Definition:Termial

Definition

Let $n \in \Z_{> 0}$ be a positive integer.

The termial of $n$ is denoted $n?$ and defined as:

$\ds n? = \sum_{k \mathop = 1}^n k = 1 + 2 + \cdots + n$

Extension to Real Numbers

Let $x \in \R$ be a real number.

The termial of $x$ is denoted $x?$ and defined as:

$x? = \dfrac {x \paren {x + 1} } 2$

Example

Termial of $\frac 1 2$

The termial of $\dfrac 1 2$ is given by:

$\dfrac 1 2 ? = \dfrac 3 8$

Historical Note

The termial was invented by Donald E. Knuth in his The Art of Computer Programming.

It was devised as an analogy of the factorial, so as to illustrate the extension of the latter to the real numbers.

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: $(9)$