
Citation: Vladimir Kaminskii, Elena Kossovich, Svetlana Epshtein, Liudmila Obvintseva, Valeria Nesterova. Activity of coals of different rank to ozone[J]. AIMS Energy, 2017, 5(6): 960-973. doi: 10.3934/energy.2017.6.960
[1] | Haiquan Wang, Hans-Dietrich Haasis, Menghao Su, Jianhua Wei, Xiaobin Xu, Shengjun Wen, Juntao Li, Wenxuan Yue . Improved artificial bee colony algorithm for air freight station scheduling. Mathematical Biosciences and Engineering, 2022, 19(12): 13007-13027. doi: 10.3934/mbe.2022607 |
[2] | Yutao Lai, Hongyan Chen, Fangqing Gu . A multitask optimization algorithm based on elite individual transfer. Mathematical Biosciences and Engineering, 2023, 20(5): 8261-8278. doi: 10.3934/mbe.2023360 |
[3] | Rami AL-HAJJ, Mohamad M. Fouad, Mustafa Zeki . Evolutionary optimization framework to train multilayer perceptrons for engineering applications. Mathematical Biosciences and Engineering, 2024, 21(2): 2970-2990. doi: 10.3934/mbe.2024132 |
[4] | Shijing Ma, Yunhe Wang, Shouwei Zhang . Modified chemical reaction optimization and its application in engineering problems. Mathematical Biosciences and Engineering, 2021, 18(6): 7143-7160. doi: 10.3934/mbe.2021354 |
[5] | Qing Wu, Chunjiang Zhang, Mengya Zhang, Fajun Yang, Liang Gao . A modified comprehensive learning particle swarm optimizer and its application in cylindricity error evaluation problem. Mathematical Biosciences and Engineering, 2019, 16(3): 1190-1209. doi: 10.3934/mbe.2019057 |
[6] | Zhenao Yu, Peng Duan, Leilei Meng, Yuyan Han, Fan Ye . Multi-objective path planning for mobile robot with an improved artificial bee colony algorithm. Mathematical Biosciences and Engineering, 2023, 20(2): 2501-2529. doi: 10.3934/mbe.2023117 |
[7] | Junhua Liu, Wei Zhang, Mengnan Tian, Hong Ji, Baobao Liu . A double association-based evolutionary algorithm for many-objective optimization. Mathematical Biosciences and Engineering, 2023, 20(9): 17324-17355. doi: 10.3934/mbe.2023771 |
[8] | Wenqiang Zhang, Chen Li, Mitsuo Gen, Weidong Yang, Zhongwei Zhang, Guohui Zhang . Multiobjective particle swarm optimization with direction search and differential evolution for distributed flow-shop scheduling problem. Mathematical Biosciences and Engineering, 2022, 19(9): 8833-8865. doi: 10.3934/mbe.2022410 |
[9] | Yijie Zhang, Yuhang Cai . Adaptive dynamic self-learning grey wolf optimization algorithm for solving global optimization problems and engineering problems. Mathematical Biosciences and Engineering, 2024, 21(3): 3910-3943. doi: 10.3934/mbe.2024174 |
[10] | Jian Si, Xiaoguang Bao . A novel parallel ant colony optimization algorithm for mobile robot path planning. Mathematical Biosciences and Engineering, 2024, 21(2): 2568-2586. doi: 10.3934/mbe.2024113 |
Special polynomials play a significantly important role in the development of several branches of mathematics, engineering, and physics by providing us with useful identities and properties. The study of special polynomials provides many useful identities, their relations, and representations associated with special numbers and polynomials. One of the powerful tools in this study is to investigate their generating functions [1,2] and connections[3,4,5,6] using the umbral calculus [7]. Furthermore, to better understand generating functions in special polynomials, the degenerate type of special polynomials has been extensively studied in many areas such as probability theory, fuzzy theory, connection problems, and other combinatorial theories in recent years by many mathematicians [8,9,10,11]. Since the introduction of degenerate versions of special polynomials and numbers by Carlitz [12], many researchers have been interested in the relationships between them. In addition, the degenerate version of umbral calculus, called λ-umbral calculus, plays a very powerful role in studying the relationships between degenerate versions of special polynomials and numbers. Recently, the Daehee polynomials and numbers were originally introduced as a new type of special polynomials by Kim and Kim [13] and thereafter their related properties and relationships with other polynomials have been extensively studied.
In this study, we derive the formulas expressing degenerate higher-order Daehee polynomials in terms of the degenerate versions of other special polynomials by making use of λ-umbral calculus. These formulas provide the degenerate Daehee polynomials by taking r=1 and the Daehee polynomials by letting λ→0. We first review the λ-analogue of umbral calculus: a class of λ-linear functionals on the polynomials, λ-differential operators based on the family of λ-linear functionals, and also λ-Sheffer sequences. See [14] and the references therein for more details on these contents.
The rest of this section briefly recalls some necessary notations and definitions that are needed throughout this paper. Throughout this paper, we assume that λ∈R∖{0} for simplicity.
The degenerate exponential function exλ(t) is defined by
exλ(t):=(1+λt)xλ=∞∑n=0(x)n,λtnn!,eλ(t):=e1λ(t)=∞∑n=0(1)n,λtnn!, (see [10, 13, 15, 16]), | (1.1) |
where (x)n,λ is a λ-analogue of the falling factorial sequence which is given by
(x)n,λ=x(x−λ)⋯(x−(n−1)λ) for n≥1 and (x)0,λ=1, (see [14, 17]). | (1.2) |
Also, the degenerate logarithm function is given by logλ(t):=1λ(tλ−1), which is the compositional inverse of eλ(t), i.e.,
logλ(eλ(t))=eλ(logλ(t))=t. |
In this study, we consider the degenerate higher-order Daehee polynomials D(r)n,λ(x) which are given by the generating function to be
(logλ(1+t)t)r(1+t)x=∞∑n=0D(r)n,λ(x)tnn!,r∈N, (see [10, 15, 16]). | (1.3) |
Especially, we call Dn,λ(x):=D(1)n,λ(x) the degenerate Daehee polynomials when r=1 and Dn,λ:=Dn,λ(0) the degenerate Daehee numbers when x=0.
