Gulliver, "Pascal matrices and

Stirling numbers," Applied Mathematics Letters, vol.

(see [15, 16, 18-23]), where [S.sub.1](n, k) is the

Stirling numbers of the first kind and [S.sub.2](n, k) is the

Stirling numbers of the second kind as follows:

Combinatorial Identities for

Stirling Numbers: The Unpublished Notes of H.W.

The numbers [W.sub.m](n,k) can be shown to be a kind of generalization of the famous

Stirling numbers of the Second kind S(n, k) when the parameter m equals to 1.

It is interesting to note that there are already classical formulas expressing the Bernoulli number in terms of

Stirling numbers such as

This computation is based on some interesting relation between the dimensions of the summands of the [E.sub.2]-page of the spectral sequence and some

Stirling numbers. Since these Euler characteristic have an exponential growth, we deduce an exponential growth of the dimension of the associated graded space of [E.sub.2].

After preliminaries they cover numerical methods for solving ordinary and partial fractional differential equations, efficient numerical methods, generalized

Stirling numbers and applications, fractional variational principles, continuous-time random walks (CTRW) and fractional diffusion models, and applications of CTRW to finance and economics.

Recently, many mathematicians have studied in the areas of the Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials,

Stirling numbers, and central factorial numbers (see [1-17]).

This expression is an alternating sum involving G-analogs of the

Stirling numbers; it appears to be new even for the deleted Shi arrangements.

They begin with a few examples, just to let students get a feel for it, then look at fundamentals of enumeration; the pigeonhole principle and Ramsey's theorem; the principle of inclusion and exclusion; generating functions and recurrence relations; Catalan, Bell, and

Stirling numbers; symmetries and the Polya-Redfield method; graph theory; coding theory; Latin squares; balanced incomplete block designs; and linear algebra methods in combinatorics.

The generalized

Stirling numbers have been considered by Tsylova [13] and Chelluri et al.

where S (n' k) are

stirling numbers of second kind.