Definition:Generating Function/Historical Note

Historical Note on Generating Function

Generating functions were introduced by Abraham de Moivre to solve the general problem of linear recurrences.

James Stirling then extended this theory in his Methodus Differentialis of $1730$, by using differentiation and integration.

Then Leonhard Paul Euler began extending their use to new fields such as combinatorics.

Pierre-Simon de Laplace took the technique into the field of probability theory in his $1812$ work Théorie Analytique des Probabilités

Many others since have developed the technique further.

A generating function is a clothesline on which we hang up a sequence of numbers for display.

-- 1990: Herbert S. Wilf: generatingfunctionology

Sources

• 1730: Jacobo Stirling: Methodus Differentialis

• 1741: Leonhard Paul Euler: Observationes Analyticae Variae de Combinationibus (Commentarii Acad. Sci. Imp. Pet. Vol. 13: pp. 64 – 93)

• 1750: Leonhard Paul Euler: De Partitione Numerorum (Novi Comment. Acad. Sci. Imp. Petropol. Vol. 3: pp. 125 – 169)

• 1812: Pierre-Simon de Laplace: Théorie Analytique des Probabilités

• 1923: E.T. Bell: Euler Algebra (Trans. Amer. Math. Soc. Vol. 25: pp. 135 – 154)  www.jstor.org/stable/1989055

• 1969: Ivan Niven: Formal Power Series (Amer. Math. Monthly Vol. 76: pp. 871 – 889)  www.jstor.org/stable/2317940

• 1974: Peter Henrici: Applied and Computational Complex Analysis: Volume $\text { 1 }$

• 1994: Herbert S. Wilf: generatingfunctionology (2nd ed.)

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.9$: Generating Functions