generating function

The concept was introduced by French mathematician Abraham de Moivre in 1730.

generating function (plural generating functions)

1. (mathematics) A formal power series with one indeterminate, whose coefficients encode a sequence that can be studied by algebraic manipulation of the series; any one of several generalizations, such as to encode more than one sequence or use more than one indeterminate.
   • 1954, George Pólya, Mathematics and Plausible Reasoning, Volume 1: Induction and Analogy in Mathematics, Princeton University Press, page 101:

     A generating function is a device somewhat similar to a bag. Instead of carrying many little objects detachedly, which could be embarrassing, we put them all in a bag, and then we have only one object to carry, the bag.

   • 1990, Herbert S. Wilf, generatingfunctionology, Academic Press, page 2:

     Most often generating functions arise from recurrence formulas. Sometimes, however, from a generating function you will find a new recurrence formula, not the one you started with, that gives new insights into the nature of your sequence.

   • 2003, Sergei K. Lando (author & translator), Lectures on Generating Functions, American Mathematical Society.

Despite the name, a generating function is not a function. As a formal power series, it is understood that its indeterminate (not a "variable") is never assigned a value and the series is never evaluated. In fact, the series is not even required to converge.

• formal power series

• exponential generating function
• Bell series
• Dirichlet series
• Lambert series
• ordinary generating function

• moment-generating function (probability, statistics)
• power series

• Examples of generating functions on Wikipedia.Wikipedia
• Polynomial sequence on Wikipedia.Wikipedia
• Generating function transformation on Wikipedia.Wikipedia