arXiv subjects

Milan Janjic

Publications and source records attributed to Milan Janjic.

23 records · Page 2Linked to original sources

arXiv2009

On Non-central Stirling Numbers of the First Kind

It is shown in this note that non-central Stirling numbers s(n,k,a) of the first kind naturally appear in the expansion of derivatives of the product of a power function and a logarithn function. We first obtain a recurrence relation for these numbers, and then, using Leibnitz rule we obtain an explicit formula for these numbers. We also obtain an explicit formula for s(n,1,a), and then derive several combinatorial identities related to these numbers.

math.CO↗

arXiv2008

On a non-combinatorial definition of Stirling numbers

In Combinatorics Stirling numbers may be defined in several ways. One such definition is given in [1], where an extensive consideration of Stirling numbers is presented. In this paper an alternative definition of Stirling numbers of both kind is given. Namely, Stirling numbers of the first kind appear in the closed formula for the n-th derivative of ln x. In the same way Stirling numbers of the second kind appear in the formula for the n-th derivative of f(e^x), where f(x) is an arbitrary smooth real function. This facts allow us to define Stirling numbers within the frame of differential calculus. These definitions may be interesting because arbitrary functions appear in them. Choosing suitable function we may obtain different properties of Stirling numbers by the use of derivatives only. Using simple properties of derivatives we obtain here three important properties of Stirling numbers. First are so called two terms recurrence relations, from which one can easily derive the combinatorial meaning of Stirling numbers. Next we obtain expansion formulas of powers into falling factorials, and vise versa. These expansions usually serve as the definitions of Stirling numbers, as in [1]. Finally, we obtain the exponential generating functions for Stirling and Bell numbers. As a by product the closed formulas for the $n$-th derivative of the functions f(e^x) and f(ln x) are obtained.

math.CO↗

arXiv2008

On a Class of Polynomials with Integer Coefficients

A class P_{n,m,p}(x) of polynomials is defined. The combinatorial meaning of its coefficients is given. Chebyshev polynomials are the special cases of P_{n,m,p}(x). It is first shown that P_{n,m,p}(x) may be expressed in terms of P_{n,0,p}(x). From this we derive that P_{n,2,2}(x) may be obtain in terms of trigonometric functions, from which we obtain some of its important properties. Some questions about orthogonality are also concerned. Furthermore, it is shown that P_{n,2,2}(x) fulfills the same three terms recurrence as Chebyshev polynomials. Some others recurrences for P_{n,m,p}(x) and its coefficients are also obtained. At the end a formula for coefficients of Chebyshev polynomials of the second kind is derived.

math.CV↗

arXiv2008

An Enumerative Function

We define an enumerative function F(n,k,P,m) which is a generalization of binomial coefficients. Special cases of this function are also power function, factorials, rising factorials and falling factorials. The first section of the paper is an introduction. In the second section we derive an explicit formula for F. From the expression for the power function we obtain a number theory result. Then we derive a formula which shows that the case of arbitrary m may be reduced to the case m=0. This formula extends Vandermonde convolution. In the second section we describe F by the series of recurrence relations with respect to each of arguments k, n, and P. As a special case of the first recurrence relation we state a binomial identity. As a consequence of the second recurrence relation we obtain relation for coefficients of Chebyshev polynomial of both kind. This means that these polynomials might be defined in pure combinatorial way.

math.CO↗

arXiv2007

Counting on rectangular areas

In the first section of this paper we prove a theorem for the number of columns of a rectangular area that are identical to the given one. In the next section we apply this theorem to derive several combinatorial identities by counting specified subsets of a finite set.

math.CO↗