\newcommand{\NAME}{DEFINITION}
and used simply by \name.
\newcommand{\gi}{\Gamma_{i}}
you get the mathematical symbol simply by \gi.
\newcommand{\gi}{\ensuremath{\Gamma_{i}}}
\newcommand{\NAME}[NARGS]{DEFINITION}
where NARGS is the number of parameters. They are used in DEFINITION as #1, #2,
\newcommand{\veca}[1]{\ensuremath{#1_1, \ldots, #1_n}}
the text
We consider the series \veca{a} and \veca{b}.
produces:
\newcommand{\defi}[1]{\textbf{#1}}
makes this logical meaning clearer than using simply \textbf. Moreover, if one decides to use italics instead and put all definitions into an index at the end, this can be done by simply changing the macro.