Simplifying coefficients in differential equations associated with higher order Bernoulli numbers of the second kind

Feng Qi; Da-Wei Niu; Bai-Ni Guo; Feng Qi; Da-Wei Niu; Bai-Ni Guo

doi:10.3934/math.2019.2.170

AIMS Mathematics

2019, Volume 4, Issue 2: 170-175. doi: 10.3934/math.2019.2.170

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Research article

Simplifying coefficients in differential equations associated with higher order Bernoulli numbers of the second kind

1.
School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China
2.
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China
3.
Department of Mathematics, East China Normal University, Shanghai 200241, China
4.
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454010, China

Received: 21 December 2018 Accepted: 14 February 2019 Published: 19 February 2019
MSC : Primary 34A05; Secondary 11A25, 11B68, 11B73, 11B83

In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.

Keywords:

Citation: Feng Qi, Da-Wei Niu, Bai-Ni Guo. Simplifying coefficients in differential equations associated with higher order Bernoulli numbers of the second kind[J]. AIMS Mathematics, 2019, 4(2): 170-175. doi: 10.3934/math.2019.2.170

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Abstract

1. Motivations

In [2,Theorem 1], it was inductively and recursively established that the family of differential equations

$\begin{equation} (-1)^n(r)_nF(t) = [\ln(1+t)]^n\sum\limits_{i = 1}^{n}a_i(n)(1+t)^iF^{(i)}(t), \quad n\in\mathbb{N} \end{equation}$

(1)

has a solution

$\begin{equation} F(t) = F(t,r) = \biggl[\frac{1}{\ln(1+t)}\biggr]^r, \quad r\in\mathbb{N}, \end{equation}$

(2)

where $a_1(n) = 1$ and

$\begin{equation} a_i(n) = \sum\limits_{k_{i-1} = 0}^{n-i}\sum\limits_{k_{i-2} = 0}^{n-i-k_{i-1}}\dotsm\sum\limits_{k_1 = 0}^{n-i-k_{i-1}-\dotsm-k_2} \prod\limits_{\ell = 2}^{i}\ell^{k_{\ell-1}}, \quad 2\le i\le n. \end{equation}$

(3)

Let

$\begin{equation*} (x)_n = \prod\limits_{\ell = 0}^{n-1}(x+\ell) = \begin{cases} x(x+1)(x+2)\dotsm(x+n-1), & n\ge1\\ 1, & n = 0 \end{cases} \end{equation*}$

and

$\begin{equation*} \langle x\rangle_n = \prod\limits_{\ell = 0}^{n-1}(x-\ell) = \begin{cases} x(x-1)(x-2)\dotsm(x-n+1), & n\ge1\\ 1,& n = 0 \end{cases} \end{equation*}$

be the rising and falling factorials of $x\in\mathbb{R}$ for $n\in\{0\}\cup\mathbb{N}$ . Let $b_n^{(r)}$ for $r\in\mathbb{N}$ , generated by

$\begin{equation*} \biggl[\frac{t}{\ln(1+t)}\biggr]^r = \sum\limits_{n = 0}^{\infty}b_n^{(r)}\frac{t^n}{n!}, \end{equation*}$

stand for the Bernoulli numbers of the second kind with order $r$ . Theorem 2 in ^[2] reads that, if $n = 0, 1, 2, \dotsc$ and $N = 1, 2, 3, \dotsc$ , then

1. for $0\le n < N+r$ ,

$\begin{equation*} (-1)^N(r)_Nb_n^{(r+N)} = \sum\limits_{i = 0}^{\min\{N-1,n\}} \sum\limits_{\ell = \max\{i,n-r+1\}}^{n}\binom{N-i}{\ell-i} \langle n-\ell-r\rangle_{N-i}\langle n\rangle_\ell a_{N-i}(N)b_{n-\ell}^{(r)}; \end{equation*}$

2. for $n\ge N+r$ ,

$\begin{equation*} (-1)^N(r)_Nb_n^{(r+N)} = \Biggl(\sum\limits_{i = 0}^{\min\{n,N-1\}}\sum\limits_{\ell = \max\{i,n-r+1\}}^{n} +\sum\limits_{i = 0}^{N-1}\sum\limits_{\ell = i}^{n-N-r+i}\Biggr)\binom{N-i}{\ell-i} \langle n-\ell-r\rangle_{N-i}\langle n\rangle_\ell a_{N-i}(N)b_{n-\ell}^{(r)}. \end{equation*}$

It is not difficult to see that the expression (3) of the quantity $a_i(n)$ is too complicated to be computed by hand and computer software. Can one find a simple, meaningful, and significant expression for the quantity $a_i(n)$ in (3)?

