper, reqno]{extarticle}
\usepackage{amsmath}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\usepackage{mathptmx}
\usepackage[lite]{mtpro2}
\usepackage{mathspec}
        \setmainfont[Scale=0.95]{Crimson Text}
        \setsansfont[Scale=0.8]{Ubuntu}
        \setmonofont[Scale=0.9]{Ubuntu Mono}
        \setmathsfont(Latin)[Uppercase=Italic,Lowercase=Italic,Scale=0.95]{Crimson Text}
        \setmathsfont(Greek)[Uppercase=Regular,Lowercase=Regular,Scale=0.85]{GFS Didot}
        \setmathrm{Crimson Text}
\usepackage{microtype}
\usepackage{amssymb}
\usepackage{amstext}
\usepackage{amsthm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
$$\sum_{i=1}^nX_iY_{n-i}$$
$\sum_i\varphi (x_i)$
\end{document}

Answer 1

With Modern Fonts

In unicode-math, you can replace these symbols with \setmathfont[range = ...]:

\documentclass{article}
\usepackage{unicode-math}
\setmathfont{Latin Modern Math}
\setmathfont[range={\sum, \prod, \coprod}, 
             Scale=MatchUppercase
            ]{STIX Two Math}
\begin{document}
[ \sum_{a \in \mathcal{S}} a = \prod_{b \in \mathcal{T}} b ]
\end{document}

With Legacy Fonts

With legacy 8-bit math fonts, you would need to look up the slot number of both the large and the small operator in the math font, load those math alphabets (of which you have only 16), and declare the command to use the appropriate large or small symbol in each of the four math modes.

The exact commands will vary from package to package, but this example imports \sum and \prod from txfonts.

\documentclass{article}
\DeclareSymbolFont{replacement-operators}{OMX}{txex}{m}{n}
\DeclareMathSymbol{\bigsum}{\mathop}{replacement-operators}{"58}
\DeclareMathSymbol{\smallsum}{\mathop}{replacement-operators}{"50}
\DeclareMathSymbol{\bigprod}{\mathop}{replacement-operators}{"59}
\DeclareMathSymbol{\smallprod}{\mathop}{replacement-operators}{"51}
\renewcommand\sum{\mathop{\mathchoice
  {\bigsum}%
  {\smallsum}%
  {\smallsum}%
  {\smallsum}}}
\renewcommand\prod{\mathop{\mathchoice
  {\bigprod}%
  {\smallprod}%
  {\smallprod}%
  {\smallprod}}}
\begin{document}
[ \sum_{a \in \mathcal{S}} a = \prod_{b \in \mathcal{T}} b ]
\end{document}

It’s trivial to change the \renewcommand declarations to \newcommand\newsum and \newcommand\newprod if you want.

Using the newtxmath symbols requires a little more digging through the source. Its extended symbol font declares an 8-bit extension of the 7-bit OMX encoding, which you must set up.

\documentclass{article}
\usepackage[T1]{fontenc}
\DeclareFontEncoding{LMX}{}{}
\DeclareFontSubstitution{LMX}{ntxexx}{m}{n}
\DeclareSymbolFont{replacement-operators}{LMX}{ntxexx}{m}{n}
\DeclareMathSymbol{\bigsum}{\mathop}{replacement-operators}{"58}
\DeclareMathSymbol{\smallsum}{\mathop}{replacement-operators}{"50}
\DeclareMathSymbol{\bigprod}{\mathop}{replacement-operators}{"59}
\DeclareMathSymbol{\smallprod}{\mathop}{replacement-operators}{"51}
\renewcommand\sum{\mathop{\mathchoice
  {\bigsum}%
  {\smallsum}%
  {\smallsum}%
  {\smallsum}}}
\renewcommand\prod{\mathop{\mathchoice
  {\bigprod}%
  {\smallprod}%
  {\smallprod}%
  {\smallprod}}}
\begin{document}
[ \sum_{a \in \mathcal{S}} a = \prod_{b \in \mathcal{T}} b ]
\end{document}

Answer 2

Interacting with such different packages as mtpro2 and mathspec is not always easy. In this particular case it turns out that you shouldn't redefine \sum and \prod, but \upsumop and \upprodop.

\documentclass[14pt, a4paper, reqno]{extarticle}
% package loading
\usepackage[lite]{mtpro2}
\usepackage{amsmath}
\usepackage{mathspec}
\usepackage{microtype}
\usepackage{amssymb}
\usepackage{amsthm}
% setup
\DeclareSymbolFont{mathptmxlargesymbols}{OMX}{ztmcm}{m}{n}
\DeclareMathSymbol{\upsumop}{\mathop}{mathptmxlargesymbols}{"50}
\DeclareMathSymbol{\upprodop}{\mathop}{mathptmxlargesymbols}{"51}
\setmainfont[Scale=0.95]{Crimson Pro}
%\setsansfont[Scale=0.8]{Ubuntu}
%\setmonofont[Scale=0.9]{Ubuntu Mono}
\setmathsfont(Latin)[Uppercase=Italic,Lowercase=Italic,Scale=0.95]{Crimson Pro}
\setmathsfont(Greek)[Uppercase=Regular,Lowercase=Regular,Scale=0.85]{GFS Didot}
\setmathrm{Crimson Pro}
\begin{document}
[\prod_{k}\sum_{i=1}^nX_iY_{n-i}]
$\prod_k\sum_i\varphi (x_i)$
\end{document}

Production note. I changed Crimson Text into Crimson Pro that I have available, to get a flavor of the final result. I commented out the call for the Ubuntu fonts because I presently don't have them on my system.