
Plastics have become vital assets for humanity; these materials are used widely in pharmaceuticals, healthcare systems, and many other applications. The rising demand and uses of articles made wholly or partly from synthetic polymers, coupled with their non-biodegradability, contributes to the massive volume of plastic wastes across cities in most developing nations. Thistrend has become an issue of significant environmental concern. However, the fight against COVID-19 would look almost impossible without personal protective equipment (PPE) primarily made from various plastics which in turn, contribute enormously to the volume of waste streams. To circumvent this present challenge, research has been recommending solutions. The existing literature primarily focuses on the most developed countries, emphasising Asian countries with less attention to other developing countries like Nigeria and African countries. This study, therefore, reviewed the personal protective equipment used in healthcare, plastic types employed for their production, and the appropriate technology for managing their associated wastes. The application of proper disposal methods can reduce the toxic effects of discarded plastics on human health and the environment. In this review, the SWOT analysis approach was employed to unveil the benefits, limitations, opportunities, and threats associated with respective waste management approaches. As the coronavirus pandemic continues to intensifier, its adverse impacts on human health and the economy are increasing; authorities are encouraged to address waste management, including medical, household, and other hazardous waste, as an urgent and critical public service to minimize potential secondary health and environmental impacts.
Citation: Wilson U. Eze, Toyese Oyegoke, Jonathan D. Gaiya, Reginald Umunakwe, David I. Onyemachi. Review of personal protective equipment and their associated wastes, life-cycle and effective management during the Covid-19 pandemic in developing nations[J]. Clean Technologies and Recycling, 2022, 2(1): 1-31. doi: 10.3934/ctr.2022001
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Plastics have become vital assets for humanity; these materials are used widely in pharmaceuticals, healthcare systems, and many other applications. The rising demand and uses of articles made wholly or partly from synthetic polymers, coupled with their non-biodegradability, contributes to the massive volume of plastic wastes across cities in most developing nations. Thistrend has become an issue of significant environmental concern. However, the fight against COVID-19 would look almost impossible without personal protective equipment (PPE) primarily made from various plastics which in turn, contribute enormously to the volume of waste streams. To circumvent this present challenge, research has been recommending solutions. The existing literature primarily focuses on the most developed countries, emphasising Asian countries with less attention to other developing countries like Nigeria and African countries. This study, therefore, reviewed the personal protective equipment used in healthcare, plastic types employed for their production, and the appropriate technology for managing their associated wastes. The application of proper disposal methods can reduce the toxic effects of discarded plastics on human health and the environment. In this review, the SWOT analysis approach was employed to unveil the benefits, limitations, opportunities, and threats associated with respective waste management approaches. As the coronavirus pandemic continues to intensifier, its adverse impacts on human health and the economy are increasing; authorities are encouraged to address waste management, including medical, household, and other hazardous waste, as an urgent and critical public service to minimize potential secondary health and environmental impacts.
Let N be the set of natural numbers with N0=N∪{0}. For n∈N0, the following harmonic-like numbers are given by partial sums:
● The classical harmonic number and its quadratic counterpart
Hn=n−1∑k=01k+1andH⟨2⟩n=n−1∑k=01(k+1)2. |
● Skew harmonic number and its quadratic counterpart
On=n−1∑k=012k+1andO⟨2⟩n=n−1∑k=01(2k+1)2. |
● Alternating harmonic number and its quadratic counterpart
ˉHn=n−1∑k=0(−1)kk+1andˉH⟨2⟩n=n−1∑k=0(−1)k(k+1)2. |
Among them, there are the following simple but useful relations:
H2n=On+Hn2,H⟨2⟩2n=O⟨2⟩n+H⟨2⟩n4;ˉH2n=On−Hn2,ˉH⟨2⟩2n=O⟨2⟩n−H⟨2⟩n4. |
Harmonic numbers and their variants have been studied since the distant past and are involved in diverse fields (cf. [1,5,6,7,11,17,18]) such as analysis of algorithms in computer science, various topics of number theory, and combinatorial analysis. In this paper, we aim at presenting several algebraic identities about infinite series involving harmonic-like numbers, the binomial coefficient (3nn) in numerators, and a free variable "y".