2. Lemmas

For answering the above question and proving our main results, we need the following lemmas.

Lemma 1. ([1,p. 134,Theorem A] and [1,p. 139,Theorem C]) For $n\ge k\ge0$ , the Bell polynomials of the second kind, or say, partial Bell polynomials, denoted by $\text{B}_{n, k}(x_1, x_2, \dotsc, x_{n-k+1})$ , are defined by

$\begin{equation*} \text{B}_{n,k}(x_1,x_2,\dotsc,x_{n-k+1}) = \sum\limits_{\substack{1\le i\le n-k+1,\, \ell_i\in\{0\}\cup\mathbb{N}\\ \sum\limits_{i = 1}^{n-k+1}i\ell_i = n,\, \sum\limits_{i = 1}^{n-k+1}\ell_i = k}}\frac{n!}{\prod\limits_{i = 1}^{n-k+1}\ell_i!} \prod\limits_{i = 1}^{n-k+1}\Bigl(\frac{x_i}{i!}\Bigr)^{\ell_i}. \end{equation*}$

The Faà di Bruno formula can be described in terms of the Bell polynomials of the second kind $\text{B}_{n, k}$ by

$\begin{equation} \frac{{\rm{d}}^n}{{\rm{d}} t^n}f\circ h(t) = \sum\limits_{k = 0}^nf^{(k)}(h(t)) \text{B}_{n,k}\bigl(h'(t),h''(t),\dotsc,h^{(n-k+1)}(t)\bigr). \end{equation}$

(4)

Lemma 2. [1,p. 135] For $n\ge k\ge0$ , we have

$\begin{equation} \text{B}_{n,k}\bigl(abx_1,ab^2x_2,\dotsc,ab^{n-k+1}x_{n-k+1}\bigr) = a^kb^n \text{B}_{n,k}(x_1,x_2,\dotsc,x_{n-k+1}) \end{equation}$

(5)

and

$\begin{equation} \text{B}_{n,k}(0!,1!,2!,\dotsc,(n-k)!) = (-1)^{n-k}s(n,k), \end{equation}$

(6)

where $a$ and $b$ are any complex numbers and $s(n, k)$ for $n\ge k\ge0$ , which can be generated by

$\begin{equation*} \frac{[\ln(1+x)]^k}{k!} = \sum\limits_{n = k}^\infty s(n,k)\frac{x^n}{n!},\quad |x| \lt 1, \end{equation*}$

stand for the Stirling numbers of the first kind.

Lemma 3. [26,p. 171,Theorem 12.1] If $b_\alpha$ and $a_k$ are a collection of constants independent of $n$ , then

$\begin{equation*} a_n = \sum\limits_{\alpha = 0}^{n}S(n,\alpha)b_\alpha \quad if \;and \;only\; if \quad b_n = \sum\limits_{k = 0}^{n}s(n,k)a_k, \end{equation*}$

where $S(n, k)$ for $n\ge k\ge0$ , which can be generated by

$\begin{equation*} \frac{(e^x-1)^k}{k!} = \sum\limits_{n = k}^\infty S(n,k)\frac{x^n}{n!}, \end{equation*}$

stand for the Stirling numbers of the second kind.

3. Main results and their proofs

Now we are in a position to answer the above question and to state and prove our main results.

Theorem 1. For $n\ge0$ and $r\in\mathbb{R}$ , the function $F(t) = F(t, r)$ defined by (2) satisfies

$\begin{equation} F^{(n)}(t) = \biggl(\frac{1}{1+t}\biggr)^{n} \Biggl[\sum\limits_{k = 0}^{n}s(n,k)\frac{\langle-r\rangle_k}{[\ln(1+t)]^k}\Biggr]F(t) \quad and \quad \sum\limits_{k = 0}^{n}S(n,k)(1+t)^kF^{(k)}(t) = \frac{\langle-r\rangle_n}{[\ln(1+t)]^n}F(t). \end{equation}$

(7)

Proof. Let $u = u(t) = \ln(1+t)$ and $r\in\mathbb{R}$ . Then, by virtue of the Faà di Bruno formula (4) and the identities (5) and (6) in sequence,

