How to broadcast pandas series with different indices?

I have two pandas series, each with a different index:

In [2]: a = pd.Series(range(5), index=pd.Index(list('abcde'), name='index'))
In [3]: b = pd.Series(range(4), index=pd.Index(list('ABCD'), name='BIG_INDEX'))

What I would like to do is something along the lines of a.mul(b, axis=1) to instruct pandas to broadcast b along the 1 axis before performing a ufunc (e.g. multiply, raise to the power of, etc). Is there a better way of doing this than using apply?

In [4]: a.apply(lambda x: x*b)
Out[4]:
BIG_INDEX  A  B  C   D
index
a          0  0  0   0
b          0  1  2   3
c          0  2  4   6
d          0  3  6   9
e          0  4  8  12

You could use numpy.outer to perform the calculation as if a and b were arrays:

In [285]: pd.DataFrame(np.outer(a, b), columns=b.index, index=a.index)
Out[285]: 
BIG_INDEX  A  B  C   D
index                 
a          0  0  0   0
b          0  1  2   3
c          0  2  4   6
d          0  3  6   9
e          0  4  8  12

This is quicker than calling a.apply(lambda x: x*b).

By the way, all NumPy ufuncs come with 5 methods: outer, accumulate, reduce, reduceat, and at. So another way to write the solution above is

In [34]: pd.DataFrame(np.multiply.outer(a, b), columns=b.index, index=a.index)
Out[36]: 
BIG_INDEX  A  B  C   D
index                 
a          0  0  0   0
b          0  1  2   3
c          0  2  4   6
d          0  3  6   9
e          0  4  8  12

And when written this way, it is clear how to apply the same idea to any NumPy ufunc. For example, to make and addition table out of a and b, call np.add's outer method:

In [37]: pd.DataFrame(np.add.outer(a, b), columns=b.index, index=a.index)
Out[37]: 
BIG_INDEX  A  B  C  D
index                
a          0  1  2  3
b          1  2  3  4
c          2  3  4  5
d          3  4  5  6
e          4  5  6  7