The degenerate Stirling numbers of the first kind S1,λ(n,m) and the second kind S2,λ(n,m) are respectively given by
1m!(logλ(1+t))m=∞∑n=mS1,λ(n,m)tnn!,(m≥0), (see [9, 18]) | (1.4) |
and
1m!(eλ(t)−1)m=∞∑n=mS2,λ(n,m)tnn!,(m≥0), (see [9, 18]). | (1.5) |
Note that the falling factorial sequence (t)n is given by
(t)n={t(t−1)(t−2)⋯(t−(n−1)) for n≥1,(t)0=1 when n=0, (see [19]), |
which provides the relation with the λ-analogue of the falling factorial sequence such as
(t)n,λ=n∑m=0S2,λ(n,m)(t)m,(n≥0). |
The main contribution of this paper is to provide various representations of the degenerate higher-order Daehee polynomials and numbers using λ-umbral calculus in terms of other well-known special polynomials and numbers. In more detail, we derive formulas for the n-th order of degenerate Daehee polynomials with the degenerate falling factorial polynomials, the degenerate type 2 Bernoulli polynomials, the degenerate Bernoulli polynomials, the degenerate Euler polynomials, the degenerate Mittag-Leffer polynomials, the degenerate Bell polynomials, and the degenerate Frobenius-Euler polynomials (see Theorems 2.1–2.7) as well as their inversion formulas. Therefore, we see that this technique enables us to represent various well-known polynomials in terms of degenerate higher-order Daehee polynomials and vice versa as a classical connection problem. In addition, to confirm the formulas, we present computational results between the degenerate higher-order Daehee polynomials and the degenerate Bernoulli polynomials for fixed variables. Moreover, we investigate the pattern of the root distribution of the polynomials.
Now, we provide brief review of λ-umbral calculus: Let P be the algebra of polynomials in t over C, i.e.,
P=C[t]={∞∑n=0antn|an∈C with an=0 for all but finite number of n}. |
and let F be the algebra of formal power series in t over the field C of complex numbers
F={f(t)=∞∑n=0antnn!|an∈C}. |
Then, the λ-linear functional ⟨f(t)|⋅⟩λ on P for f(t)=∑∞n=0antnn!∈F is given by
⟨f(t)|(x)n,λ⟩λ=an,(n≥0), (see [14]), | (2.1) |
and it satisfies
⟨tk|(x)n,λ⟩λ=n!δn,k, (see [14]), | (2.2) |
where δn,k is the Kronecker delta.
Note that the order of the formal power series for a nontrivial f(t), o(f(t)), is represented by the smallest integer k for which ak does not vanish. Especially, we call f(t) a delta series when o(f(t))=1, and also we say f(t) an invertible series when o(f(t))=0, (see [1,7,14] for details).
For a non-negative integer order k, the λ-differential operator (tk)λ on P is defined by
(tk)λ(x)n,λ={(n)k(x)n−k,λ if 0≤k≤n,0, if k>n, (see [14, 20]). | (2.3) |
In general, for f(t)=∑∞k=0aktkk!∈F, the λ-differential operator (f(t))λ is satisfied with
(f(t))λ(x)n,λ=n∑k=0(nk)ak(x)n−k,λ. | (2.4) |
Or equivalently, one can express (f(t))λ as
(f(t))λ=∞∑k=0akk!(tk)λ. |
For a delta series f(t) and an invertible series g(t), i.e., o(f(t))=1 and o(g(t))=0, there exists a unique sequence pn,λ(x) of polynomials deg(pn,λ(x))=n satisfying the orthogonality condition
⟨g(t)(f(t))k|pn,λ(x)⟩λ=n!δn,k,(n,k≥0). | (2.5) |
Here, pn,λ(x) is called the λ-Sheffer sequence for (g(t),f(t)) denoted by pn,λ(x)∼(g(t),f(t))λ.
We recall that pn,λ(x)∼(g(t),f(t))λ if and only if
1g(ˉf(t))exλ(ˉf(t))=∞∑n=0pn,λ(x)tnn!,(see [7, 20]). | (2.6) |
Here ˉf(t) represents the compositional inverse of f(t), i.e., f(ˉf(t))=ˉf(f(t))=t.
For given a pair of λ-Sheffer sequences pn,λ(x)∼(g(t),f(t))λ and qn,λ(x)∼(h(t),ℓ(t))λ, we have the relation:
pn,λ(x)=n∑k=0μn,kqk,λ(x), | (2.7) |
where μn,k is obtained by
μn,k=1k!⟨h(ˉf(t))g(ˉf(t))(ℓ(ˉf(t)))k|(x)n,λ⟩λ. |
Likewise, if qn,λ(x) is expressed in terms of pn,λ(x) as
qn,λ(x)=n∑k=0νn,kpk,λ(x), | (2.8) |
then νn,k can be obtained by
νn,k=1k!⟨g(ˉℓ(t))h(ˉℓ(t))(f(ˉℓ(t)))k|(x)n,λ⟩λ. |
It is easily shown that for f(t),g(t)∈F and p(x)∈P,
⟨f(t)g(t)|p(x)⟩λ=⟨g(t)|(f(t))λp(x)⟩λ=⟨f(t)|(g(t))λp(x)⟩λ,(see [14]). |
We also note that from (x)n,λ∼(1,t)λ, any λ-Sheffer sequence pn,λ(x)∼(g(t),f(t))λ is represented by
pn,λ(x)=n∑k=01k!⟨1g(ˉf(t))(ˉf(t))k|(x)n,λ⟩λ(x)k,λ. | (2.9) |
Now, we want to present representations of the degenerate higher-order Daehee polynomials D(r)n,λ(x) by using the algebraic properties of λ-Sheffer sequences.
From (1.3), we have that ∑∞n=0D(r)n,λ(x)tnn!=(logλ(1+t)t)rexλ(logλ(1+t)), so that we consider f(t)=eλ(t)−1,ˉf(t)=logλ(1+t), and g(t)=eλ(t)−1t in the view of (2.6) to obtain
D(r)n,λ(x)∼((eλ(t)−1t)r,eλ(t)−1)λ. | (2.10) |
If we let pn,λ(x)=∑nℓ=0μℓD(r)ℓ,λ(x), then, by using (2.5) we have
⟨(eλ(t)−1t)r(eλ(t)−1)k|pn,λ(x)⟩λ=n∑ℓ=0μℓ⟨(eλ(t)−1t)r(eλ(t)−1)k|D(r)ℓ,λ(x)⟩λ=n∑ℓ=0μℓℓ!δk,ℓ=k!μk, |
which implies
μk=1k!⟨(eλ(t)−1t)r(eλ(t)−1)k|pn,λ(x)⟩λ. |
Thus, for pn,λ(x)∈P we have
pn,λ(x)=n∑k=01k!⟨(eλ(t)−1t)r(eλ(t)−1)k|pn,λ(x)⟩λD(r)k,λ(x). |
Then, the formula between D(r)n,λ(x) and (x)n,λ is obtained.
