Hello :)

I have some expressions for a sum, in Wolfram is possible to find a recurrence equation for these expressions ?

for example ,

for m = 0 i have this expression,

(f[x1, x2, \[Alpha]] P.P (24 \[CapitalDelta] (ma1 mb2 + 
      2 \[CapitalDelta]) + 
   P.P (-18 \[CapitalDelta] + 
      f[x1, x2, \[Alpha]]^2 (12 ma1 mb2 + 26 \[CapitalDelta] + 
         3 (-3 + f[x1, x2, \[Alpha]]^2) P.P))))/(24 \[CapitalDelta]^4)

for m = 1,

1/(48 \[CapitalDelta]^4) (16 \[CapitalDelta]^2 (ma1 mb2 - 
      4 \[CapitalDelta]) + 
   P.P (8 \[CapitalDelta] (-2 \[CapitalDelta] + 
         f[x1, x2, \[Alpha]] (3 ma1 mb2 + 
            6 \[CapitalDelta] - (5 ma1 mb2 + 13 \[CapitalDelta]) f[x1,
               x2, \[Alpha]])) + 
      2 f[x1, x2, \[Alpha]] (-9 \[CapitalDelta] + 
         f[x1, x2, \[Alpha]] (17 \[CapitalDelta] + 
            f[x1, x2, \[Alpha]] (6 ma1 mb2 + 
               13 \[CapitalDelta] - (6 ma1 mb2 + 
                  17 \[CapitalDelta]) f[x1, x2, \[Alpha]]))) P.P - 
      3 (-1 + f[x1, x2, \[Alpha]]) f[x1, 
        x2, \[Alpha]]^3 (-3 + f[x1, x2, \[Alpha]]^2) (P.P)^2))

for m = 2

1/(96 \[CapitalDelta]^4) (-16 \[CapitalDelta]^2 (-2 ma1 mb2 + 
      8 \[CapitalDelta] + (3 ma1 mb2 - 15 \[CapitalDelta]) f[x1, 
        x2, \[Alpha]]) + 
   4 \[CapitalDelta] (8 \[CapitalDelta] + 
      f[x1, x2, \[Alpha]] (6 ma1 mb2 - 3 \[CapitalDelta] - 
         4 (5 ma1 mb2 + 13 \[CapitalDelta]) f[x1, 
           x2, \[Alpha]] + (14 ma1 mb2 + 45 \[CapitalDelta]) f[x1, 
           x2, \[Alpha]]^2)) P.P + 
   2 (-1 + f[x1, x2, \[Alpha]]) f[x1, 
     x2, \[Alpha]] (9 \[CapitalDelta] + 
      f[x1, x2, \[Alpha]] (-25 \[CapitalDelta] + 
         f[x1, x2, \[Alpha]] (-6 ma1 mb2 - 13 \[CapitalDelta] + 
            3 (2 ma1 mb2 + 7 \[CapitalDelta]) f[x1, 
              x2, \[Alpha]]))) (P.P)^2 + 
   3 (-1 + f[x1, x2, \[Alpha]])^2 f[x1, 
     x2, \[Alpha]]^3 (-3 + f[x1, x2, \[Alpha]]^2) (P.P)^3)

m = 3

1/(192 \[CapitalDelta]^4) (-1 + 
   f[x1, x2, \[Alpha]]) (-48 \[CapitalDelta]^2 (ma1 mb2 - 
      3 \[CapitalDelta] - 
      2 (ma1 mb2 - 6 \[CapitalDelta]) f[x1, x2, \[Alpha]]) - 
   12 \[CapitalDelta] (4 \[CapitalDelta] + 
      f[x1, x2, \[Alpha]] (2 ma1 mb2 - 7 \[CapitalDelta] + 
         f[x1, x2, \[Alpha]] (-8 ma1 mb2 - 
            22 \[CapitalDelta] + (6 ma1 mb2 + 23 \[CapitalDelta]) f[
              x1, x2, \[Alpha]]))) P.P - 
   2 (-1 + f[x1, x2, \[Alpha]]) f[x1, 
     x2, \[Alpha]] (9 \[CapitalDelta] + 
      f[x1, x2, \[Alpha]] (-33 \[CapitalDelta] + 
         f[x1, x2, \[Alpha]] (-6 ma1 mb2 - 
            13 \[CapitalDelta] + (6 ma1 mb2 + 25 \[CapitalDelta]) f[
              x1, x2, \[Alpha]]))) (P.P)^2 - 
   3 (-1 + f[x1, x2, \[Alpha]])^2 f[x1, 
     x2, \[Alpha]]^3 (-3 + f[x1, x2, \[Alpha]]^2) (P.P)^3)

m = 4

1/(384 \[CapitalDelta]^4) (-1 + 
   f[x1, x2, \[Alpha]])^2 (-32 \[CapitalDelta]^2 (-2 ma1 mb2 + 
      2 \[CapitalDelta] + 
      5 (ma1 mb2 - 7 \[CapitalDelta]) f[x1, x2, \[Alpha]]) + 
   8 \[CapitalDelta] (8 \[CapitalDelta] + 
      f[x1, x2, \[Alpha]] (3 ma1 mb2 - 23 \[CapitalDelta] - 
         2 (7 ma1 mb2 + 20 \[CapitalDelta]) f[x1, 
           x2, \[Alpha]] + (11 ma1 mb2 + 49 \[CapitalDelta]) f[x1, 
           x2, \[Alpha]]^2)) P.P + 
   2 (-1 + f[x1, x2, \[Alpha]]) f[x1, 
     x2, \[Alpha]] (9 \[CapitalDelta] + 
      f[x1, x2, \[Alpha]] (-41 \[CapitalDelta] + 
         f[x1, x2, \[Alpha]] (-6 ma1 mb2 - 
            13 \[CapitalDelta] + (6 ma1 mb2 + 29 \[CapitalDelta]) f[
              x1, x2, \[Alpha]]))) (P.P)^2 + 
   3 (-1 + f[x1, x2, \[Alpha]])^2 f[x1, 
     x2, \[Alpha]]^3 (-3 + f[x1, x2, \[Alpha]]^2) (P.P)^3)

thanks!

Attachments:

recurrence.nb