Calculator Inputs
Example Data Table
| x | k | Falling Factorial | Interpretation |
|---|---|---|---|
| 9 | 4 | 3,024 | 9 × 8 × 7 × 6 |
| 10 | 3 | 720 | Standard permutation case, equal to 10P3 |
| 5.5 | 3 | 86.625 | Product remains valid with a decimal start value |
| -2.5 | 2 | 8.75 | Negative starts are supported through descending multiplication |
| 7.2 | 2.5 | Gamma-based | Uses continuation through the gamma function |
Formula Used
xk = x(x − 1)(x − 2)...(x − k + 1)
xk = Γ(x + 1) / Γ(x − k + 1)
For integers n and r with n ≥ r ≥ 0, nr = nP r = n! / (n − r)!
Ratio = xk / xk, when xk is real and nonzero.
The page uses direct multiplication for integer orders and gamma continuation for fractional orders. This combination supports permutation analysis, symbolic comparison, and smooth graphing across many cases.
How to Use This Calculator
- Enter the starting value x.
- Enter the order k.
- Select Auto, Exact product, or Gamma continuation.
- Choose the decimal precision for displayed results.
- Set the graph max order to compare descending products.
- Enable step-by-step output when you want multiplication details.
- Press the calculate button.
- Review the result block above the form, then export CSV or PDF.
FAQs
1) What is a falling factorial?
A falling factorial multiplies a value by the next lower values. For integer order k, it uses k descending terms. It appears in permutations, finite differences, polynomial bases, and combinatorics.
2) When does it equal a permutation?
It matches nPr when both inputs are nonnegative integers and n is at least r. In that case, the descending product and the permutation formula produce the same value.
3) Why does the calculator offer gamma continuation?
Gamma continuation extends the idea beyond integer orders. It lets you evaluate many decimal inputs smoothly with Γ(x+1) divided by Γ(x−k+1), as long as the expression avoids poles.
4) Can x be negative?
Yes. Negative starting values can still be multiplied directly for integer orders. The sign changes according to the number of negative factors in the descending sequence.
5) What happens when k is zero?
The result equals 1. This follows the usual empty-product rule and keeps the falling factorial consistent with combinatorial identities and recurrence relations.
6) Why is a ratio to x^k shown?
That ratio shows how a descending product differs from an ordinary power. It is useful for growth comparisons, asymptotic checks, and teaching the structural difference between the two expressions.
7) Why might gamma mode be undefined?
Gamma mode becomes undefined at poles, which occur at zero or negative integers inside the gamma function. The calculator warns you and may switch to direct multiplication when appropriate.
8) What does the graph show?
The graph plots falling factorial values against order. It can also compare them with ordinary powers and permutation values, which helps visualize how quickly the descending product changes.