Theorem 2.1. For n∈N∪{0} and r∈N, we have
D(r)n,λ(x)=n∑k=0(n∑ℓ=k(nℓ)S1,λ(ℓ,k)D(r)n−ℓ,λ)(x)k,λ. |
Reversely, we have the inversion formula given by
(x)n,λ=n∑k=0(n∑ℓ=kr∑j=0(nℓ)(rj)(−1)r−jS2,λ(ℓ,k)(j)n+r−ℓ,λ(n−ℓ+r)r)D(r)k,λ(x). |
Proof. Let D(r)n,λ(x)=∑nk=0μn,k(x)k,λ. Then, by (1.4), (2.3), and (2.9), we can obtain
μn,k=1k!⟨(logλ(1+t)t)r(logλ(1+t))k|(x)n,λ⟩λ=⟨(logλ(1+t)t)r|(1k!(logλ(1+t))k)λ(x)n,λ⟩λ=1ℓ!∞∑ℓ=kS1,λ(ℓ,k)⟨(logλ(1+t)t)r|(tℓ)λ(x)n,λ⟩λ=n∑ℓ=k(nℓ)S1,λ(ℓ,k)⟨(logλ(1+t)t)r|(x)n−ℓ,λ⟩λ=n∑ℓ=k(nℓ)S1,λ(ℓ,k)D(r)n−ℓ,λ, |
which shows the first formula.
For the inversion formula, we first note that
(eλ(t)−1t)r=r∑j=0(rj)(−1)r−j∞∑m=0(j)m,λtm−rm!, |
which implies
⟨(eλ(t)−1t)r|(x)n−ℓ,λ⟩λ=r∑j=0(rj)(−1)r−j(j)n+r−ℓ,λ(n−ℓ)!(n−ℓ+r)!=r∑j=0(rj)(−1)r−j(j)n+r−ℓ,λ1(n−ℓ+r)r. | (2.11) |
Now, let (x)n,λ=∑∞k=0νn,kD(r)k,λ(x). Then, from (1.5), (2.7), and (2.11), νk satisfies
νn,k=1k!⟨(eλ(t)−1t)r(eλ(t)−1)k|(x)n,λ⟩λ=⟨(eλ(t)−1t)r|(1k!(eλ(t)−1)k)λ(x)n,λ⟩λ=n∑ℓ=k(nℓ)S2,λ(ℓ,k)⟨(eλ(t)−1t)r|(x)n−ℓ,λ⟩λ=n∑ℓ=k(nℓ)r∑j=0(rj)(−1)r−jS2,λ(ℓ,k)(j)n+r−ℓ,λ(n−ℓ+r)r, |
which shows the second result.
Next, we consider the degenerate Bernoulli polynomials βn,λ(x), which is defined by the generating function to be
teλ(t)−1exλ(t)=∞∑n=0βn,λ(x)tnn!, (see [21]). |
Then, the connection formulas between D(r)n,λ(x) and βn,λ(x) are as follows.
Theorem 2.2. For n∈N∪{0}, we have
D(r)n,λ(x)=n∑k=0(k+r−1)r−1(n+r−1)r−1S1,λ(n+r−1,k+r−1)βk,λforr∈N. |
As the inversion formula, we have
βn,λ(x)=n∑k=0(n∑ℓ=kr−1∑j=0(nℓ)(r−1j)(−1)r−1−j(j)n−ℓ+r−1,λ(n−ℓ+r−1)r−1S2,λ(ℓ,k))D(r)k,λ(x)forr>1, |
and
βn,λ(x)=n∑k=0S2,λ(n,k)Dk,λ(x)forr=1. |
Proof. First, note that βn,λ(x) is the λ-Sheffer sequence for
βn,λ(x)∼(eλ(t)−1t,t)λ. | (2.12) |
Let us consider D(r)ℓ,λ(x)=n∑k=0μn,kβk,λ(x). By (1.4), (2.9), (2.10) and (2.12), we obtain
μn,k=1k!⟨tlogλ(1+t)(tlogλ(1+t))r(logλ(1+t))k|(x)n,λ⟩λ=1k!⟨t1−r(logλ(1+t))k+r−1|(x)n,λ⟩λ=⟨k+1⟩r−1(k+r−1)!⟨t1−r(logλ(1+t))k+r−1|(x)n,λ⟩λ=(k+r−1)r−1(n+r−1)r−1S1,λ(n+r−1,k+r−1), |
which implies the first formula.
To find the inversion formula, we first note that from (1.1) for r>1
(eλ(t)−1t)r−1=t1−rr−1∑j=0(r−1j)(−1)r−1−jejλ(t)=r−1∑j=0(r−1j)(−1)r−1−j∞∑m=0(j)m,λtm+1−rm!. | (2.13) |
Thus, by (2.2) and (2.13)
⟨(eλ(t)−1t)r−1|(x)n−ℓ,λ⟩λ=r−1∑j=0(r−1j)(−1)r−1−j(j)n−ℓ+r−1,λ(n−ℓ)!(n−ℓ+r−1)!={r−1∑j=0(r−1j)(−1)r−1−j(j)n−ℓ+r−1,λ(n−ℓ+r−1)r−1 if r>1,δn,ℓ if r=1. | (2.14) |
Now, if we consider βn,λ(x)=∞∑k=0νn,kD(r)k,λ(x), then by (1.5), (2.7), and (2.14), νn,k satisfies
νn,k=1k!⟨(eλ(t)−1t)reλ(t)−1t(eλ(t)−1)k|(x)n,λ⟩λ=n∑ℓ=k(nℓ)S2,λ(ℓ,k)⟨(eλ(t)−1t)r−1|(x)n−ℓ,λ⟩λ={n∑ℓ=k(nℓ)S2,λ(ℓ,k)r−1∑j=0(r−1j)(−1)r−1−j(j)n−ℓ+r−1,λ(n−ℓ+r−1)r−1 if r>1,S2,λ(n,k) if r=1, |
which provides the formula.
Next, we consider the degenerate type 2 Bernoulli polynomials bn,λ(x), which are defined by the generating functions to be
teλ(t)−e−1λ(t)exλ(t)=∞∑n=0bn,λ(x)tnn!, (see [22]). |
Note that bn,λ(x) satisfies
bn,λ(x)∼(eλ(t)−e−1λ(t)t,t)λ. | (2.15) |
Then, we can have the following relation between D(r)n,λ(x) and bn,λ(x).
