RootSum result manipulation/simplification

Question

Consider the sum

sum1 = Sum[ k/( k^7 - 2 k + 3), {k, Infinity}]

-RootSum[ 2 + 5 #1 + 21 #1^2 + 35 #1^3 + 35 #1^4 + 21 #1^5 + 7 #1^6 + #1^7 &,
          ( PolyGamma[0, -#1] + PolyGamma[0, -#1] #1)/( 5 + 42 #1 + 105 #1^2 + 140 #1^3 
             + 105 #1^4 + 42 #1^5 + 7 #1^6)& ]

Back in the day of Mathematica 3.0 this same sum gave

sum2 = RootSum[( 3 - 2 #1 + #1^7) & , -(( PolyGamma[0, -#1] #1)/(-2 + 7 #1^6))& ]

-RootSum[ 3 - 2 #1 + #1^7 &, ( PolyGamma[0, -#1] #1)/(-2 + 7 #1^6)& ]

They are numerically equal

(sum1 - sum2) // N // Chop

0

• Are the two RootSum expressions equivalent? and If so:
• How do I manipulate/simplify sum1 result to the simpler form of sum2?
• Why is the current result more complicated?

Answer 1

PolyGamma was modified in ver.6 (see Summary of New Features in 6.0) as well as Sum last modified in ver.7 therfore the results of these sums are symbolically different. Perhaps sum1 is simpler than sum2 with respect to internal algebraic manipulations.

To verify equivalence you can make use of symbolic functionality, e.g.

RootApproximant[ sum1 - sum2 ]

0     

This may be enforced by numerical approximations to arbitrary order e.g :

N[sum1, 3000] === N[sum2, 3000]

True