For n∈N0 and an indeterminate x, the shifted factorials (cf. [3, §1.1])) are usually defined by
(x)0=1and(x)n=x(x+1)⋯(x+n−1)forn∈N. |
Denote by [xm]ϕ(x) the coefficient of xm in the formal power series ϕ(x). Then we can express harmonic-like numbers (see [11,18]) by extracting the initial coefficients of x from the quotients of shifted factorials as below:
[x](1+x)nn!=Hn,[x2](1+x)nn!=H2n−H⟨2⟩n2,[x]n!(1−x)n=Hn,[x2]n!(1−x)n=H2n+H⟨2⟩n2,[y](12+y)n(12)n=2On,[y2](12+y)n(12)n=2(O2n−O⟨2⟩n),[y](12)n(12−y)n=2On,[y2](12)n(12−y)n=2(O2n+O⟨2⟩n). |
There exist many infinite series identities involving central binomial coefficients (see [12,14,19,25]). However, the closed form evaluations for the known series with (3nn) are rare. In a recent paper, Sun [23] reported several remarkable infinite series identities, detected by numerical experimentation. Among them, the following three (see [23, Eqs (2.12), (2.13), and (2.15)]) seem rather challenging:
∞∑n=1(227)n(3nn)H2n=1+√32{√3ln2−3ln(3−√3)}, | (1.1) |
∞∑n=1(227)n(3nn)H3n=1+√34ln(4(7+4√3)3)−√3ln2, | (1.2) |
∞∑n=1(3+√554)n(3nn){H3n−H2n}=1+√52ln(9−3√52). | (1.3) |
When making efforts to confirm these three formulae, two substantial novelties emerged. First, not only the harmonic numbers H3n and H2n are separable in the third series, but also independent series involving Hn and On can be evaluated explicitly in closed form (cf. Corollaries 6 and 10). Another surprising novelty (see Theorems 1 and 5) lies in the fact that these series are particular cases of more general algebraic identities containing a free variable "y", corresponding to the convergent rates of these numerical series. To our knowledge, most of the infinite series identities presented in this paper have not appeared previously in the literature.
The rest of the paper is organized as follows. As preliminaries, we shall evaluate, in closed form, three series containing binomial coefficient (3nn), harmonic number Hn and a free variable "y" by integrating Lambert's series. Then in Sections 3–5, three classes of infinite series with harmonic-like numbers will be examined by applying the "coefficient extraction" method (see [8,13,24]) to cubic transformations for the 3F2-series, that result in several algebraic identities about these series and confirm or refine, as special numerical examples, three conjectured identities made recently by Sun [23]. Finally, the paper will end with Section 6, where concluding remarks and further problems are provided.
In classical analysis and enumerative combinatorics, Lambert's binomial series are well-known (see Riordan [22, §4.5] and [9,10,16]), which can be reproduced as follows: Let T and τ be the two variables related by the equation τ=T/(1+T)β. Then
(1+T)α=∞∑k=0αα+kβ(α+kβk)τkand(1+T)α+11+T−Tβ=∞∑k=0(α+kβk)τk. | (2.1) |
As preliminaries, we are going to evaluate, in closed form, three series containing ternary coefficient (3nn) and a free variable "y" by integrating across the above two equations, that will be crucial for demonstrating infinite series identities involving harmonic numbers in the subsequent sections.
For "α=0 and β=3", performing the replacement τ→xy/(1+y)3 in Lambert's second series, we can state the resulting equality as
ϕ(T):=1+T1−2T=∞∑n=0(3nn)(xy)n(1+y)3n:x=T(1+y)3y(1+T)3. | (2.2) |
Since the rightmost function is monotone around T=0, it determines an implicit inverse function T=T(x) in the neighborhood of x=0 with particular values "T(0)=0 and T(1)=y".
By expressing the harmonic number Hn in terms of the integral
Hn=∫101−xn1−xdx, |
we can insert Hn in the series (2.2) as follows:
∞∑n=0(3nn)Hnyn(1+y)3n=∫10ϕ(T(1))−ϕ(T(x))1−xdx=(1+y)3y∫y01−2T(1+T)41+y1−2y−1+T1−2T1−T(1+y)3y(1+T)3dT=(1+y)3∫y01−(1−2T)(1+y)(1+T)(1−2y)T(1+y)3−y(1+T)3dT, |
where the change of variable x=T(1+y)3y(1+T)3 has been made so that the integral becomes explicit.