Theorem 2.3. For n∈N∪{0} and r∈N, we have
D(r)n,λ(x)={n∑k=0n∑m=k(nm)S1,λ(m,k)(D(r−1)n−m,λ+n−m∑ℓ=0(−1)ℓ(n−m)ℓD(r−1)n−m−ℓ,λ)bk,λforr>1.S1,λ(n,k)+n∑m=k(nm)S1,λ(m,k)(−1)n−m(n−m)n−mforr=1. |
For the inversion formula, we have
bn,λ(x)=12n∑k=0(n∑ℓ=kr−1∑j=0n+r−1∑m=0(nℓ)(r−1j)(n+r−1m)S2,λ(ℓ,k)(−1)r−1−j(j)m,λ(n+r−1)r−1×En+r−1−m,λ(12))D(r)k,λ(x)forr>1 |
and
bn,λ(x)=12n∑k=0(n∑m=k(nm)S2,λ(m,k)En−m,λ(12))Dk,λ(x)forr=1. |
Proof. Let us consider D(r)n,λ(x)=∑nk=0μn,kbk,λ(x). By (1.4), (2.9), (2.10), and (2.15), we get
μn,k=1k!⟨1+t−11+tlogλ(1+t)(tlogλ(1+t))r(logλ(1+t))k|(x)n,λ⟩λ=⟨(t(t+2)t+1)(logλ(1+t)t)r1k!(logλ(1+t))k−1|(x)n,λ⟩λ=⟨(t+2t+1)(logλ(1+t)t)r−11k!(logλ(1+t))k|(x)n,λ⟩λ=n∑m=k(nm)S1,λ(m,k)⟨(logλ(1+t)t)r−1|(1+∞∑ℓ=0(−t)ℓ)λ(x)n−m,λ⟩λ. | (2.16) |
From (2.3), it is noted that
(1+∞∑ℓ=0(−t)ℓ)λ(x)n−m,λ=(x)n−m,λ+n−m∑ℓ=0(−1)ℓ(n−m)ℓ(x)n−m−ℓ,λ. | (2.17) |
By applying the note (2.17) in (2.16), we have
μn,k={n∑m=k(nm)S1,λ(m,k)(D(r−1)n−m,λ+n−m∑ℓ=0(−1)ℓ(n−m)ℓD(r−1)n−m−ℓ,λ) for r>1,S1,λ(n,k)+n∑m=k(nm)S1,λ(m,k)(−1)n−m(n−m)n−m for r=1. |
To find the inversion formula, let us consider bn,λ(x)=∑∞k=0νn,kD(r)k,λ(x). From (1.5), (2.7), and (2.15), νn,k satisfies
νn,k=1k!⟨(eλ(t)−1t)reλ(t)−e−1λ(t)t(logλ(t+1))k|(x)n,λ⟩λ=n∑ℓ=k(nℓ)S2,λ(ℓ,k)⟨eλ(t)eλ(t)+1(eλ(t)−1t)r−1|(x)n−ℓ,λ⟩λ=12n∑ℓ=k(nℓ)S2,λ(ℓ,k)⟨2e12λ(t)e12λ(t)+e−12λ(t)(eλ(t)−1t)r−1|(x)n−ℓ,λ⟩λ. | (2.18) |
Since exλ(t)−1t=∑∞n=0(x)n+1,λn+1tnn!, we have that for r>1
(eλ(t)−1t)r−1=t1−rr−1∑j=0(r−1j)ejλ(t)(−1)r−1−j=t1−rr−1∑j=0(r−1j)(−1)r−1−j∞∑m=0(j)m,λtmm!. | (2.19) |
Then, (2.19) implies that for r>1
2e12λ(t)e12λ(t)+e−12λ(t)(eλ(t)−1t)r−1=t1−rr−1∑j=0(r−1j)(−1)r−1−j(∞∑m=0(j)m,λtmm!)(∞∑k=0Ek,λ(12)tkk!)=t1−rr−1∑j=0(r−1j)(−1)r−1−j∞∑m=0m∑k=0(mk)(j)k,λEm−k,λ(12)tmm!=r−1∑j=0(r−1j)(−1)r−1−j∞∑m=0m∑k=0(mk)(j)k,λEm−k,λ(12)tm+1−rm!, | (2.20) |
where En,λ(x) are the type 2 degenerate Euler polynomials defined by the following generating function
2e12λ(t)+e−12λ(t)exλ(t)=∞∑n=0En,λ(x)tnn!,(see [23]). | (2.21) |
Here we call En,λ:=En,λ(0) the type 2 degenerate Euler numbers if x=0. Thus, for m=n−ℓ+r−1 in (2.20) for r>1
⟨2e12λ(t)e12λ(t)+e−12λ(t)(eλ(t)−1t)r−1|(x)n−ℓ,λ⟩λ=r−1∑j=0(r−1j)(−1)r−1−jn−ℓ+r−1∑k=0(n−ℓ+r−1k)(j)k,λ×En−ℓ+r−1−k,λ(12)n!(n+r−1)!=r−1∑j=0(r−1j)(−1)r−1−jn−ℓ+r−1∑k=0(n−ℓ+r−1k)(j)k,λ×En−ℓ+r−1−k,λ(12)1(n−ℓ+r−1)r−1, |
and
⟨2e12λ(t)e12λ(t)+e−12λ(t)|(x)n−ℓ,λ⟩λ=En−ℓ,λ(12) for r=1, |
which provides the inversion formula with (2.18).
We consider the degenerate Euler polynomials Ek,λ that is defined by the generating function to be
(2eλ(t)+1)exλ(t)=∞∑n=0En,λ(x)tnn!,(see [12, 21]), |
which satisfies that
En,λ(x)∼(eλ(t)+12,t)λ. | (2.22) |
Then, the representation formula between D(r)n,λ(x) and En,λ(x) holds true.
