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# Pochhammer

 Pochhammergives the Pochhammer symbol .
• 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.
Compute a sum involving Pochhammer:
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Compute a sum involving Pochhammer:
 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:
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:
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:
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:
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:
Plot Pochhammer at infinity:
Plot Pochhammer for complex arguments:
Capelli's sum (binomial theorem with Pochhammer symbols):
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