For the sake of brevity, the notation "ρ" will be fixed, throughout the paper, for the algebraic number
ρ:=32(√2+1)1/3−32(√2−1)1/3−1≈−0.105892543. | (2.3) |
The convergence region of the preceding series is determined by
−1<27y4(1+y)3<1,ρ<y<12. |
This can be illustrated graphically as below:
By decomposing the integrand into partial fractions
1−(1−2T)(1+y)(1+T)(1−2y)T(1+y)3−y(1+T)3=3(1+T)(1+y)2(1−2y)−32(1+y)2(1−2y)√4+y×{√4+y+√yT+√y(3+y)−(1+y)√4+y2√y+√4+y−√yT+√y(3+y)+(1+y)√4+y2√y}, |
we can explicitly compute the integral
∫y01−(1−2T)(1+y)(1+T)(1−2y)T(1+y)3−y(1+T)3dT=3ln(1+y)(1+y)2(1−2y)+3(√y−√4+y)2(1+y)2(1−2y)√4+yln((1+y)(2−2y−y2+y√y2+4y)2)−3(√y+√4+y)2(1+y)2(1−2y)√4+yln((1+y)(2−2y−y2−y√y2+4y)2). |
By cancelling the three terms containing ln(1+y) and then making substitution, we find the closed formula for the series as in the theorem below:
Theorem 1 (ρ<y<1/2).
∞∑n=1(3nn)Hnyn(1+y)3n=(1+y)3∫y01−(1−2T)(1+y)(1+T)(1−2y)T(1+y)3−y(1+T)3dT=3(1+y)ln(1−2y)2(2y−1)+3(1+y)√y2(1−2y)√4+yln(2−y(2+y−√y2+4y)2−y(2+y+√y2+4y)). |
For "α=1 and β=3", performing the replacement τ→xy/(1+y)3, we can rewrite Lambert's first series as
ψ(T):=1+T=∞∑n=0(3nn)(xy)n(1+2n)(1+y)3n:x=T(1+y)3y(1+T)3. | (2.4) |
Analogously, by putting the harmonic number Hn inside the series (2.4), we have
∞∑n=0(3nn)Hnyn(1+2n)(1+y)3n=∫10ψ(T(1))−ψ(T(x))1−xdxx=T(1+y)3y(1+T)3=(1+y)3y∫y0(y−T)(1−2T)dT(1+T)4{1−T(1+y)3y(1+T)3}=∫y0(1+y)3(T−y)(1−2T)dT(1+T){T(1+y)3−y(1+T)3}=∫y0(1+y)3(1−2T)dT(1+T)(1−T2y−Ty2−3Ty). |
By decomposing the integrand into partial fractions
(1+y)3(1−2T)dT(1+T)(1−T2y−Ty2−3Ty)=31+y1+T−(1+y)(3√y+√4+y)2√y(T+(3+y)√y+(1+y)√4+y2√y)−(1+y)(3√y−√4+y)2√y(T+(3+y)√y−(1+y)√4+y2√y), |
we can explicitly compute the integral
∫y0(1+y)3(1−2T)dT(1+T)(1−T2y−Ty2−3Ty)=3(1+y)ln(1+y)−(1+y)(3√y+√4+y)2√yln((1+y)(3√y+√4+y)(3+y)√y+(1+y)√4+y)−(1+y)(3√y−√4+y)2√yln((1+y)(3√y−√4+y)(3+y)√y−(1+y)√4+y), |
which is simplified into the closed formula as in the theorem below:
Theorem 2 (ρ<y<1/2).
