Yesterday I said,
Looks like there's a bilateral version, too. sum((i-a)!*(i-b)!*(i-c)!/((i+a+1)!*(i+b+1)!*(i+c+1)!),i,minf,inf) =(-a-1)!*(-b-1)!*(-c-1)!*(c+b+a)!/((b+a)!*(c+a)!*(c+b)!)
oo ==== \ (i - a)! (i - b)! (i - c)!
-------------------------------------- =
/ (i + a + 1)! (i + b + 1)! (i + c + 1)! ==== i =-oo (- a - 1)! (- b - 1)! (- c - 1)! (c + b + a)! ---------------------------------------------. (b + a)! (c + a)! (c + b)! Alternatively, as c -> -oo, sum((-1)^i*(i-a)!*(i-b)!/((i+a+1)!*(i+b+1)!),i,0,inf) =box((-a-1)!*(-b-1)!/2)*(1/(a!*b!)-1/((b+a)!)) inf ==== i \ (- 1) (i - a)! (i - b)! (- a - 1)! (- b - 1)! 1 1
------------------------- = --------------------- (----- - --------). / (i + a + 1)! (i + b + 1)! 2 a! b! (b + a)! ==== i = 0
These things are seriously weird. There are many *two term*, *inhomogeneous* relations among the contiguous sums, but none, so far, that would yield a closed form for one in isolation. --rwg