Pochhammer

Pochhammer[a,n]

gives the Pochhammer symbol TemplateBox[{a, n}, Pochhammer].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • TemplateBox[{a, n}, Pochhammer]=a (a+1) ... (a+n-1)=TemplateBox[{{a, +, n}}, Gamma] /TemplateBox[{a}, Gamma].
  • 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

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Basic Examples  (7)

Evaluate numerically:

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:

Series expansion at a singular point:

Scope  (36)

Numerical Evaluation  (7)

Evaluate numerically:

Evaluate for halfinteger arguments:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Pochhammer threads elementwise over lists:

Pochhammer can be used with Interval and CenteredInterval objects:

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:

Values at zero:

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 :

Plot the real part of TemplateBox[{z, n}, Pochhammer]:

Plot the imaginary part of TemplateBox[{z, n}, Pochhammer]:

Function Properties  (11)

Real domain of Pochhammer:

Complex domain:

Function range of Pochhammer[x,n] for various fixed values of n:

Pochhammer has the mirror property TemplateBox[{{z, }, 2}, Pochhammer]=(TemplateBox[{z, 2}, Pochhammer]):

TemplateBox[{x, 3}, Pochhammer] is an analytic function of x:

TemplateBox[{x, 3}, Pochhammer] is neither non-decreasing nor non-increasing:

TemplateBox[{x, 3}, Pochhammer] is not injective:

TemplateBox[{x, 3}, Pochhammer] is surjective:

TemplateBox[{x, 3}, Pochhammer] is neither non-negative nor non-positive:

TemplateBox[{x, 3}, Pochhammer] does not have either singularity or discontinuity:

TemplateBox[{x, 3}, Pochhammer] is neither convex nor concave:

TraditionalForm formatting:

Differentiation  (2)

First derivative of TemplateBox[{a, n}, Pochhammer] with respect to :

First derivative of TemplateBox[{a, n}, Pochhammer] with respect to :

Higher derivatives of TemplateBox[{a, n}, Pochhammer] with respect to :

Plot the higher derivatives with respect to a when n=5:

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:

Function Identities and Simplifications  (2)

Functional identity:

Recurrence relations:

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 ^(th) 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 TemplateBox[{a, n}, Pochhammer]=TemplateBox[{a, n, {-, 1}}, FactorialPower3] for integer :

Verify an expansion of Pochhammer in terms of FactorialPower for the first few cases:

Sums involving Pochhammer:

Solve recurrence relations:

The generating function is divergent:

Use Borel regularization:

Consider the generating function as a formal power series:

Formal 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 highprecision 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):

Wolfram Research (1988), Pochhammer, Wolfram Language function, https://reference.wolfram.com/language/ref/Pochhammer.html (updated 2022).

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_2024_pochhammer, author="Wolfram Research", title="{Pochhammer}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Pochhammer.html}", note=[Accessed: 18-July-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_pochhammer, organization={Wolfram Research}, title={Pochhammer}, year={2022}, url={https://reference.wolfram.com/language/ref/Pochhammer.html}, note=[Accessed: 18-July-2024 ]}