Funny Factorials and Slick Sums

[Albert Chan]

(08-05-2019 12:21 AM)Albert Chan Wrote:  From problem 12, Σ(x^m, x=0 to n-1) = n^m+1 / (m+1)

We can extend this to negative integer powers (but, m≠-1)
Note that x^m = perm(x,m) = x / (x-m)!       (*)

Cas> lfac(x,m) := factor(simplify(x!/(x-m)!))
Cas> lfac(x,3)       → x*(x-1)*(x-2)
Cas> ifac(x,0)       → 1
Cas> ifac(x,-3)      → 1/((x+1)*(x+2)*(x+3))

Issues to watch out for.

1). just like ∫(1/x,x=1..n) = ln(n), quoted formula does not work for m == -1
Note: Psi(n+1) = 1/n + Psi(n) = H_n + Psi(1) = H_n - euler_gamma

Cas> sum(lfac(x,-1), x=0..n-1)       → Psi(n+1) + euler_gamma

2). Other negative integer powers, we have to evaluate for lower limit.
Note: "!" has higher precedence than unary minus, parentheses necessary.

m<-1 : 0 ^m+1 / (m+1) = 1/(-(m+1))!/(m+1)

Example:

Cas> m := -3
Cas> s1 := sum(lfac(x,m), x=0..n-1)

1/4 + 1/(2*n+2) + 1/(2*n+4) + 1/(-n-1)

Cas> s2 := lfac(n,m+1)/(m+1) - 1/(-(m+1))!/(m+1)

1/4 − 1/2/((n+1)*(n+2))

Cas> simplify(s1-s2)

0

(*) not quite. Numerical version for negative m not yet implemented.

Cas> perm(x,-3)       → x!/(x+3)!
Cas> perm(0,-3)       → 0 // should be 1/6