Theorem 2.4. For n∈N∪{0} and r∈N, we have
D(r)n,λ(x)=12n∑k=0(n∑ℓ=kS1,λ(ℓ,k)(nℓ)((n−ℓ)D(r)n−ℓ−1,λ+2D(r)n−ℓ,λ))Ek,λ(x). |
As the inversion formula, we have
En,λ(x)=n∑k=0(n∑ℓ=kr∑j=0n+r∑l=0(nℓ)(rj)(n+rl)S2,λ(ℓ,k)(−1)r−j(j)l,λ(n+r)rEn+r−l,λ)Dk,λ(x). |
Proof. Let D(r)n,λ(x)=∑nk=0μn,kEk,λ. Then, By (1.4), (2.9), (2.10), and (2.22), we can obtain
μn,k=1k!⟨(1+t)+12(tlogλ(1+t))r(logλ(1+t))k|(x)n,λ⟩λ=⟨t+22(logλ(1+t)t)r1k!(logλ(1+t))k|(x)n,λ⟩λ=12∞∑ℓ=k(nℓ)S1,λ(ℓ,k)⟨(t+2)(logλ(1+t)t)r|(x)n−ℓ,λ⟩λ, | (2.23) |
where
⟨t(logλ(1+t)t)r|(x)n−ℓ,λ⟩λ=⟨n∑k=0D(r)k,λtk+1k!|(x)n−ℓ,λ⟩λ=D(r)n−ℓ−1,λ(n−ℓ) | (2.24) |
and
⟨n∑k=0D(r)k,λtkk!|(x)n−ℓ,λ⟩λ=D(r)n−ℓ,λ. | (2.25) |
Therefore, combining (2.24) and (2.25) to (2.23), we have
μn,k=12n∑ℓ=k(nℓ)S1,λ(ℓ,k)((n−ℓ)D(r)n−ℓ−1,λ+2D(r)n−ℓ,λ). |
To find the inversion formula, let En,λ(x)=∑∞k=0νn,kD(r)k,λ(x), where νn,k satisfies
νn,k=1k!⟨(eλ(t)−1t)reλ(t)+12(eλ(t)−1)k|(x)n,λ⟩λ=n∑ℓ=k(nℓ)S2,λ(ℓ,k)⟨(eλ(t)−1t)r2eλ(t)+1|(x)n,λ⟩λ. | (2.26) |
We note that
(eλ(t)−1t)r=t−rr∑j=0(rj)ejλ(t)(−1)r−j=t−rr∑j=0(rj)(−1)r−j∞∑m=0(j)m,λtmm!=r∑j=0(rj)(−1)r−j∞∑m=0(j)m,λtm−rm! |
and
(eλ(t)−1t)r2eλ(t)+1=r∑j=0(rj)(−1)r−j∞∑m=0(m∑l=0(ml)(j)l,λEm−l,λ)tm−rm!. |
Thus,
⟨(eλ(t)−1t)r2eλ(t)+1|(x)n,λ⟩λ=r∑j=0(rj)(−1)r−jn+r∑l=0(n+rl)(j)l,λ(n+r)rEn+r−l,λ. | (2.27) |
Combining (2.27) to (2.26) gives
νn,k=n∑ℓ=kr∑j=0n+r∑l=0(nℓ)(rj)(n+rl)S2,λ(ℓ,k)(−1)r−j(j)l,λ(n+r)rEn+r−l,λ. |
The degenerate Mittag-Leffer polynomials Mn,λ(x) are given by the generating function to be
exλ(logλ(1+t1−t))=∞∑n=0Mn,λ(x)tnn!,(see [24]). |
It is noted that
Mn,λ(x)∼(1,eλ(t)−1eλ(t)+1)λ. | (2.28) |
Then, we have the representation formulas between D(r)n,λ(x) and Mn,λ(x).
Theorem 2.5. For n∈N∪{0} and r∈N, we have
D(r)n,λ(x)=n∑k=0(1k!n∑m=0D(r)m,λ(nm)(n−m−1n−m−k)(−1)n−m−k2n−m(n−m)!)Mk,λ. |
As the inversion formula, we have
Mn,λ(x)=∞∑k=0(n∑m=0n!k!m!Km(λ)2m+k+r−1)D(r)k,λ(x), |
where Kn(x|λ) are the Korobov polynomials of the first kind given by the generating function
λt(1+t)λ−1(1+t)x=∞∑n=0Kn(x|λ)tnn!,(see[25]). |
In particular, when x=0 Kn(λ):=Kn(0|λ) are called Korobov numbers of the first kind, that is,
tlogλ(1+t)=∞∑n=0Kn(λ)tnn!. |
Proof. Let D(r)ℓ,λ(x)=∑nk=0μn,kMk,λ. Then, by (1.4), (2.9), (2.10), and (2.28), we can obtain
μn,k=1k!⟨1((1+t)−1logλ(1+t))r(1+t−11+t+1)k|(x)n,λ⟩λ=1k!⟨(logλ(1+t)t)r(tt+2)k|(x)n,λ⟩λ=1k!n∑m=0D(r)m,λ(nm)⟨(tt+2)k|(x)n−m,λ⟩λ, |
where
⟨(tt+2)k|(x)n−m,λ⟩λ=⟨∞∑ℓ=0(−1)ℓ2k+ℓ(k+ℓ−1ℓ)tk+ℓ|(x)n−m,λ⟩λ=(−1)n−m−k2n−m(n−m−1n−m−k)(n−m)!. | (2.29) |
Thus,
μn,k=1k!n∑m=0D(r)m,λ(nm)(n−m−1n−m−k)(−1)n−m−k2n−m(n−m)!. |
To find the inversion formula, let Mn,λ(x)=∑∞k=0νn,kD(r)k,λ(x), where
νn,k=1k!⟨(1+t1−t−1)rlogλ(1+t1−t)(1+t1−t−1)k|(x)n,λ⟩λ=1k!⟨(1+t−(1−t)1−t)rlogλ(1+t1−t)(1+t−1+t1−t)k|(x)n,λ⟩λ=1k!⟨(2t1−t)rlogλ(1+t1−t)(2t1−t)k|(x)n,λ⟩λ=1k!n∑m=01m!Km(λ)⟨(2t1−t)m+k+r−1|(x)n,λ⟩λ, |
where
⟨(2t1−t)m+k+r−1|(x)n,λ⟩λ=2m+k+r−1⟨∞∑ℓ=0tℓ+m+k+r−1|(x)n,λ⟩λ=2m+k+r−1n!. |
Thus,
νn,k=1k!n∑m=0n!m!Km(λ)2m+k+r−1. |
Next, let us consider the degenerate Bell polynomials Beln,λ(x), which are defined by the generating function to be
ex(eλ(t)−1)λ=∞∑n=0Beln,λ(x)tnn!,(see [26, 27]). |
Note that Beln,λ(x) are the λ-Sheffer sequences of
Beln,λ(x)∼(1,logλ(1+t))λ, | (2.30) |
which satisfies that
Beln,λ(x)=n∑k=0S2,λ(n,k)(x)k,λ, (see [26]). |
Then, we have the representation formulas between D(r)n,λ(x) and Beln,λ(x).
