06 Nov '17 08:598 edits

is there any way to simplify either

( ∑[k=c, p] (k – 1)! / ( k^m (k – c)! ) ) / ∑[k=c, p] ( (k – 1)! / k^(m-1) (k – c)! )

where

c, m, p ∈ ℕ,

1≤c≤m,

c≤p

or

( ∑[k=c, ∞] (k – 1)! / ( k^m (k – c)! ) ) / ∑[k=c, ∞] (k – 1)! / ( k^(m-1) (k – c)! )

where

c, m ∈ ℕ,

1≤c<m>2,

c ∈ [1, m–2],

m ∈ [3, ∞ )

?

I really hope there is as I use them often in my book I am writing + I find their evaluation is currently horribly computatively inefficient.

In each case the numerator and denominator are almost identical except m in the sum for the numerator is replaced with m-1 in the sum for the denominator.

( ∑[k=c, p] (k – 1)! / ( k^m (k – c)! ) ) / ∑[k=c, p] ( (k – 1)! / k^(m-1) (k – c)! )

where

c, m, p ∈ ℕ,

1≤c≤m,

c≤p

or

( ∑[k=c, ∞] (k – 1)! / ( k^m (k – c)! ) ) / ∑[k=c, ∞] (k – 1)! / ( k^(m-1) (k – c)! )

where

c, m ∈ ℕ,

1≤c<m>2,

c ∈ [1, m–2],

m ∈ [3, ∞ )

?

I really hope there is as I use them often in my book I am writing + I find their evaluation is currently horribly computatively inefficient.

In each case the numerator and denominator are almost identical except m in the sum for the numerator is replaced with m-1 in the sum for the denominator.