BUILT-IN MATHEMATICA SYMBOL

Pochhammer

gives the Pochhammer symbol

.

MORE INFORMATION

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.

EXAMPLES

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Basic Examples (2)

Compute a sum involving Pochhammer:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

In[3]:=

Out[3]=

Compute a sum involving Pochhammer:

In[1]:=

Out[1]=

Scope (8)

Evaluate for large arguments:

Evaluate for half-integer arguments:

Evaluate numerically:

Evaluate for complex arguments:

Evaluate to high precision:

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

Series expansion at a generic point:

TraditionalForm formatting:

Generalizations & Extensions (4)

Infinite arguments give symbolic results:

Pochhammer threads element-wise 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 (5)

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:

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):