# 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

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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 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 :

Plot the imaginary part of :

### 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 : is an analytic function of x: is neither non-decreasing nor non-increasing: is not injective: is surjective: is neither non-negative nor non-positive: does not have either singularity or discontinuity: is neither convex nor concave:

### Differentiation(2)

First derivative of with respect to :

First derivative of with respect to :

Higher derivatives of 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  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:

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):