I made it with the following Gnuplot code:

# set the output
set terminal svg
set output "legendre_poly.svg"

set title "Legendre Polynomials"

# axis properties
set xrange [-1:1]
set yrange [-1:2]
set xlabel "x"
set ylabel "P_n (x)"
plot 1 ti "n = 0" with lines linewidth 2,\
     x ti "n = 1" with lines linewidth 2,\
     (3*x**2-1)/2 ti "n = 2" with lines linewidth 2,\
     (5*x**3-3*x)/2 ti "n = 3" with lines linewidth 2,\
     (35*x**4-30*x**2+3)/8 ti "n = 4" with lines linewidth 2,\
     (63*x**5-70*x**3+15*x)/8 ti "n = 5" with lines linewidth 2