Stirling Numbers

STIRLING NUMBERS, in mathematics. In the year 1730 James Stirling, in his Methodus Differentialis introduced into analysis two sets of numbers which, because of their uses in vari ous branches of analysis, their properties, and the methods used in their computation, have continued to attract the attention of mathematicians.

One of the latest writers on the subject, Professor Nielsen of Copenhagen has named them "Stirling numbers of the first and second kind" in honour of their discoverer.

Definitions.

The Stirling numbers of order n of the first kind may be defined as the coefficients in the expansion of (1 x) ( + 2x) • • • ( = I +7,Six-f-nS2x2+7,S3x3+ . • in ascending powers of x, while the Stirling numbers of the second kind are found in Stirling used the numbers now named after him as a tool for expressing A'n as a series of factorials.

i)(x-2)+3x(x—i)-F-x, 0(x-2)+7x(x—i)-i-x, iox(x— i)(x —2)(x -1-25.7c(x — i)(x-2)-FI5x(x—I)-Fx.

The coefficients of the various factorials (i•I ; 1.3.1; 1.6.7.1; 1.10.25.15.1) are the Stirling numbers of the second kind. For about 15o years mathematicians considered it important to be able to express algebraic expressions in form of sequences of factorials. The result was a rich literature consisting of memoirs dealing with the peculiarities of this method of expression. In the year 1846 Weierstrass proved the general futility of this mode of notation, and since then the subject of "factorial notation" has slowly but surely been eliminated as a topic of discussion in mathematical literature. At present it is used only in certain problems of finite summation, in theories of interpolation, and as an example illustrating some of the uses of the Stirling numbers.

Applications.

George Boole in his treatise on the calculus of finite differences calls attention to the "system of numbers expressed by An O'n (differences of nothing)" without indicat ing whether he realized that these numbers were already used by Stirling. The formula given by Boole for An Om may well be

used for the calculation of the above mentioned coefficients T; i.e., the Stirling numbers of the second kind, The second formula is evidently obtained from the first by interchanging the T's and the S's.

It has also been observed (Amer. Math. Monthly, 1928) that the Stirling numbers of the first kind can be obtained by perform ing the algebraic divisions, while the Stirling numbers of the second kind appear in the quotients of The Stirling numbers have many important relationships to the Bernoulli numbers (q.v.), Euler numbers (q.v.), and the tangen tial coefficients, and there are numerous formulas connecting them. In fact, there is a way of regarding the Bernoulli numbers as a species of Stirling numbers; that is (ex— always gives, when expanded, Stirling numbers; and when we let n= — 1, we obtain a well-known expansion giving the Bernoulli numbers.

The consideration of (ex— leads to what may be called ultra-Stirling numbers, a subject which, like the ultra-Bernoulli and ultra-Euler numbers has been very little studied.

BIBLIOGRAPHY.—Chr. Kramp, Analyse des refractions astronomiques et terrestres (Leipzig, 1799) ; v. Ettinghausen, Die combinatorische Analysis (Vienna, 1826) ; Schlafli, "Sur les coefficients du developpement du product (I-Fx) (1-1-2x) . . . [I+ (n—I)x] suivant les puissances ascendantes de x," Journal far die reine and angewandte Mathematik (1852) ; N. Nielsen, Handbuch der Theorie der Gammafunktion (Leip zig, 1906) ; E. Netto, Lehrbuch der Combinatonik (Leipzig, 1927) ; J.

F. Steffenson, Interpolation (Baltimore, 1927). (J. GO