Problems with Versions > 5.0

In the Mathematica version 5.1 and 5.2, a change in the way the Function Series works has been made, which causes some problems with the use of Series for expanding Hypergeometric functions. The change is illustrated in the following.


Mathematica 5.0
 Mathematica 5.1, 5.2

In[1]:= f /: Series[f[x_], {x_, 0, n_}] := g[n]

In[2]:= Series[f[x], {x, 0, 2}]

Out[2]= g[2]

In[3]:= Series[1 + f[x], {x, 0, 2}]

Out[3]= 1 + g[2]
In[1]:= f /: Series[f[x_], {x_, 0, n_}] := g[n]

In[2]:= Series[f[x], {x, 0, 2}]

Out[2]= g[2]

In[3]:= Series[1 + f[x], {x, 0, 2}]

                                       2
                               f''[0] x        3
Out[3]= (1 + f[0]) + f'[0] x + --------- + O[x]
                                   2


The newer versions of Mathematica do not use the definition of the series expansion but use the Tailor expansion.

As a consequence, the use of Series with HypExp might fail if the hypergeometric function or its derivative are not defined for vanishing expansion parameter.

What works:

- Series of a single hypergeometric function
- Series of  a single hypergeometric function times a coefficient
- a sum containing one or more hypergeometric function times a coefficient

In[1]:= <<HypExp`

In[2]:= Series[3 e+ 1/e Hypergeometric2F1[1+e,1-2 e,2+3 e,x],{e,0,0}]

                                                     2
          Log[1 - x]     -3 Log[1 - x]   2 Log[1 - x]    3 PolyLog[2, x]
Out[2]= -(----------) + (------------- - ------------- - ---------------) +  O[e]
             x e               x               x                x
  


What does not work:

- Series of  powers of hypergeometric functions
- Series of  functions of hypergeometric functions

The function HypExp is not concerned by this problem and works for all 5.* versions.


--- last update:  10 Jan 2006