Theorem 2.6. For n∈N∪{0} and r∈N, it holds:
D(r)n,λ(x)=n∑k=0(n∑m=kn∑ℓ=m(nℓ)S1,λ(m,k)S1,λ(ℓ,m)D(r)n−ℓ,λ)Belk,λ(x). |
Also the inversion formula are established
Beln,λ(x)=n∑k=0(n∑m=kn∑ℓ=0r∑j=0(rj)S2,λ(m,k)S2,λ(n,ℓ)(−1)r−jm!(j)ℓ+r−m,λ)D(r)k,λ(x). |
Proof. Let D(r)ℓ,λ(x)=∑nk=0μn,kBelk,λ(x). Then, by (1.4), (2.9), (2.10), and (2.30), we can obtain
μn,k=1k!⟨1((1+t)−1logλ(1+t))r(logλ(1+logλ(1+t)))k|(x)n,λ⟩λ=1k!⟨(logλ(1+t)t)r(logλ(1+logλ(1+t)))k|(x)n,λ⟩λ=⟨(logλ(1+t)t)r|((1k!logλ(1+logλ(1+t)))k)λ(x)n,λ⟩λ=n∑m=kS1,λ(m,k)n∑ℓ=m(nℓ)S1,λ(ℓ,m)⟨(logλ(1+t)t)r|(x)n−ℓ,λ⟩λ=n∑m=kn∑ℓ=m(nℓ)S1,λ(m,k)S1,λ(ℓ,m)D(r)n−ℓ,λ. |
To find the inversion formula, let Beln,λ(x)=∑nk=0νn,kD(r)k,λ(x), then νn,k satisfies
νn,k=1k!⟨(eλ(t)−1t)r(eλ(t)−1)k|Beln,λ(x)⟩λ=∞∑m=kS2,λ(m,k)⟨(eλ(t)−1t)rtmm!|n∑ℓ=0S2,λ(n,ℓ)(x)ℓ,λ⟩λ=n∑m=kS2,λ(m,k)n∑ℓ=0S2,λ(n,ℓ)⟨(eλ(t)−1t)rtmm!|(x)ℓ,λ⟩λ, |
where
⟨(eλ(t)−1t)rtmm!|(x)ℓ,λ⟩λ=⟨r∑j=0(rj)(−1)r−j∞∑l=0(j)l,λtl+m−rl!m!|(x)ℓ,λ⟩λ=r∑j=0(rj)(−1)r−j(j)ℓ+r−m,λ1m!. |
Therefore, we have
νn,k=n∑m=kn∑ℓ=0r∑j=0(rj)S2,λ(m,k)S2,λ(n,ℓ)(−1)r−jm!(j)ℓ+r−m,λ. |
The degenerate Frobenius-Euler polynomials h(α)n,λ(x|u) of order α are defined by the generating function as
(1−ueλ(t)−u)αexλ(t)=∞∑n=0h(α)n,λ(x|u)tnn!,u(≠1)∈C. |
When x=0, h(α)n,λ(u):=h(α)n,λ(0|u) are called the degenerate Frobenius-Euler numbers.
We note that h(α)n,λ(x|u) satisfy
h(α)n,λ(x|u)∼((eλ(t)−u1−u)α,t)λ. | (2.31) |
Then, we have the representation formulas between D(r)n,λ(x) and h(α)n,λ(x|u).
Theorem 2.7. For n∈N∪{0} and r∈N, the representation holds:
D(r)n,λ(x)=n∑k=0(n∑m=kα∑ℓ=0(nm)(αℓ)(n−m)ℓ(1−u)ℓS1,λ(m,k)D(r)n−m−ℓ,λ)h(α)k,λ(x|u). |
As the inversion formula, we have
h(α)n,λ(x|u)=n∑k=0(n∑ℓ=0(nk+ℓ)(r+k)!k!(ℓ+r+k)!S2,λ(r+k+ℓ,r+k)h(α)n−k−ℓ,λ(u))D(r)k,λ(x). |
Proof. Let D(r)n,λ(x)=∑nk=0μn,kh(α)k,λ(x|u). By (1.4), (2.9), (2.10), and (2.31), we have
μn,k=1k!⟨(1+t−u1−u)α(tlogλ(1+t))r(logλ(1+t))k|(x)n,λ⟩λ=⟨(logλ(1+t)t)r(1+t1−u)α|(1k!(logλ(1+t))k)λ(x)n,λ⟩λ=n∑m=k(nm)S1,λ(m,k)⟨(logλ(1+t)t)r(1+t1−u)α|(x)n−m,λ⟩λ=n∑m=k(nm)S1,λ(m,k)⟨(logλ(1+t)t)r|(α∑ℓ=0(αℓ)(t1−u)ℓ)λ(x)n−m,λ⟩λ=n∑m=k(nm)S1,λ(m,k)α∑ℓ=0(αℓ)(n−m)ℓ(1−u)ℓ⟨(logλ(1+t)t)r|(x)n−m−ℓ,λ⟩λ=n∑m=k(nm)S1,λ(m,k)α∑ℓ=0(αℓ)(n−m)ℓ(1−u)ℓD(r)n−m−ℓ,λ, |
which implies the first formula.
Conversely, we assume that h(α)n,λ(x|u)=∑nk=0nun,kD(r)k,λ(x). Then, νn,k satisfies
νn,k=1k!⟨(eλ(t)−1t)r(eλ(t)−u1−u)α(eλ(t)−1)k|(x)n,λ⟩λ=1k!⟨(1−ueλ(t)−u)α(eλ(t)−1)r+ktr|(x)n,λ⟩λ=n∑ℓ=0(nk+ℓ)(r+k)!k!(ℓ+r+k)!S2,λ(r+k+ℓ,r+k)⟨(1−ueλ(t)−u)α|(x)n−k−ℓ,λ⟩λ=n∑ℓ=0(nk+ℓ)(r+k)!k!(ℓ+r+k)!S2,λ(r+k+ℓ,r+k)h(α)n−k−ℓ,λ(u), |
which shows the second assertion.
In this subsection, we present the pattern of the zeros of the polynomials. The understanding of patterns of zeros of degenerate polynomials can provide useful information about the original polynomials which can be obtained as limit of λ approaches zero. For example, the first three consecutive degenerate higher-order Daehee polynomials of degree r are given by
D(r)1,λ(x)=x+r(λ−1)2,D(r)2,λ(x)=x2+(r(λ−1)−1)x+112r(λ−1)(3r(λ−1)+λ−5),D(r)3,λ(x)=x3+12(3r(λ−1)−9)x2+14(8+r(λ−1)(3r(λ−1)+λ−11))x+18r(λ−1)(r(λ−1)−2)(r(λ−1)+λ−3), |
which approach to the higher-order Daehee polynomials of degree r as λ→0. We observe the patterns of roots by the changing parameters λ and r on the polynomials. In order to do this, we fix the degree of the polynomials as n=40, and compute the roots of D(r)40,λ(x) with fixed r=3 and six different parameters λ=±1, ±10, ±100 with the help of the Mathematica tool. The results are displayed in Figure 1. Next, we increase the degree of the polynomials and investigate the distribution of the roots of the polynomials.
For further investigation, we computed the roots of the polynomials by increasing the degree n of polynomials from 1 to 40 in Figure 2.
Finally, we investigated the distribution of the roots of D(r)40,λ(x) with a fixed λ=10 and three different parameters r=3, 4, 5 and the results were displayed in Figure 3.
In this subsection, we provide the explicit formulas presented in Theorem 2.2 that show the representations of the degenerate higher-order Daehee polynomials in terms of the degenerate Bernoulli polynomials and vice versa. To better understand, we present the graphs of D(r)n,λ(x) with λ=0.1 and r=3 for n=0, 1, ⋯, 5 and of D(r)n,λ(x) with λ=0.1 for various orders r=1, 2, ⋯, 5 in Figure 4.