∞∑n=1(3nn)Hnyn(1+2n)(1+y)3n=∫y0(1+y)3(1−2T)dT(1+T)(1−T2y−Ty2−3Ty)=(1+y)√4+y2√yln(2−y(2+y+√y2+4y)2−y(2+y−√y2+4y))−32(1+y)ln(1−2y). |
Alternatively, rewriting Lambert's first series
(1+T)α−1α=∞∑n=11α+nβ(α+nβn)τn:τ=T/(1+T)β |
and letting "α→0 and β=3", we deduce the identity
σ(T):=3ln(1+T)=∞∑n=1(3nn)(xy)nn(1+y)3n:x=T(1+y)3y(1+T)3. | (2.5) |
By inserting Hn in the above series, we obtain an integral expression
∞∑n=1(3nn)Hn(xy)nn(1+y)3n=∫10σ(T(1))−σ(T(x))1−xdxx→T(1+y)3y(1+T)3=3∫y0ln(1+T1+y)(1−2T)(1+y)3(1+T)(T−y)(1−T2y−Ty2−3Ty)dT=3∫y0ln(1+T1+y){1T−y−31+T+y2+3y+2TyT2y+Ty2+3Ty−1}dT. |
In view of the partial fraction decomposition
y2+3y+2TyT2y+Ty2+3Ty−1=1T+√y(3+y)+(1+y)√4+y2√y)+1T+√y(3+y)−(1+y)√4+y2√y), |
it suffices to evaluate, for a parameter λ subject to T+λ≠0 for T∈(0,y), the parametric integral
J(λ,y)=∫y0ln(1+T1+y)dTT+λ=Li2(1+y1−λ)−Li2(11−λ)+ln(λλ−1)ln(1+y), |
with particular cases
J(1,y)=−12ln2(1+y)andJ(−y,y)=π26+ln(1+y)ln(y1+y)−Li2(11+y). |
Hereafter, the dilogarithm function (see Lewin [20, §1.1]) is defined for all complex values of z by
Li2(z)=∫z0ln(1−τ)−τandLi2(z)=∞∑n=1znn2for|z|≤1. |
By making substitution and then simplifying the terms concerning logarithms and dilogarithms in the resulting expression, we find the explicit formula as in the following theorem:
Theorem 3 (ρ<y<1/2).
∞∑n=1(3nn)Hnynn(1+y)3n=∫y0−3dT1+Tln(1−T(1+y)3y(1+T)3)=π22+3Θ(y)+3ln(1+y)ln(y√(1+y)3), |
where, for simplicity, Θ(y) denotes the sum of five dilogarithmic values
Θ(y)=Li2(y−√y2+4y2)+Li2(y+√y2+4y2)−Li2(y−√y2+4y2+2y)−Li2(y+√y2+4y2+2y)−Li2(11+y). |
According to Bailey [3], the classical hypergeometric series reads as
1+pFp[a0,a1,a2,⋯,apb1,b2,⋯,bp|z]=∞∑k=0zkk!(a0)k(a1)k(a2)k⋯(ap)k(b1)k(b2)k⋯(bp)k. |
We shall confine ourselves only to examining the series with |z|<1 and none of the parameters in the numerator and denominator being a non-positive integer, which imply that the corresponding series is well-defined and convergent.
There are many transformation and summation formulae for hypergeometric series (see Bailey [3] and Brychkov [4, Chapter 8]) in the literature. In 1928, Bailey [2] discovered an important cubic transformation (see the next section). Half a century later, Gessel and Stanton [15, Eq (5.4)] found the following counterpart
3F2[1+a3,2+a3,3+a31+a+c2,2+a−c2|27y4(1+y)3]=(1+y)1+a1−2y×4F3[1+a3,a,c,1−ca3,1+a+c2,2+a−c2|−y4]. | (3.1) |
Under the parameter replacements "a→6ax,c→2cx", this transformation can be reformulated as
∞∑n=0(y4(1+y)3)n(1+6ax)3nn!(1+3ax−cx)n(12+3ax+cx)n=(1+y)1+6ax1−2y{1+6cx∞∑k=1(−y4)k(k+2ax)(1+6ax)k−1(1+2cx)k−1(1−2cx)kk!(1+3ax−cx)k(12+3ax+cx)k}, | (3.2) |
where "y∈(ρ,12)", or equivalently, 27y4(1+y)3∈(−1,1) so that the series are convergent.
The above transformation is an analytic equality with respect to x in the neighborhood of x=0. Equating the constant terms on both sides yields the algebraic identity below:
∞∑n=0(3nn)yn(1+y)3n=1+y1−2y. |
As illustrated in the introduction, harmonic-like numbers can be expressed in terms of x-coefficients from quotients of shifted factorials. By examining the coefficients of x across the above equation, we find that the corresponding equation involving harmonic numbers can be written as
∞∑n=1yn(1+y)3n(3nn){6aH3n−(3a−c)Hn−2(3a+c)On}=6a1+y1−2yln(1+y)+6c1+y1−2y∞∑k=1(k−1)!(12)k(−y4)k. |
The rightmost series admits the closed form expression, as in the lemma below:
Lemma 4 (−4<y<4).
