Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2022.019941

ARTICLE

Partial Bell Polynomials, Falling and Rising Factorials, Stirling Numbers, and Combinatorial Identities

1College of Mathematics and Physics, Inner Mongolia Minzu University, Tongliao, 028043, China

2School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, 454010, China

3School of Mathematical Sciences, Tiangong University, Tianjin, 300387, China

*Corresponding Authors: Bai-Ni Guo. Email: bai.ni.guo@gmail.com; Feng Qi. Email: qifeng618@gmail.com

Received: 25 October 2021; Accepted: 24 January 2022

Abstract: In the paper, the authors collect, discuss, and find out several connections, equivalences, closed-form formulas, and combinatorial identities concerning partial Bell polynomials, falling factorials, rising factorials, extended binomial coefficients, and the Stirling numbers of the first and second kinds. These results are new, interesting, important, useful, and applicable in combinatorial number theory.

Keywords: Connection; equivalence; closed-form formula; combinatorial identity; partial Bell polynomial; falling factorial; rising factorial; binomial coefficient; Stirling number of the first kind; Stirling number of the second kind; problem

In this paper, we use the notation

The partial Bell polynomials, also known as the Bell polynomials of the second kind, in combinatorics can be denoted and defined by

for

for

The double factorial of negative odd integers −(2k+1) is defined by

The falling factorial

and

respectively. It is easy to verify that

and

See page 167 in [2] and related texts in the paper [3].

The Stirling numbers of the first kind s(n, k) for

and can be explicitly computed (see Corollary 2.3 in [4]) by

for

and can be explicitly computed (see TheoremA on page 204 in [1]) by

For more information on the Stirling numbers of the first and second kinds s(n, k) and S(n, k), please refer to the papers [5,6] and the monographs [7,8].

The extended binomial coefficient

in terms of the falling factorial

On page 206 in [1] and on page 165 in [7], there are two relations

and

The falling factorial

for

In this paper, we will collect, discuss, and find out several connections, equivalences, closed-form formulas, and combinatorial identities concerning partial Bell polynomials

Among the Stirling numbers of the first and second kinds s(n, k) and S(n, k), the falling factorial

Theorem 2.1. For

and

Proof. In Remark 3.1 of [11], the formula

for

for

The formulas (11) and (12) can be rewritten respectively as

and

for

Considering the formulas (6), (7), and (8) in the formulas (12) or (13), we can derive

for

Combining (12), (13), and (14) results in

for

3 Simpler Closed-Form Formulas

When taking

Theorem 3.1. For

and

Proof. By the definition (1), we can easily deduce that, for

where we used the relation (15). See also pages 167–168 in [2]. The identity (16), which recovers the first one in (28) below, is thus proved.

In Theorem 5.1 of [13] and in Section 3 of [14], the formula

was established for

for

In the proof of Theorem 3.2 in [2], it was obtained that

for

Replacing

for

in Section 1.5 of [15] and in Theorem 1.2 of [16], we derive

for

Employing the relation (25) and using the identities

and

in Sections 1.3 and 1.9 of [15] and in Lemma 6 of [17], we acquire

and

The identities (19) and (20) are thus derived. The proof of Theorem 3.1 is complete.

Remark 3.1. We can regard those identities from (17) to (20) in Theorem 3.1 as generalizations of the orthogonality relations

listed on page 171 in [7].

Theorem 3.2. For

and

Proof. Combining (12) and (14) yields

Accordingly, similar to arguments in Lemma 2.2 of [18], we acquire

for

which is a special case x = 0 and

in the formula (1.48) on pages 27–28 of [8], we used the relation (7) twice, and we used the equality

which is a special case r = 1 and p = m of the identity

in the formula (X.5) on page 132 of [8]. Further applying relations in (15), we conclude those relations in (29).

Replacing

Remark 3.2. The last equality in (29) can be rewritten as

Theorem 12.1 on page 171 of [7] reads that, if

Applying Theorem 12.1 on page 171 in [7] to the second equality in (33), we find

Considering the explicit formula (6) and utilizing (31) and (32), we arrive at

for

which is a recovery of the well-known relation

in the equation (1.27) on page 19 of [10].

4 Several Combinatorial Identities

In items (3.163) and (3.164) on pages 91–92 of [8], we find two identities

and

Lemma 2.2 in [18] reads that

We can also find some discussions and alternative proofs for these three identities at the sites https://math.stackexchange.com/q/1098257 and https://math.stackexchange.com/q/4235171.

Theorem 4.1. For

and the identity (34) are valid.

Proof. For the case

For the case

for all

The proof of Theorem 4.1 is complete.

Remark 4.1. The identity (35) can be simplified as

for

The identities (36), (37), and (39) in Theorem 4.1 are probably new.

Theorem 4.2. For

and

Proof. From (29), we conclude that

for

Substituting (21) into (46) gives

The identity (39) is thus proved.

Substituting (23) into (46) results in

The identity (40) is verified.

Utilizing the relations (2) and (5), we can reformulate the identity (26) as

Substituting this equality into (46) arrives at

The formula (41) follows.

Utilizing the relations (2) and (5), we can reformulate the identity (27) as

Substituting this equality into (46) and employing (31) reveal

and

The fourth equality (42) in Theorem 4.2 is thus proved.

Employing (31), we can rearrange the identity (43) as

The equality (43) is deduced.

In Theorem 3.2 of [2], on page 5 in [15], and in Theorem 4.2 of [19], there is the equality

for

for

Combining the last one with the relation

which is obtained by applying

Substituting (24) into (46) leads to

which is a recovery of the formula (44).

For

Letting Sn = ( −1)n22n and sn = ( −1)n2n(n+1), considering (40), applying the inversion theorem expressed by (47), and simplifying figure out the identity (45).

Remark 4.2. The formula (44) is also alternatively established in the proof of Theorem 3.2 in [18] and in Remark 5.3 of [21].

Remark 4.3. The identity (34) established in Lemma 2.2 of [18] and recovered in Theorem 4.1, the identity (36) in Theorem 4.1, and the formula (43) in Theorem 4.2 were announced at https://math.stackexchange.com/a/4268339 and https://math.stackexchange.com/a/4268341 online.

Remark 4.4. In Remark 3.4 of [18], applying the inversion theorem expressed by (47), we obtained

and

5 Several Problems and Numerical Demonstrations

Can one find out simpler closed-form formulas like those in Theorem 3.1 for the quantities

for

By the methods used in this paper, can one find out more combinatorial identities like those in Theorems (4.1) and 4.2?

In general, can one find explicit and closed-form formulas of the quantities

for some special values

For better understanding the above problems, by the Wolfram Mathematica 12, we numerically compute the quantity

for

If fixing k = 4, 5 and n = 7, 8 and regarding

In this paper, we collected, discussed, and found out significant connections, equivalences, closed-form formulas, and combinatorial identities concerning partial Bell polynomials

Acknowledgement: The authors thank anonymous referees for their careful corrections, helpful suggestions, and valuable comments on the original version of this paper.

Funding Statement: This work was supported in part by the National Natural Science Foundation of China (Grant No.12061033), by the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region (Grants No. NJZY20119), and by the Natural Science Foundation of Inner Mongolia (Grant No. 2019MS01007), China.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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