Next, we compute the combinatorial results of μn,k and νn,k presented in the proof of Theorem 2.2 to confirm the connection formulas presented. To do this, we compute D(r)n,λ(x) and βn,λ(x) for r=3, λ=0.1, and n=0, 1, ⋯, 5 and expand them using the coefficients μn,k and νn,k which are computed with two decimal place accuracy. The expressions presented confirm the results of Theorem 2.2.
The degenerate higher-order Daehee polynomials D(r)n,λ(x) with λ=0.1, r=3 for n=1, 2, ⋯, 5 are expressed in terms of βn,λ(x) as follows:
D(3)5,λ(x)=x5−674x4+4194x3−30083100x2+386814310000x−68088951400000=β5,λ(x)−272β4,λ(x)+69β3,λ(x)−8195β2,λ(x)+174984310000β1,λ(x)−6312807100000β0,λ(x),D(3)4,λ(x)=x4−575x3+1794x2−34941500x+174984350000=β4,λ(x)−9β3,λ(x)+141350β2,λ(x)−8883250β1,λ(x)+70742750000β0,λ(x),D(3)3,λ(x)=x3−14120x2+59340x−88831000=β3,λ(x)−275β2,λ(x)+35140β1,λ(x)−80372000β0,λ(x),D(3)2,λ(x)=x2−3710x+11740=β2,λ(x)−2710β1,λ(x)+309200β0,λ(x),D(3)1,λ(x)=x−2720=β1,λ(x)−910β0,λ(x). |
Conversely, the degenerate Bernoulli polynomials βn,λ(x) with λ=0.1, r=3 for n=1, 2, ⋯, 5 are represented in terms of D(r)n,λ(x):
β5,λ(x)=x5−134x4+6720x3−2625x2−350x+9009400000=D(3)5,λ(x)+272D(3)4,λ(x)+1052D(3)3,λ(x)+6579100D(3)2,λ(x)+13527625D(3)1,λ(x)+28983125D(3)0,λ(x),β4,λ(x)=x4−125x3+4125x2−625x−2673100000,=D(3)4,λ(x)+9D(3)3,λ(x)+101750D(3)2,λ(x)+45940D(3)1,λ(x)+57516250D(3)0,λ(x),β3,λ(x)=x3−3320x2+1320x−994000,=D(3)3,λ(x)+275D(3)2,λ(x)+1161200D(3)1,λ(x)+910D(3)0,λ(x),β2,λ(x)=x2−x+33200=D(3)2,λ(x)+2710D(3)1,λ(x)+177200D(3)0,λ(x),β1,λ(x)=x−920=D(3)1,λ(x)+910D(3)0,λ(x). |
The study of special polynomials provides useful tools in differential equations, fuzzy theory, probability, orthogonal polynomials, and special functions and numbers. These researches are conducted using various tools, including generating functions, p-adic analysis, combinatorial methods, and umbral calculus. Recently, degenerate versions of special polynomials and numbers have been investigated using λ-analogues of these methods, and their arithmetical and combinatorial properties and relations have been studied by several mathematicians. These degenerate versions of special polynomials and numbers have been applied in differential equations and probability theories, providing new applications. In this paper, we explore the connection problems between the degenerate higher-order Daehee polynomials and other degenerate types of special polynomials. We present explicit formulas for representations with the help of umbral calculus and vice versa. In addition, we illustrate the results with some explicit examples. In order to better understanding the polynomials, the distribution of roots are presented.
The authors declare there is no conflict of interest.
[1] | Speight J (2012) The chemistry and technology of coal. CRC Press, New York. |
[2] |
Puente GDL, Iglesias MJ, Fuente E, et al. (1998) Changes in the structure of coals of different rank due to oxidation-effects on pyrolysis behaviour. J Anal Appl Pyrolysis 47: 33–42. doi: 10.1016/S0165-2370(98)00087-4
![]() |
[3] | Nelson MI, Chen XD (2007) Survey of experimental work on the self-heating and spontaneous combustion of coal. Reviews in Engineering Geology. Geological Society of America, 31–83. |
[4] |
De SK, Prabu V (2017) Experimental studies on humidified/water influx O2 gasification for enhanced hydrogen production in the context of underground coal gasification. Int J Hydrogen Energ 42: 14089–14102. doi: 10.1016/j.ijhydene.2017.04.112
![]() |
[5] |
Pan CX, Liu HL, Liu Q, et al. (2017) Oxidative depolymerization of Shenfu subbituminous coal and its thermal dissolution insoluble fraction. Fuel Process Technol 155: 168–173. doi: 10.1016/j.fuproc.2016.05.017
![]() |
[6] | Boron DJ, Taylor SR (1985) Mild oxidations of coal.1. Hydrogen peroxide oxidation. Fuel 64: 209–211. |
[7] |
Yu J, Jiang Y, Tahmasebi A, et al. (2014) Coal oxidation under mild conditions: current status and applications. Chem Eng Technol 37: 1635–1644. doi: 10.1002/ceat.201300651
![]() |
[8] |
Hayatsu R, Winans RE, McBeth RL (1984) Oxidative degradation studies and modern concepts of the formation and transformation of organic constituents of coals and sedimentary rocks. Org Geochem 6: 463–471. doi: 10.1016/0146-6380(84)90069-X
![]() |
[9] |
Wang YG, Wei XY, Yan HL, et al. (2013) Mild oxidation of Jincheng No.15 anthracite. J Fuel Chem Technol 41: 819–825. doi: 10.1016/S1872-5813(13)60035-3
![]() |
[10] | Rozhkova NN, Gorlenko LE, Emelyanova GI, et al. (2009) Effect of ozone on the structure and physicochemical properties of ultradisperse diamond and shungite nanocarbon elements. Pure Appl Chem 81: 2093–2105. |
[11] | Semenova SA, Fedyaeva ON, Patrakov YF (2006) Liquid-phase ozonation of highly metamorphized coal. Chem Sustain Dev 14: 43–48. |
[12] |
Semenova S, Patrakov Y, Batina M (2009) Preparation of oxygen-containing organic products from bed-oxidized brown coal by ozonation. Russ J Appl Chem 82: 80–85. doi: 10.1134/S1070427209010157
![]() |
[13] |
Obvintseva LA, Sukhareva IP, Epshtein SA, et al. (2017) Interaction of coals with ozone at low concentrations. Solid Fuel Chem 51: 155–159. doi: 10.3103/S0361521917030077
![]() |
[14] |
Wu F, Wang M, Lu Y, et al. (2017) Catalytic removal of ozone and design of an ozone converter for the bleeding air purification of aircraft cabin. Build Environ 115: 25–33. doi: 10.1016/j.buildenv.2017.01.007
![]() |
[15] |
Guo W, Ke P, Zhang S (2015) Effects of environment control system operation on ozone retention inside airplane cabin. Procedia Eng 121: 396–403. doi: 10.1016/j.proeng.2015.08.1084