W(y)=∞∑k=1(k−1)!(12)k(−y4)k=−√4y4+yArcTanh√y4+y=√4y4+yln√4+y−√y2. |
Proof. Recall the beta function (see [21, §16])
B(a,b)=∫10xa−1(1−x)b−1dx=Γ(a)Γ(b)Γ(a+b),whereℜ(a)>0andℜ(b)>0. |
We can express W(y) as a definite integral
W(y)=∞∑k=1(k−1)!(12)k(−y4)k=∞∑k=1B(k,12)(−y4)k=∞∑k=1(−y4)k∫10xk−1dx√1−x=∫10dxx√1−x∞∑k=1(−xy4)k=∫10−ydx√1−x(4+xy)=∫10−2y4+y−T2ydT, |
where the last line is justified by the change of variable √1−x→T. Then the formula in the lemma follows by evaluating the last integral explicitly.
Therefore, we find the general formula below:
∞∑n=1yn(1+y)3n(3nn){6aH3n−(3a−c)Hn−2(3a+c)On}=6a1+y1−2yln(1+y)+6c1+y1−2yW(y). |
By specifying two parameters, a and c, we deduce four independent series
a=0:∞∑n=1yn(1+y)3n(3nn)ˉH2n=31+y1−2yW(y),c=0:∞∑n=1yn(1+y)3n(3nn){H3n−H2n}=1+y1−2yln(1+y),c=3a:∞∑n=1yn(1+y)3n(3nn){H3n−2On}=1+y1−2yln(1+y)−31+y1−2yW(y),c=−3a:∞∑n=1yn(1+y)3n(3nn){H3n−Hn}=1+y1−2yln(1+y)+31+y1−2yW(y). |
Combining the above formulae with Theorem 1 and then keeping in mind the equalities
H2n=ˉH2n+Hn,On=ˉH2n+Hn2,H3n=(H3n−Hn)+Hn, |
we establish further closed formulae for four series with a free variable "y".
Theorem 5 (ρ<y<1/2).
(a)∞∑n=1(3nn)ˉH2nyn(1+y)3n=3+3y1−2y√4y4+yln√4+y+√y2,(b)∞∑n=1(3nn)H2nyn(1+y)3n=3+3y2−4y√y4+yln(2(1+y+√y2+4y)2−y(2+y+√y2+4y))−(3+3y)ln(1−2y)2(1−2y),(c)∞∑n=1(3nn)Onyn(1+y)3n=3+3y2−4y√y4+yln(1+y+√y2+4y√1−2y)−(3+3y)ln(1−2y)4(1−2y),(d)∞∑n=1(3nn)H3nyn(1+y)3n=1+y2−4yln((1+y)2(1−2y)3)+3+3y2−4y√y4+yln(2(1+y+√y2+4y)2−y(2+y+√y2+4y)). |
Numerous infinite series identities with different convergence rates can be derived by assigning particular values for y in Theorems 1 and 5. Five groups of representative examples are recorded as corollaries. In particular, the second formula (b) and the last formula (e) displayed in Corollary 6 confirm (1.1) and (1.2) conjectured by Sun [23, Eqs (2.12) and (2.13)].
Corollary 6 (y→3√3−52 in Theorems 1 and 5).
(a)∞∑n=1(227)n(3nn)ˉH2n=√3−32ln(√3−1),(b)∞∑n=1(227)n(3nn)H2n=1+√32{√3ln2−3ln(3−√3)},(c)∞∑n=1(227)n(3nn)Hn=3−√34ln(43)+√3ln(2+√33),(d)∞∑n=1(227)n(3nn)On=3+√32{ln(1+√3√2)−√34ln3},(e)∞∑n=1(227)n(3nn)H3n=1+√34ln(4(7+4√3)3)−√3ln2. |
Corollary 7 (y→√5−2 in Theorems 1 and 5).
(a)∞∑n=1(18)n(3nn)ˉH2n=3(1−√5)√5ln(√5−12),(b)∞∑n=1(18)n(3nn)H2n=3(2+√5)√5ln(3+√52)−3ln5√5,(c)∞∑n=1(18)n(3nn)Hn=32ln(3+√52)+3√5ln(11+5√510),(d)∞∑n=1(18)n(3nn)On=3(1+√5)2√5ln(2+√5)−3ln52√5,(e)∞∑n=1(18)n(3nn)H3n=ln(11+5√5)+3√5ln(4+2√55). |
Corollary 8 (y→√5−2√5 in Theorems 1 and 5).
