Pochhammer

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

- Mathematical function, suitable for both symbolic and numerical manipulation.
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- 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 allclose 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 (33)
Numerical Evaluation (6)
Evaluate for half‐integer arguments:
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
Pochhammer can be used with Interval and CenteredInterval objects:
Specific Values (5)
Values of Pochhammer at fixed points:
Pochhammer for symbolic n:
Find a value of x for which Pochhammer[x,2]=15:
Evaluate the associated Pochhammer[x,4] polynomial for integer n:
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:
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 (4)
Generalizations & Extensions (4)
Infinite arguments give symbolic results:
Pochhammer threads elementwise over lists:
Pochhammer can be applied to a power series:
Applications (3)
Obtain elementary and special functions from infinite sums:
Plot Pochhammer:
The average number of runs of length or larger in a sequence of zeros and ones:
Properties & Relations (7)
Use FullSimplify to simplify expressions involving Pochhammer:
Use FunctionExpand to expand in Pochhammer in terms of Gamma functions:
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 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):
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