| ||Computer Modeling in Engineering & Sciences || |
Partial Bell Polynomials, Falling and Rising Factorials, Stirling Numbers, and Combinatorial Identities
Siqintuya Jin1, Bai-Ni Guo2,* and Qi Feng3,*
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: firstname.lastname@example.org; Feng Qi. Email: email@example.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 . See TheoremA on page 134 in . The partial Bell polynomials satisfy the identity
for . See page 135 in .
The double factorial of negative odd integers −(2k+1) is defined by
The falling factorial and the rising factorial (z)n for and can be defined by
respectively. It is easy to verify that
See page 167 in  and related texts in the paper .
The Stirling numbers of the first kind s(n, k) for can be analytically generated (see page 51 in ) by
and can be explicitly computed (see Corollary 2.3 in ) by
for . The Stirling numbers of the second kind S(n, k) for can be analytically generated (see page 51 in ) by
and can be explicitly computed (see TheoremA on page 204 in ) 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 for is defined in  by
in terms of the falling factorial , which is defined by (3), and the classical Euler’s gamma function , which can be defined (see Chapter3 in ) by
On page 206 in  and on page 165 in , there are two relations
The falling factorial and the rising factorial (z)n can be represented by
for and .
In this paper, we will collect, discuss, and find out several connections, equivalences, closed-form formulas, and combinatorial identities concerning partial Bell polynomials , falling factorials , rising factorials (z)n, extended binomial coefficients , and the Stirling numbers of the first and second kinds s(n, k) and S(n, k).
Among the Stirling numbers of the first and second kinds s(n, k) and S(n, k), the falling factorial , and extended binomial coefficient , there are the following beautiful equivalences.
Theorem 2.1. For and , we have
Proof. In Remark 3.1 of , the formula
for and was concluded. An equivalence of the formula (11) is
for and , which was proved in Theorems 2.1 and 4.1 of .
The formulas (11) and (12) can be rewritten respectively as
for and , as done in Remark 7 of .
Considering the formulas (6), (7), and (8) in the formulas (12) or (13), we can derive
for and .
Combining (12), (13), and (14) results in
for and . The equivalences in (9) and (10) are thus proved. The proof of Theorem 2.1 is complete.
3 Simpler Closed-Form Formulas
When taking in (9) and (10) respectively, we can derive several simpler closed-form combinatorial identities.
Theorem 3.1. For , we have
Proof. By the definition (1), we can easily deduce that, for ,
where we used the relation (15). See also pages 167–168 in . The identity (16), which recovers the first one in (28) below, is thus proved.
In Theorem 5.1 of  and in Section 3 of , the formula
was established for , where we assumed and for . Making use of the identity (2), we can derive from the formula (22) that
for . The identity (17) is thus verified.
In the proof of Theorem 3.2 in , it was obtained that
for . The identity (18) is thus proved.
Replacing by in (14) and utilizing (5) give
for and . Taking in the formula (25) and making use of the identity
in Section 1.5 of  and in Theorem 1.2 of , we derive
for . The identity (18) is proved again.
Employing the relation (25) and using the identities
in Sections 1.3 and 1.9 of  and in Lemma 6 of , we acquire
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 .
Theorem 3.2. For and , we have
Proof. Combining (12) and (14) yields
Accordingly, similar to arguments in Lemma 2.2 of , we acquire
for , where we used the relation
which is a special case x = 0 and of the identity
in the formula (1.48) on pages 27–28 of , 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 . Further applying relations in (15), we conclude those relations in (29).
Replacing by in (29), using the identities (4), (5), and (2) in sequence, and simplifying lead to (30). The proof of Theorem 3.2 is thus complete.
Remark 3.2. The last equality in (29) can be rewritten as
Theorem 12.1 on page 171 of  reads that, if and ak are a collection of constants independent of n, then
Applying Theorem 12.1 on page 171 in  to the second equality in (33), we find
Considering the explicit formula (6) and utilizing (31) and (32), we arrive at
for . Therefore, we obtain
which is a recovery of the well-known relation
in the equation (1.27) on page 19 of .
4 Several Combinatorial Identities
In items (3.163) and (3.164) on pages 91–92 of , we find two identities
Lemma 2.2 in  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 , the identities
and the identity (34) are valid.
Proof. For the case , these identities follow from Theorem 3.1, the equivalence (10), and simplifying.
For the case , making use of the relation 8 and utilizing the explicit formula (6), we acquire
for all and , where 00 was regarded as 1. Therefore, it is clear that
The proof of Theorem 4.1 is complete.
Remark 4.1. The identity (35) can be simplified as
The identities (36), (37), and (39) in Theorem 4.1 are probably new.
Theorem 4.2. For , we have
Proof. From (29), we conclude that
for and .
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
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 , on page 5 in , and in Theorem 4.2 of , there is the equality
for . Hence, we obtain
for , where we assumed 00 = 1 and used (7), (31), and (32). Hence, we acquire (44) and
Combining the last one with the relation
which is obtained by applying to the second equality in (29), yields the identity (43) again.
Substituting (24) into (46) leads to
which is a recovery of the formula (44).
For , let sk and Sk be two sequences independent of n such that . Theorem 4.4 on page 528 in  reads that
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  and in Remark 5.3 of .
Remark 4.3. The identity (34) established in Lemma 2.2 of  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 , applying the inversion theorem expressed by (47), we obtained
5 Several Problems and Numerical Demonstrations
Can one find out simpler closed-form formulas like those in Theorem 3.1 for the quantities
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 and list their values as
If fixing k = 4, 5 and n = 7, 8 and regarding as a real variable on the interval [ −9, 9], then the graphs plotted by the Wolfram Mathematica 12 are showed in Figs. 1 and 2.
Figure 1: The graphs of and for
Figure 2: The graphs of and for
In this paper, we collected, discussed, and found out significant connections, equivalences, closed-form formulas, and combinatorial identities concerning partial Bell polynomials , falling factorials , rising factorials (z)n, extended binomial coefficients , and the Stirling numbers of the first and second kinds s(n, k) and S(n, k). These results are new, interesting, important, useful, and applicable in combinatorial number theory and other areas, as done in the papers [22–27] and closely related references therein.
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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