[Tex/LaTex] How to resize \Sum and \Prod commands

Question

I'd like to make \sum and \prod symbols slightly smaller than they usually are. There are several related questions, but none of them does exactly what I need. I do not want them to be as small as \scriptsize or smaller; also, I want limits in textstyle and displaystyle to be as they originally are (that is, up and down in math mode; and next to $\sum$ and $\prod$ when inline). I have tried many things, but could not get what I desire. A (rather useless, I admit) MWE would be:

\documentclass{article}
\let\oldsum\sum
\renewcommand{\sum}{\oldsum}

\let\oldprod\prod
\renewcommand{\prod}{\oldprod}

\begin{document}
\begin{equation}
\sum \prod
\end{equation}
\end{document}

How could I resize sum and prod keeping the limits as they usually are? Thank you all in advance for your time.

Answer 1

Here's a solution compatible with amsmath:

\documentclass{article}
\usepackage{amsmath}
\usepackage{etoolbox}

\makeatletter
\newcommand{\changeoperator}[1]{%
  \csletcs{#1@saved}{#1@}%
  \csdef{#1@}{\changed@operator{#1}}%
}
\newcommand{\changed@operator}[1]{%
  \mathop{%
    \mathchoice{\textstyle\csuse{#1@saved}}
               {\csuse{#1@saved}}
               {\csuse{#1@saved}}
               {\csuse{#1@saved}}%
  }%
}
\makeatother

\changeoperator{sum}
\changeoperator{prod}

\begin{document}

Display style: $\displaystyle \sum_{k=1}^n a_k+\prod_{k=1}^n b_k$

\bigskip

Text style: $\sum_{k=1}^n a_k+\prod_{k=1}^n b_k$

\bigskip

Script style: $\scriptstyle \sum_{k=1}^n a_k+\prod_{k=1}^n b_k$

\bigskip

Scriptscript style: $\scriptscriptstyle \sum_{k=1}^n a_k+\prod_{k=1}^n b_k$

\end{document}