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:

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:

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:

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 a symbolic expression for Pochhammer at negative integers:

Plot Pochhammer at infinity:

Plot Pochhammer for complex arguments:

Capelli's sum (binomial theorem with Pochhammer symbols):