Nota Bene :

1. It’s important to note that this notation is not commonly used and should therefore be used in the right context.
2. The question mark notation is distinct from the Minkowski question mark function.
3. I have not found a translation for “Termial” in French.

Definition :

Termial is presented by Knuth as a function analogous to the factorial function “!”.

The factorial function is defined $\forall n \in \mathbb{N}$ by :

$$n!=\prod^n_{i=1}i= 1 \times 2 \times \ldots \times (n-1) \times n$$

The “termial” function is defined by :

$$n?=\sum^n_{i=1}i = 1 + 2 + \ldots + (n-1) + n$$

This series is grossly divergent.

Extension for non-integer n :

As with the factorial function using the gamma function, it is possible to extend the “Termial” function to non-integer values by taking :

$$n?=\sum^n_{i=1}i = \frac{n(n+1)}{2}$$

Thus :

$$0.5? = \frac{0.5(0.5+1)}{2} = \frac{3}{8}$$

Sources :

1. DONALD (“DON”) ERVIN KNUTH
2. Donald E. Knuth (1997). The Art of Computer Programming: Volume 1: Fundamental Algorithms. 3rd Ed. Addison Wesley Longman, U.S.A. p. 48.
3. en.wikipedia.org - Termial