Abstract

In the short note, by virtue of several formulas and identities for special values of the Bell polynomials of the second kind, the authors provide an alternative proof of a closed formula for central factorial numbers of the second kind. Moreover, the authors pose two open problems on closed form of a special Bell polynomials of the second kind and on closed form of a finite sum involving falling factorials.

2010 Mathematics Subject Classification. Primary 11B83; Secondary 11B75, 33B10.

1. Introduction

In mathematics, a closed formula is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, four arithmetic operations, and elementary functions, but usually no limit.

The Bell polynomials of the second kind, denoted by for are defined ^{1, 2, 3, 4, 5, 6, 7} by

The central factorial numbers of the second kind for can be generated ^{8, 9, 10} by

(1.1)

The central factorial numbers of the first kind for can be generated ^{8, 9} by

(1.2)

In [ ⁸, Proposition 2.4, (xii)], the authors established the closed formula

(1.3)

In this short note, by virtue of several formulas and identities for special values of the Bell polynomials of the second kind , we will provide an alternative proof of the closed formula (1.3).

2. Lemmas

For alternatively proving the closed formula (1.3), we need the following lemmas.

Lemma 2.1 ( ^{1, 2, 11, 12, 13, 14}). The Faà di Bruno formula can be described in terms of by

(2.1)

For , the Bell polynomials of the second kind satisfy the identity

(2.2)

is valid, where

Lemma 2.2 ^{10, 11, 12, 14, 15, 16, 17, 18}. For , the Bell polynomials of the second kind satisfy the closed formula

(2.3)

where is regarded as and denote the associate Stirling numbers of the second kind or weighted Stirling numbers which can be generated by

3. An Alternative Proof of the Closed Formula (1.3)

The closed formula (1.3) can be rewritten in terms of the associate Stirling numbers of the second kind or weighted Stirling numbers as follows.

Theorem 3.1. For the central factorial numbers of the second kind satisfy

(3.1)

Proof. The equation (1.1) implies that

(3.2)

Let . By virtue of the Faà di Bruno formula (2.1), we obtain

where the quantity

is called ^{2, 19} the falling factorial of x.

Since

for as , it follows that

where we used the identity (2.2) and the closed formula (2.3). Consequently, the formula (3.1) follows immediately. The proof of Theorem 3.1 is complete.

4. Two Open Problems

In this section, we pose two open problems.

For alternatively and similarly finding a closed formula for the central factorial numbers of the first kind generated in (1.2), we need to solve the following open problem.

Open Problem 4.1. Can one find a closed formula of the Bell polynomials of the second kind

for ?

For and , the falling factorial is defined by

(4.1)

In ^{12, 20} and closely related references therein, the following conclusions were obtained.

Theorem 4.1 ([ ²⁰, Theorem 3.1] and [ ¹², Section 1.4]). For , we have

(4.2)

Theorem 4.2 ([ ²⁰, Theorem 3.2] and [ ¹², Section 1.5]). For , we have

(4.3)

where the double factorial of negative odd integers for is defined by

The formula (4.3) has been applied in [ ²¹, Theorem 1.1].

Motivated by the above conclusions, one can naturally pose the following open problem.

Open Problem 4.2. For and , can one find an explicit, elementary, simple, and general formula of the type as in (4.2) and (4.3) for the finite sum

In particular, how about special cases for in (4.4)?

References