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Pochhammer Symbol — Definition, Formula & Examples

The Pochhammer symbol (x)n(x)_n is a compact notation for a product of nn consecutive values starting at xx, each incremented by 1. It appears frequently in series expansions of hypergeometric functions and in formulas involving binomial coefficients.

The Pochhammer symbol (rising factorial) is defined as (x)n=x(x+1)(x+2)⋯(x+n−1)=∏k=0n−1(x+k)(x)_n = x(x+1)(x+2)\cdots(x+n-1) = \prod_{k=0}^{n-1}(x+k) for a positive integer nn, with (x)0=1(x)_0 = 1 by convention. Equivalently, (x)n=Γ(x+n)Γ(x)(x)_n = \frac{\Gamma(x+n)}{\Gamma(x)} when xx is not a non-positive integer.

Key Formula

(x)n=∏k=0n−1(x+k)=Γ(x+n)Γ(x)(x)_n = \prod_{k=0}^{n-1}(x+k) = \frac{\Gamma(x+n)}{\Gamma(x)}
Where:
  • xx = The starting value (can be any real or complex number)
  • nn = The number of factors in the product (a non-negative integer)
  • Γ\Gamma = The gamma function, which generalizes the factorial

Worked Example

Problem: Evaluate the Pochhammer symbol (3)4(3)_4.
Write out the product: Starting at 3, multiply four consecutive integers.
(3)4=3⋅4⋅5⋅6(3)_4 = 3 \cdot 4 \cdot 5 \cdot 6
Compute: Multiply step by step: 3⋅4=123 \cdot 4 = 12, then 12⋅5=6012 \cdot 5 = 60, then 60⋅6=36060 \cdot 6 = 360.
(3)4=360(3)_4 = 360
Answer: (3)4=360(3)_4 = 360

Why It Matters

The Pochhammer symbol is essential notation in the theory of hypergeometric series, where coefficients involve ratios of rising factorials. It also simplifies expressions in combinatorial identities, probability distributions (such as the beta-binomial), and solutions to differential equations involving special functions.

Common Mistakes

Mistake: Confusing the rising factorial with the falling factorial. Some combinatorics texts use (x)n(x)_n to mean x(x−1)(x−2)⋯(x−n+1)x(x-1)(x-2)\cdots(x-n+1) (the falling factorial), which is the opposite convention.
Correction: Always check which convention your source uses. In analysis and special functions, (x)n(x)_n almost always denotes the rising factorial. The falling factorial is often written xn‾x^{\underline{n}} or (x)(n)(x)_{(n)} to avoid ambiguity.