On generalized Stirling numbers and polynomials
by
Nenad P. Cakic
Faculty of Electrical Engineering, University of Belgrade, Belgrade, SERBIA & MONTENEGRO

The Stirling numbers of the second kind and the corresponding polynomials, so called, single variable Bell polynomials are defined by

S(n, k)=\frac1k! k å i=0  (-1)k-i\bincokiin = \frac(-1)kk!\Deltak0n

(\theequation)

and

An(x)= n å k=0  S(n, k)xk,

(\theequation)

respectively.

Singh , Sinha and Dhawan, and Shrivastava , studied the generalized Stirling numbers and polynomials defined, respectively, by

S(\a)(n, k, r) = \frac(-1)kk! k å j=0  (-1)j\bincokj(\a + rj)n

(\theequation)

and

Tn(\a)(x, r, -p) = n å k=0  S(\a)(n, k, r)pkxrk.

(\theequation)

It is easily verified from (4) that

Tn(\a)(x, r, p)=x-\aepxr(xD)nx\ae-pxr

i.e the generalized Truesdell polynomials .

Singh's generalization was motivated by the generalization of Hermite polynomials of Gould-Hopper given by

Hn(r)(x, \a, p)=(-1)nx-\aepxrDn(x\ae-pxr).

Singh Chandel introduced the following generalizations of the Stirling numbers and polynomials :

S(\a, \l)(n, k, r)=\frac(-1)kk! k å j=0  (-1)j\bincokj(\a + rj)(\l-1, n),

(\theequation)

and

Tn(\a, \l)(x, r, -p) = xn(\l -1) n å k=o  S(\a, \l)(n, k, r)pkxrk

(\theequation)

where

a(\l -1, n)=( \fraca\l -1)n(\l -1)n.

Evidently, when \l --> 1, equations (5) and (6) would reduce to (3) and (4) respectively, which, in turns, will yield (1) and (2), respectively for r-1=\a = 0.

In this paper we prove that the generalizations (3) and (4) and related properties are an old result, published in 1887. by d'Ocagne . Also, we prove that Singh Chandel result is the consequence of the fundametal results of Toscano and Chak .

The new explicit expressions for numbers (3) and (5) are also given.

Date received: May 6, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cakq-33.