(a)∞∑n=1(564)n(3nn)ˉH2n=6(√5−1)11ln(5−√52),(b)∞∑n=1(564)n(3nn)H2n=6√511ln(25+9√522)−311ln(23−3√510),(c)∞∑n=1(564)n(3nn)Hn=−3√511ln(267−119√52)−611ln(7+√510),(d)∞∑n=1(564)n(3nn)On=3(1+√5)11ln(4+√511)+3√511ln5,(e)∞∑n=1(564)n(3nn)H3n=6√511ln(4(4+√5)11)+111ln(200(2667−73√5)116). |
Corollary 9 (y→−110 in Theorems 1 and 5).
(a)∞∑n=1(−100729)n(3nn)ˉH2n=−38√313ArcCos(161200),(b)∞∑n=1(−100729)n(3nn)H2n=−98ln(65)−38√313ArcCos(6780),(c)∞∑n=1(−100729)n(3nn)Hn=−98ln(65)+38√313ArcCos(79878000),(d)∞∑n=1(−100729)n(3nn)On=−916ln(65)−316√313ArcCos(720),(e)∞∑n=1(−100729)n(3nn)H3n=−38ln(3215)−38√313ArcCos(6780). |
Corollary 10 (y→7−3√52 in Theorems 1 and 5).
(a)∞∑n=1(3+√554)n(3nn)ˉH2n=√6√5−6√5ln(1+√15+6√53+√5),(b)∞∑n=1(3+√554)n(3nn)H2n=3(1+√5)4ln(2+√53)+√3/2√5+√5ln(3√5−1+√30−6√54),(c)∞∑n=1(3+√554)n(3nn)Hn=3(1+√5)4ln(2+√53)+√3/2√5+√5ln(11√3+√185−82√511√3−√185−82√5),(d)∞∑n=1(3+√554)n(3nn)On=3(1+√5)8ln(2+√53)+√3/8√5+√5ln(3√5−4+2√15−6√5),(e)∞∑n=1(3+√554)n(3nn)H3n=1+√54ln(11+5√56)+√3/2√5+√5ln(3√5−1+√30−6√54). |
We remark that the second formula (b) and the last series (e) in this corollary refine identity (1.3) conjectured by Sun [23, Eq (2.15)]:
∞∑n=1(3+√554)n(3nn){H3n−H2n}=1+√52ln(9−3√52). |
Now, we turn to examine Bailey's cubic transformation (Bailey [2, Eq (4.05)])
3F2[a3,1+a3,2+a31+a+c2,2+a−c2|27y4(1+y)3]=(1+y)a×3F2[a,c,1−c1+a+c2,2+a−c2|−y4]. | (4.1) |
Under the parameter replacements "a→1+6ax,c→2cx", it can be reformulated as
∞∑n=0(y4(1+y)3)n(1+6ax)3nn!(1+3ax+cx)n(32+3ax−cx)n=(1+y)1+6ax×{1+2cx∞∑k=1(−y4)k(1+6ax)k(1+2cx)k−1(1−2cx)kk!(1+3ax+cx)k(32+3ax−cx)k}, | (4.2) |
where, as before, both series converge for "y∈(ρ,12)", or equivalently, 27y4(1+y)3∈(−1,1).
When x=0, the above series becomes the following simpler one:
∞∑n=0(3nn)yn(1+2n)(1+y)3n=1+y. |
Instead, the coefficients of x across (4.2) result in the general formula as below:
∞∑n=1yn(1+y)3n(3nn){6aH3n−(3a+c)Hn−2(3a−c)On1+2n+4(3a−c)n(1+2n)2}=6a(1+y)ln(1+y)+2c(1+y)∞∑k=1(k−1)!(32)k(−y4)k. |
For brevity, we introduce the notations U(y) and V(y) for the two series:
U(y)=∞∑k=1(k−1)!(32)k(−y4)kandV(y)=∞∑n=0yn(1+y)3n(3nn)n(1+2n)2. |
They are evaluated in closed form by the following lemma:
Lemma 11.