![]() |
[16] |
Ondarts M, Outin J, Reinert L, et al. (2015) Removal of ozone by activated carbons modified by oxidation treatments. Eur Phys J-Spec Top 224: 1995–1999. doi: 10.1140/epjst/e2015-02516-6
![]() |
[17] |
Gorlenko LE, Emelyanova GI, Kharlanov AN, et al. (2006) Low-temperature oxidative modification of lignites and lignite-based cokes. Russ J Phys Chem 80: 878–881. doi: 10.1134/S0036024406060069
![]() |
[18] |
Semenova SA, Patrakov YF (2007) Ozonation of coal vitrinites of different metamorphism degrees in gas and liquid phases. Solid Fuel Chem 41: 15–18. doi: 10.3103/S0361521907010041
![]() |
[19] |
Patrakov YF, Semenova SA (2012) Chemical composition of various petrographic constituents of brown coal from the Balakhtinskoe deposit. Solid Fuel Chem 46: 1–6. doi: 10.3103/S0361521912010119
![]() |
[20] |
Patrakov Y, Fedyaeva O, Semenova S, et al. (2006) Influence of ozone treatment on change of structural-chemical parameters of coal vitrinites and their reactivity during the thermal liquefaction process. Fuel 85: 1264–1272. doi: 10.1016/j.fuel.2005.11.005
![]() |
[21] | Ksenofontova MM, Kudryavtsev AV, Mitrofanova AN, et al. (2005) Ozone application for modification of humates and lignins. In: Perminova IV, Hatfield K, Hertkorn N Editors, Use of Humic Substances to Remediate Polluted Environments: From Theory to Practice, Springer Netherlands, 473–484. |
[22] | Lunin VV, Popovich MP, Tkachenko SN (1998) Physical chemistry of ozone, Moscow. Moscow State University Publishing, 480. |
[23] | Batakliev T, Georgiev V, Anachkov M, et al. (2014) Ozone decomposition. Interdiscip Toxicol 7: 47–59. |
[24] |
Oyama ST (2000) Chemical and catalytic properties of ozone. Catal Rev 42: 279–322. doi: 10.1081/CR-100100263
![]() |
[25] |
Deitz VR, Bitner JL (1973) Interaction of ozone with adsorbent charcoals. Carbon 11: 393–401. doi: 10.1016/0008-6223(73)90079-1
![]() |
[26] | World Health Organization (2006) WHO Air quality guidelines for particulate matter, ozone, nitrogen dioxide and sulfur dioxide: global update 2005: summary of risk assessment. Geneva: World Health Organization: 1–22. |
[27] |
Elansky NF (2012) Russian studies of atmospheric ozone in 2007–2011. Izv Atmos Ocean Phy 48: 281–298. doi: 10.1134/S0001433812030024
![]() |
[28] |
Epshtein SA, Kossovich EL, Kaminskii VA, et al. (2017) Solid fossil fuels thermal decomposition features in air and argon. Fuel 199: 145–156. doi: 10.1016/j.fuel.2017.02.084
![]() |
[29] | Korovushkin VV, Epshtein SA, Durov NM, et al. (2015) Mineral and valent forms of iron and their effects on coals oxidation and self-ignition. Gornyi Zhurnal 2015: 70–74. |
[30] | Epshtein SA, Kossovich EL, Dobryakova NN, et al. (2016) New approaches for coal oxidization propensity estimation. XVIII International Coal Preparation Congress. Springer International Publishing, Cham, 483–487. |
[31] | Epshtein SA, Gavrilova DI, Kossovich EL, et al. (2016) Thermal methods exploitation for coals propensity to oxidation and self-ignition study. Gornyi Zhurnal 2016: 100–104. |
[32] | Belikov IB, Zhernikov KV, Obvintseva LA, et al. (2008) Analyzer of atmospheric gas impurities based on semiconductor sensors. Instrum Exp Tech 2008: 139–140. |
[33] | Obvintseva LA, Zhernikov KV, Belikov IB, et al. (2008) Semiconductor sensors and sensor containing gas analyzer for ozone monitoring in the atmosphere. Proceedings of the Eurosensors XXII Conference, 1594–1598. |
[34] | Ito O, Seki H, Iino M (1988) Diffuse reflectance spectra in near-i.r. region of coals; a new index for degrees of coalification and carbonization. Fuel 67: 573–578. |
[35] |
Maroto-Valer MM, Love GD, Snape CE (1994) Relationship between carbon aromaticities and HC ratios for bituminous coals. Fuel 73: 1926–1928. doi: 10.1016/0016-2361(94)90224-0
![]() |
[36] |
Maroto-Valer MM (1998) Verification of the linear relationship between carbon aromaticities and H/C ratios for bituminous coals. Fuel 77: 783–785. doi: 10.1016/S0016-2361(97)00227-5
![]() |
[37] |
Gan H, Nandi SP, Walker PL (1972) Nature of the porosity in American coals. Fuel 51: 272–277. doi: 10.1016/0016-2361(72)90003-8
![]() |
[38] |
Nie B, Liu X, Yang L, et al. (2015) Pore structure characterization of different rank coals using gas adsorption and scanning electron microscopy. Fuel 158: 908–917. doi: 10.1016/j.fuel.2015.06.050
![]() |
[39] | Mavor MJ, Owen LB, Pratt TJ (1990) Measurement and evaluation of coal sorption isotherm data. Proceedings of SPE Annual Technical Conference and Exhibition, Society of Petroleum Engineers, SPE 20728. |
[40] |
Rodrigues CF, Sousa MJLD (2002) The measurement of coal porosity with different gases. Int J Coal Geol 48: 245–251. doi: 10.1016/S0166-5162(01)00061-1
![]() |
1. | Eduardo G. Pardo, Sergio Gil-Borrás, Antonio Alonso-Ayuso, Abraham Duarte, Order Batching Problems: taxonomy and literature review, 2023, 03772217, 10.1016/j.ejor.2023.02.019 | |
2. | Daniel Alejandro Rossit, Fernando Tohmé, Máximo Méndez-Babey, Mariano Frutos, Diego Broz, Diego Gabriel Rossit, Special Issue: Mathematical Problems in Production Research, 2022, 19, 1551-0018, 9291, 10.3934/mbe.2022431 | |
3. | Giorgia Casella, Andrea Volpi, Roberto Montanari, Letizia Tebaldi, Eleonora Bottani, Trends in order picking: a 2007–2022 review of the literature, 2023, 11, 2169-3277, 10.1080/21693277.2023.2191115 | |
4. | Fabio Maximiliano Miguel, Mariano Frutos, Máximo Méndez, Fernando Tohmé, Begoña González, Comparison of MOEAs in an Optimization-Decision Methodology for a Joint Order Batching and Picking System, 2024, 12, 2227-7390, 1246, 10.3390/math12081246 |