From Johnson, Kotz and Balakrishnan (1995) [2nd ed.; formula (31.15), p.516],
the non-central t density is
f(x, df, ncp) =
df^(df/2) * exp(-.5*ncp^2) /
(sqrt(pi)*gamma(df/2)*(df+x^2)^((df+1)/2)) *
sum_{k=0}^Inf gamma((df + k + df)/2)*ncp^k /
prod(1:k)*(2*x^2/(df+x^2))^(k/2)
The functional relationship
f(x, df, ncp) = df/x *
(F(sqrt((df+2)/df)*x, df+2, ncp) - F(x, df, ncp))
is used to evaluate the density at x != 0 and
f(0, df, ncp) = exp(-.5*ncp^2) /
(sqrt(pi)*sqrt(df)*gamma(df/2))*gamma((df+1)/2)
is used for x=0.