Hitherto, the argument which is passed to a function was treated in a straight forward manner. Functions pass the argument given to the compound functions, these functions are evaluated and afterwards, an operation is performed using the result of the single evaluation of the compound functions, for instance an addition. For the summation an equivalent problem occurs. As a first example, which shall introduce the use of unnamed functions, a binary function is given. A compound function shall be written which yields the following result: .
Due to the possibility of altering the argument of the quantity functions, functions are not restricted to the evaluation of one single argument. As a consequence it can also be sensible to use a binary second order function. Here, the unnamed functions and represent the following dependences:
If the second-order quantity function is evaluated with respect to the first argument it can be written shortly as or only as . An evaluation with the second-order function is written by the abbreviation . Using this notation, the following expressions hold true:
As for all other elements of this calculus, also the summations are second order functions. Hitherto, it has been implicitly assumed, that the base element of the summation is the argument passed to the sum function. As two arguments are available, it has to be specified which argument is used. In order to treat sums as any other second order function, a function is passed to the summation function, which determines the base element of the summation from the arguments passed.