Pochhammer Symbol (Rising Factorial) Calculator

Compute the Pochhammer symbol, the rising factorial (x)ₙ=x(x+1)…(x+n-1). Also shows the falling factorial and term expansion, with exact integer results for integer inputs.

Input

Enter a real number x and a non-negative integer n to compute the Pochhammer symbol (rising factorial) (x)ₙ=x(x+1)…(x+n-1).

x (real number)

Starting value. Integer or decimal.

n (non-negative integer)

Number of factors to multiply.

Result

Rising factorial (3)_4

360

Exact value (integer input)

Falling factorial 3^(4)

Term expansion

3 × 4 × 5 × 6

Partial products (x)_k

k	Value of (x)_k
0	1
1	3
2	12
3	60
4	360

How it works

The Pochhammer symbol (rising factorial) is defined as (x)ₙ = x(x+1)(x+2)…(x+n-1), with the empty product giving 1 when n=0. Using the gamma function it equals Γ(x+n)/Γ(x).
The falling factorial is x^(n) = x(x-1)(x-2)…(x-n+1), multiplying successive integers decreased by one each step. For a positive integer x with n=x it equals x! (x factorial).
When x is an integer the products are computed exactly with BigInt, so even results with many digits are returned without rounding error. For non-integer x the continued product is approximated in floating point.
Rising and falling factorials are linked by (x)ₙ = (-1)ⁿ (-x)^(n). The binomial coefficient can also be written with the falling factorial as C(x,n)=x^(n)/n!.
Pochhammer symbols appear constantly in hypergeometric functions, series expansions, and combinatorics. Conventions differ between sources, so this tool reports the rising factorial as the primary result.

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