$\begin{align*} F^{(n)}(t)& = \sum\limits_{k = 0}^{n}\bigl(u^{-r}\bigr)^{(k)} \text{B}_{n,k}\biggl(\frac{0!}{1+t}, -\frac{1!}{(1+t)^2},\dotsc,(-1)^{n-k}\frac{(n-k)!}{(1+t)^{n-k+1}}\biggr)\\ & = \sum\limits_{k = 0}^{n}\frac{\langle-r\rangle_k}{u^{r+k}} \biggl(\frac{1}{1+t}\biggr)^{n} (-1)^{n+k} \text{B}_{n,k}(0!,1!,\dotsc,(n-k)!)\\ & = \sum\limits_{k = 0}^{n}\frac{\langle-r\rangle_k}{[\ln(1+t)]^{r+k}} \biggl(\frac{1}{1+t}\biggr)^{n} (-1)^{n+k} (-1)^{n-k}s(n,k) \end{align*}$

for $n\ge0$ . Thus, the first identity in (7) is proved.

Applying Lemma 3 to the first equality in (7) leads to

$\begin{equation*} \frac{\langle-r\rangle_n}{[\ln(1+t)]^n}F(t) = \sum\limits_{k = 0}^{n}S(n,k)(1+t)^kF^{(k)}(t) \end{equation*}$

which can be rewritten as the second equality in (7). The required proof is complete.

Corollary 1. Comparing (1) with two equalities in (7) reveals that

$\begin{equation} a_i(n) = S(n,i), \quad n\ge i\ge0. \end{equation}$

(8)

This implies that the second identity in (7) is more meaningful, more significant, more computable than (1).

4. Remarks

In this section, we give several remarks and some explanation about our main results.

Remark 1. Theorem 1 extends the range of $r$ from $\mathbb{N}$ to $\mathbb{R}$ .

Remark 2. By virtue of the expression (8), all the above mentioned results in the paper ^[2] can be reformulated simpler, more meaningfully, and more significantly. For the sake of saving the space and shortening the length of this paper, we do not rewrite them in details here.

Remark 3. Currently we can see that the method used in this paper is simpler, shorter, nicer, more meaningful, and more significant than the inductive and recursive method used in ^[2] and closely related references therein.

Remark 4. In the papers ^[5,8,24,25], there are some new results about the Bernoulli numbers of the second kind.

Remark 5. In the papers ^{[3,4,6,7,9,10,11,12,13,14,15,16,17,18,20,21,22,23,25,27]}, there are similar ideas, methods, techniques, and purposes to this paper.

Remark 6. This paper is a slightly revised version of the preprint ^[19].

