[Tex/LaTex] Dante Monotype typeface and fractions

Question

I am handling a text which is natively written in a proprietary LaTeX documentclass which implements Dante Monotype fontface. In this text there is a lot of maths and I would say that it works, overall, even if I would not advise in particular Dante font for maths applications.

Even exponents ($e^{x^2}$) are, say, well proportioned to my eye while what I do not like at all are fractions. As in the following picture:

the fraction circled in red, as well as the following ones, has numerator and denominator that are very far from the fraction bar.

How can I improve this in particular? Any package related to fractions may work currently with this font? Any command similar to \dfrac or \tfrac may help decreasing (rather than increasing) the total height of the fractions?

No further solutions come to my mind to fix such "length".

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Addendum:

I have MikTeX 2.9 on Windows 7 OS. The editors I use are TeXstudio and TeXnicCenter and I compile with pdfLaTeX. The behaviour of the fractions is the same when using TeXshop on Mac OS. Thank you.

The code for the picture above is:

$$y_0(t)=1$$
$$y_1(t)=1+\int^t_0 \mt\,\md\mt=1+\frac{t^2}{2}$$
$$y_2(t)=1+\int^t_0 \mt\left(1+\frac{\mt^2}{2}\right)\md\mt=1+\frac{t^2}{2}+\frac{t^4}{8}$$
\vspace{1pt}
$$\vdots$$
\vspace{1pt}
$$y_n(t)=1+\int^t_0 \mt\,y_{n-1}(\mt)\,\md\mt=1+\frac{t^2}{2}+\frac{1}{2!}\left(\frac{t^2}{2}\right)^2+\ldots+\frac{1}{n!}\left(\frac{t^2}{2}\right)^n$$

where \md and \mt were defined in my preamble. Thank you.

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Addendum (2):

I appreciated both the answers posted so far and I tried to define a newcommand as in lines 2 to 15 of the code by Steven B. (I preferred this approach because I never put my hands on LuaLaTeX before and this was meant to be "just fine tuning"). Furthermore I tried all of the following three:

\renewcommand{\frac}[2]{\displaystyle{\frac{\lower2pt\hbox{#1}}{\raise2pt\hbox{#2}}}}

and:

\renewcommand{\frac}[2]{\dfrac{\lower2pt\hbox{#1}}{\raise2pt\hbox{#2}}}

or:

\newcommand\newfrac[2]{\frac{\lower2pt\hbox{#1}}{\raise2pt\hbox{#2}}}

but all of these four candidate solutions generated several errors in my system (class). Do these three lines of me suggest something useful?

I'll post updates.

Answer 1

I don't have access to your font, but (as indicated in my comment), my answer at A special fraction macro with raised fraction bar can address your issue. Here, I create it as \newfrac.

I modified the dimensions from the cited answer to further squeeze the fractions together, for demonstration purposes. The optional arguments for \stackunder and \stackon are the gap distances to adjust to suit your needs. This MWE, of course, is using the CM fonts.

\documentclass{article}
\usepackage{stackengine}
\usepackage{ifthen}
\newlength\testwida
\newlength\testwidb
\newlength\mywidth
\newcommand\newfrac[2]{%
  \setlength\testwida{\widthof{$#1$}}%
  \setlength\testwidb{\widthof{$#2$}}%
  \ifthenelse{\lengthtest{\testwida>\testwidb}}%
    {\mywidth=\testwida}%
    {\mywidth=\testwidb}%
  \raisebox{.5ex}{%
    \stackunder[2pt]{\stackon[2pt]{\rule{\mywidth}{.4pt}}{$#1$}}{$#2$}}%
}
\def\mt{\mathbf{t}}
\def\md{\mathbf{d}}
%\let\newfrac\frac% UNCOMMENT TO SEE ORIGINAL \frac VERSION
\begin{document}
$$y_0(t)=1$$
$$y_1(t)=1+\int^t_0 \mt\,\md\mt=1+\newfrac{t^2}{2}$$
$$y_2(t)=1+\int^t_0 \mt\left(1+\newfrac{\mt^2}{2}\right)\md\mt=1+\newfrac{t^2}{2}+\newfrac{t^4}{8}$$
\vspace{1pt}
$$\vdots$$
\vspace{1pt}
$$y_n(t)=1+\int^t_0 \mt\,y_{n-1}(\mt)\,\md\mt=1+\newfrac{t^2}{2}+\newfrac{1}{2!}\left(\newfrac{t^2}{2}\right)^2+\ldots+\newfrac{1}{n!}\left(\newfrac{t^2}{2}\right)^n$$

\end{document}

For comparison, here is the original \frac version.