Indeterminate expression 0^0 encountered

Question

I'm new to Wolfram Mathematica so the the question may seem quite simple.

I define the following function:

d[i_, j_] := 
 Sum[α^(i + j - 2 k) (-1)^(-k) Sqrt[i!] Sqrt[j!]/((i - k)! (j - k)! k!), {k, 0, Min[i, j]}]

Then, when I set α = 0 and try to evaluate d[0, 0] I get the following warning:

Power::indet: Indeterminate expression 0^0 encountered.

How can I input the condition Power[0,0]=1 in my function ?

Please help!

Answer 1

Although it is best to avoid this when possible, using the undocumented function Internal`InheritedBlock you can temporarily add a rule to Power:

α = 0;

d[i_, j_] := Internal`InheritedBlock[{Power},
  Unprotect[Power];
  Power[0, 0] = 1;
  Sum[α^(i + j - 2 k) (-1)^(-k) Sqrt[i!] Sqrt[j!]/((i - k)! (j - k)! k!), {k, 0, 
    Min[i, j]}]
 ]

d[0, 0]

1

Be aware that user rules for System functions may be ignored when working with packed arrays. See: Block attributes of Equal

Generally better is to use a proxy replacement for Power with your own behavior:

Attributes[myPower] = {Listable};
myPower[0, 0] = 1;
myPower[x_, y_] := x^y;

d[i_, j_] :=
 Sum[myPower[α, (i + j - 2 k)] (-1)^(-k) Sqrt[
    i!] Sqrt[j!]/((i - k)! (j - k)! k!), {k, 0, Min[i, j]}]

The Listable attribute makes sure that zeroes in vectors and matrices are also handled in the way we defined:

In[2]:= myPower[0, {{1, 1}, {1, 0}}]

Out[2]= {{0, 0}, {0, 1}}

Answer 2

Here is a variation of @Daniel's answer from his comment:

α = 0;
Block[{α}, d[0, 0]]

1