Natural numbers: \(n\), \(k\)
Real numbers: \(x\)
Real numbers: \(x\)
Factorial of \(n\): \(n!\)
Gamma function: \(\Gamma \left( x \right)\)
Gamma function: \(\Gamma \left( x \right)\)
- The factorial of a non-negative integer \(n\) is called the product of all positive integers less than or equal to the number \(n.\) The factorial of \(n\) is denoted as \(n!\)
\(n! =\) \( 1 \cdot 2 \cdot 3 \ldots \left( {n – 1} \right) \cdot n\) - factorial of zero by definition is equal to\(1:\) \(0! = 1\)
- Factorials of the numbers \(1-10\)
- Recursive formula \(\left( {n + 1} \right)! = n! \cdot \left( {n + 1} \right)\)
- Extension of the factorial function to non-negative real numbers
The factorial of a non-negative real number \(x\) is expressed through the gamma function by the formula \(x! = \Gamma \left( {x + 1} \right),\) which allows to calculate the factorial of any real numbers \(x \ge 0\). - Rate of increase
The factorial function increases faster than the exponential function. The inequality \(n! \gt \exp \left( n \right)\) holds for all \(n \ge 6\). When \(n \ge 1,\) the following relation is valid: \(n \le n! \le {n^n}.\) - Stirling formula
For large \(n\) the approximating factorial value can be determined by the Stirling formula:
\(n! \approx\) \( {n^n}\sqrt {2\pi n} \,\exp \left( { – n} \right) \cdot\) \(\Big[ {1 + {\large\frac{1}{{12n}}\normalsize} + {\large\frac{1}{{288{n^2}}}\normalsize} }-\) \({ {\large\frac{{139}}{{51840{n^3}}}\normalsize} – \ldots } \Big]\).
Taking into account only the first term in the expansion, this formula takes the form
\(n! \approx {n^n}\sqrt {2\pi n} \,\exp \left( { – n} \right)\). - Double factorial
The double factorial is the product of all odd integers from \(1\) to some odd positive integer \(n\). The double factorial is denoted by \(n!!\)
\(\left( {2k + 1} \right)!! =\) \( 1 \cdot 3 \cdot 5 \cdots \left( {2k – 1} \right) \cdot\) \( \left( {2k + 1} \right).\)
Sometimes, the double factorial is considered for even integers. We can define it as
\(\left( {2k} \right)!! =\) \( 2 \cdot 4 \cdot 6 \cdots \left( {2k – 2} \right) \cdot 2k\)
It is implied here that \(0!! = 1.\)