5. Acknowledgements

The authors are thankful to Professors Dmitry V. Dolgy (Far Eastern State University, Russia) and Taekyun Kim (Kwangwoon University, South Korea) for their supplying an electronic version of the paper ^[2] through e-mail on 3 August 2017. The authors are grateful to anonymous referees for their careful corrections to and valuable comments on the original version of this paper.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	L. Comtet, Advanced Combinatorics: The art of finite and infinite Expansions, Revised and Enlarged Edition, Dordrecht and Boston: D. Reidel Publishing Co., 1974.
[2]	D. V. Dolgy, L. C. Jang, D. S. Kim, et al. Differential equations associated with higher-order Bernoulli numbers of the second kind revisited, J. Anal. Appl. 14 (2016), 107-121.
[3]	B. N. Guo, F. Qi, Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math., 272 (2014), 251-257. doi: 10.1016/j.cam.2014.05.018
[4]	B. N. Guo, F. Qi, Some identities and an explicit formula for Bernoulli and Stirling numbers, J. Comput. Appl. Math., 255 (2014), 568-579. doi: 10.1016/j.cam.2013.06.020
[5]	F. Qi, A new formula for the Bernoulli numbers of the second kind in terms of the Stirling numbers of the first kind, Publ. I. Math., 100 (2016), 243-249. doi: 10.2298/PIM150501028Q
[6]	F. Qi, A simple form for coefficients in a family of nonlinear ordinary differential equations, Adv. Appl. Math. Sci., 17 (2018), 555-561.
[7]	F. Qi, A simple form for coefficients in a family of ordinary differential equations related to the generating function of the Legendre polynomials, Adv. Appl. Math. Sci., 17 (2018), 693-700.
[8]	F. Qi, Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind, Filomat, 28 (2014), 319-327. doi: 10.2298/FIL1402319O
[9]	F. Qi, Notes on several families of differential equations related to the generating function for the Bernoulli numbers of the second kind, Turkish J. Anal. Number Theory, 6 (2018), 40-42. doi: 10.12691/tjant-6-2-1
[10]	F. Qi, Simple forms for coefficients in two families of ordinary differential equations, Glob. J. Math. Anal., 6 (2018), 7-9.
[11]	F. Qi, B. N. Guo, Simplification of coefficients in two families of nonlinear ordinary differential equations, Turkish J. Anal. Number Theory, 6 (2018), 116-119. doi: 10.12691/tjant-6-4-2
[12]	F. Qi, Simplifying coefficients in a family of nonlinear ordinary differential equations, Acta Commentat. Univ. Tartuensis Math., 22 (2018), 293-297.
[13]	F. Qi, Simplifying coefficients in a family of ordinary differential equations related to the generating function of the Laguerre polynomials, Appl. Appl. Math., 13 (2018), 750-755.
[14]	F. Qi, Simplifying coefficients in differential equations related to generating functions of reverse Bessel and partially degenerate Bell polynomials, Bol. Soc. Parana. Mat., 39 (2021), in press.
[15]	F. Qi, B. N. Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Appl. Anal. Discrete Math., 12 (2018), 153-165. doi: 10.2298/AADM170405004Q
[16]	F. Qi, B. N. Guo, Explicit formulas and recurrence relations for higher order Eulerian polynomials, Indagat. Math., 28 (2017), 884-891. doi: 10.1016/j.indag.2017.06.010
[17]	F. Qi, D. Lim, B. N. Guo, Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations, RACSAM, 113 (2019), 1-9. doi: 10.1007/s13398-017-0427-2
[18]	F. Qi, D. Lim, B. N. Guo, Some identities related to Eulerian polynomials and involving the Stirling numbers, Appl. Anal. Discrete Math., 12 (2018), 467-480. doi: 10.2298/AADM171008014Q
[19]	F. Qi, D. W. Niu, B. N. Guo, Simplifying coefficients in differential equations associated with higher order Bernoulli numbers of the second kind, Preprints, 2017. Available from: http://dx.doi.org/10.20944/preprints201708.0026.v1.
[20]	F. Qi, D. W. Niu, B. N. Guo, Some identities for a sequence of unnamed polynomials connected with the Bell polynomials, RACSAM, 112 (2018), in press. Available from: https://doi.org/10.1007/s13398-018-0494-z.
[21]	F. Qi, J. L. Wang, B. N. Guo, Notes on a family of inhomogeneous linear ordinary differential equations, Adv. Appl. Math. Sci., 17 (2018), 361-368.
[22]	F. Qi, J. L. Wang, B. N. Guo, Simplifying and finding ordinary differential equations in terms of the Stirling numbers, Korean J. Math., 26 (2018), 675-681.
[23]	F. Qi, J. L.Wang, B. N. Guo, Simplifying differential equations concerning degenerate Bernoulli and Euler numbers, Trans. A. Razmadze Math. Inst., 172 (2018), 90-94. doi: 10.1016/j.trmi.2017.08.001
[24]	F. Qi, X. J. Zhang, An integral representation, some inequalities, and complete monotonicity of the Bernoulli numbers of the second kind, B. Korean Math. Soc., 52 (2015), 987-998. doi: 10.4134/BKMS.2015.52.3.987
[25]	F. Qi, J. L. Zhao, Some properties of the Bernoulli numbers of the second kind and their generating function, B. Korean Math. Soc., 55 (2018), 1909-1920.
[26]	J. Quaintance, H. W. Gould, Combinatorial Identities for Stirling Numbers: The Unpublished Notes of H. W. Gould, Singapore: World Scientific Publishing Co. Pte. Ltd., 2016.
[27]	J. L. Zhao, J. L. Wang, F. Qi, Derivative polynomials of a function related to the Apostol-Euler and Frobenius-Euler numbers, J. Nonlinear Sci. Appl., 10 (2017), 1345-1349. doi: 10.22436/jnsa.010.04.06

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AIMS Mathematics

Simplifying coefficients in differential equations associated with higher order Bernoulli numbers of the second kind

Related Papers:

Abstract

1. Motivations

2. Lemmas

3. Main results and their proofs

4. Remarks

5. Acknowledgements

Conflict of interest

References

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AIMS Mathematics

Simplifying coefficients in differential equations associated with higher order Bernoulli numbers of the second kind

Related Papers:

Abstract

1. Motivations

2. Lemmas

3. Main results and their proofs

4. Remarks

5. Acknowledgements

Conflict of interest

References

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