Use Nest in a recurrence function

 Use Nest in a recurrence function

I have following recurrence function:

al0[k_, ee_] := (k d[k, ee] + (k + 1) d[k + 1, ee]) (-1)^k ;
al1[k_, ee_] := 
  al0[k, ee] - (k + 1)/(2 k + 3) al0[k + 1, ee] - k/(2 k - 1) al0[k - 1, ee];
al2[k_, ee_] := 
  al1[k, ee] - (k + 1)/(2 k + 3) al1[k + 1, ee] - k/(2 k - 1) al1[k - 1, ee];
al3[k_, ee_] := 
  al2[k, ee] - (k + 1)/(2 k + 3) al2[k + 1, ee] - k/(2 k - 1) al2[k - 1, ee];

Could I use Nest to make it simpler and faster? If not nest, anything else i can use?

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