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

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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 (33)

Numerical Evaluation (6)

Evaluate numerically:

Evaluate for half‐integer 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 (5)

Values of Pochhammer at fixed points:

Pochhammer for symbolic n:

Values at zero:

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 :

Plot the real part of :

Plot the imaginary part of :

Function Properties (11)

Real domain of Pochhammer:

Complex domain:

Function range of Pochhammer:

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:

TraditionalForm formatting:

Differentiation (2)

First derivative with respect to a:

First derivative with respect to n:

Higher derivatives with respect to a:

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

Series Expansions (4)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

Function Identities and Simplifications (2)

Functional identity:

Recurrence relations:

Generalizations & Extensions (4)

Infinite arguments give symbolic results:

Pochhammer threads elementwise over lists:

Pochhammer can be applied to a power series:

Series expansion at infinity:

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:

Count runs in a random binary sequence:

Compare with the theoretical average:

Properties & Relations (7)

Use FullSimplify to simplify expressions involving Pochhammer:

Use FunctionExpand to expand in Pochhammer in terms of Gamma functions:

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 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):

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Pochhammer

Details

Examples

Basic Examples (7)