Symbol: Put a smile symbol under a plus

Here is an easy solution (I think) with stackengine:

\documentclass{article}
\usepackage{stackengine}
\newcommand\pluss{\mathop{\stackMath\stackinset{c}{0pt}{c}{-1ex}{{\smile}}{{+}}}}

\begin{document}

\[ \pluss_{k} f(k) \]

\end{document} 

With some low level TeX programming:

\documentclass{article}
\usepackage{amsmath}

\makeatletter
\DeclareRobustCommand{\pluss}{\mathbin{\text{\pluss@}}}
\newcommand{\pluss@}{%
  \vtop{%
    \offinterlineskip\m@th
    \halign{\hfil##\hfil\cr$+$\cr$\smile$\cr}%
  }%
}
\makeatother

\begin{document}

$3\pluss 4+5$

$\scriptstyle 3\pluss 4+5$

\end{document}

With a smaller \smile:

\documentclass{article}
\usepackage{amsmath}

\makeatletter
\DeclareRobustCommand{\pluss}{\mathbin{\text{\pluss@}}}
\newcommand{\pluss@}{%
  \vtop{%
    \offinterlineskip\m@th
    \halign{\hfil##\hfil\cr$+$\cr$\scriptstyle\smile$\cr}%
  }%
}
\makeatother

\begin{document}

$3\pluss 4+5$

$\scriptstyle 3\pluss 4+5$

\end{document}

I'm not sure exactly what spacing you want, so this is an adjustable approach using xparse around your original design.

\documentclass[11pt]{article}

\usepackage{amsmath,xparse}  

\NewDocumentCommand{\pluss}{O{-.5ex} O{0.5ex}}{%
    \raisebox{#1}{\,$\overset{\textstyle{\raisebox{#2}{$+$}}}{\smash{\smile}}$}\,%
}

\begin{document}

$3 \pluss 4 $

$3 \pluss[0.ex][0.2ex] 4 $

$3 \pluss[-0.3ex][0.2ex] 4 $

\end{document}

You can adjust it until you find the defaults you like.