Pochhammer[a,n]
gives the Pochhammer symbol .


Pochhammer
Pochhammer[a,n]
gives the Pochhammer symbol .
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
.
- For certain special arguments, Pochhammer automatically evaluates to exact values.
- Pochhammer can be evaluated to arbitrary numerical precision.
- Pochhammer automatically threads over lists.
- Pochhammer can be used with Interval and CenteredInterval objects. »
Examples
open all close allBasic Examples (7)
Evaluate symbolically with respect to n:
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Scope (36)
Numerical Evaluation (7)
Evaluate for half‐integer arguments:
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:
Or compute average-case statistical intervals using Around:
Compute the elementwise values of an array:
Or compute the matrix Pochhammer function using MatrixFunction:
Specific Values (6)
Values of Pochhammer at fixed points:
Obtain the polynomial representation Pochhammer[x,n] for integer values of n:
Expand Pochhammer[x,n] for a fixed value of x:
Infinite arguments give symbolic results:
Find a value of x for which Pochhammer[x,2]=15:
Visualization (3)
Plot the Pochhammer function for various orders:
Plot Pochhammer as a function of its parameter :
Function Properties (11)
Real domain of Pochhammer:
Function range of Pochhammer[x,n] for various fixed values of n:
Pochhammer has the mirror property :
is neither non-decreasing nor non-increasing:
is neither non-negative nor non-positive:
does not have either singularity or discontinuity:
is neither convex nor concave:
TraditionalForm formatting:
Differentiation (2)
Series Expansions (5)
Find the Taylor expansion using Series:
Plots of the first three approximations around :
Find a series expansion at Infinity:
Find a series expansion for an arbitrary symbolic direction :
Taylor expansion at a generic point:
Pochhammer can be applied to a power series:
Applications (4)
Obtain elementary and special functions from infinite sums:
Plot Pochhammer for various values of a parameter:
The average number of runs of length or larger in a sequence of zeros and ones:
Count runs in a random binary sequence:
Compare with the theoretical average:
Define a negative hypergeometric distribution:
Find the probability that black balls were sampled without replacement before a
white ball was drawn from an urn initially filled with
black and
white balls:
Alternatively, compute the probability of drawing a white ball provided that there were black balls in the previous
samplings without replacement:
Properties & Relations (10)
Use FullSimplify to simplify expressions involving Pochhammer:
Use FunctionExpand to expand in Pochhammer in terms of Gamma functions:
Pochhammer can be expressed in terms of a single FactorialPower expression:
Verify the identity for integer
:
Verify an expansion of Pochhammer in terms of FactorialPower for the first few cases:
Sums involving Pochhammer:
The generating function is divergent:

Consider the generating function as a formal power series:
Pochhammer can be represented as a DifferenceRoot:
The exponential generating function for Pochhammer:
Possible Issues (3)
Large arguments can give results too large to be computed explicitly:


Machine-number inputs can give high‐precision results:
As a bivariate function, Pochhammer is not continuous in both variables at negative integers:
Use FunctionExpand to obtain a symbolic expression for Pochhammer at negative integers:
Neat Examples (3)
Plot Pochhammer at infinity:
Plot Pochhammer for complex arguments:
Capelli's sum (binomial theorem with Pochhammer symbols):
See Also
Beta Binomial Gamma Factorial FactorialPower Hypergeometric0F1 Hypergeometric1F1 Hypergeometric2F1 QPochhammer DiscreteRatio AppellF1 AppellF2 AppellF3 AppellF4
Function Repository: FactorialSeriesExpansion
Tech Notes
Related Guides
Related Links
History
Introduced in 1988 (1.0) | Updated in 2021 (13.0) ▪ 2022 (13.1)
Text
Wolfram Research (1988), Pochhammer, Wolfram Language function, https://reference.wolfram.com/language/ref/Pochhammer.html (updated 2022).
CMS
Wolfram Language. 1988. "Pochhammer." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Pochhammer.html.
APA
Wolfram Language. (1988). Pochhammer. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Pochhammer.html
BibTeX
@misc{reference.wolfram_2025_pochhammer, author="Wolfram Research", title="{Pochhammer}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Pochhammer.html}", note=[Accessed: 11-August-2025]}
BibLaTeX
@online{reference.wolfram_2025_pochhammer, organization={Wolfram Research}, title={Pochhammer}, year={2022}, url={https://reference.wolfram.com/language/ref/Pochhammer.html}, note=[Accessed: 11-August-2025]}