(a)U(y)=2+2√4+y√yln√4+y−√y2,−4<y<4;(b)V(y)=√(1+y)3√yln(√y+√1+y)−(1+y),ρ<y<1/2. |
Proof. The first series can be evaluated by computing the integral as follows:
U(y)=∞∑k=1(k−1)!(32)k(−y4)k=∞∑k=1B(k,32)(−y4)k=∞∑k=1(−y4)k∫10xk−1√1−xdx=∫10√1−xdxx∞∑k=1(−xy4)k=∫10−y√1−xdx4+xy=∫10−2yT24+y−T2ydT√1−x→T=2+2√4+y√yln√4+y−√y2. |
For the second series V(y), by shifting forward the summation index n→n+1, we can rewrite it as
V(y)=∞∑n=0yn(1+y)3n(3nn)n(1+2n)2=y(1+y)3∞∑n=0yn(1+y)3n(3n+3n)13+2n. |
Then, for "α=β=3", perform the replacement τ→xy/(1+y)3 in Lambert's second series
∞∑n=0xnyn(1+y)3n(3n+3n)=(1+T)41−2T. |
Now multiplying this equation by x12 and then integrating across over x∈[0,1], we can proceed with
∞∑n=0yn(1+y)3n(3n+3n)23+2n=∫10x12(1+T)41−2Tdxx=T(1+y)3y(1+T)3=√(1+y)9y√y∫y0√T√(1+T)3dT. |
By means of integration by parts, we can further compute
∫y0√T√(1+T)3dT=−2√y√1+y+∫y01√T(1+T)dT=2ln(√y+√1+y)−2√y√1+y. |
After substitution, the second formula in Lemma 11 follows accordingly.
Therefore, we obtain the following reduced transformation:
∞∑n=0yn(1+y)3n(3nn){6aH3n−(3a+c)Hn−2(3a−c)On1+2n}=6a(1+y)ln(1+y)+2c(1+y)U(y)−4(3a−c)V(y). | (4.3) |
By specifying two parameters, a and c in the above equality, we deduce four independent series:
a=0:∞∑n=1yn(1+y)3n(3nn){ˉH2n1+2n}=(1+y)U(y)+2V(y),c=0:∞∑n=1yn(1+y)3n(3nn){H3n−H2n1+2n}=(1+y)ln(1+y)−2V(y),c=3a:∞∑n=1yn(1+y)3n(3nn){H3n−Hn1+2n}=(1+y)ln(1+y)+(1+y)U(y),c=−3a:∞∑n=1yn(1+y)3n(3nn){H3n−2On1+2n}=(1+y)ln(1+y)−(1+y)U(y)−4V(y). |
For the above four equations, first replacing U(y) and V(y) by their algebraic values in Lemma 11, and then putting the resulting equations in conjunction with that in Theorem 2, we establish, after some routine simplifications, four closed formulae as in the following theorem:
Theorem 12 (ρ<y<1/2).
(a)∞∑n=1yn(1+y)3n(3nn)ˉH2n1+2n=2√(1+y)3√yln(√y+√1+y)+2(1+y)√4+y√yln(√4+y−√y2),(b)∞∑n=1yn(1+y)3n(3nn)H2n1+2n=2√(1+y)3√yln(√y+√1+y)−3+3y2ln(1−2y)+(1+y)√4+y2√yln(2+2y−2√y2+4y2−y(2+y−√y2+4y)),(c)∞∑n=1yn(1+y)3n(3nn)On1+2n=2√(1+y)3√yln(√y+√1+y)−3+3y4ln(1−2y)+(1+y)√4+y2√yln(1+y−√y2+4y√1−2y),(d)∞∑n=1yn(1+y)3n(3nn)H3n1+2n=2(1+y)+1+y2ln((1+y)2(1−2y)3)+(1+y)√4+y2√yln(2+2y−2√y2+4y2−y(2+y−√y2+4y)). |
By assigning particular values for y in Theorems 2 and 12, we can deduce numerous infinite series identities with different convergence rates. In order to show the variety of implications, we record five representative classes of infinite series values.
Corollary 13 (y→√5−2 in Theorems 2 and 12).
(a)∞∑n=1(18)n(3nn)ˉH2n1+2n=2√2ln(3+√101+√2)−(3+√5)ln(1+√52),(b)∞∑n=1(18)n(3nn)H2n1+2n=2√2ln(3+√101+√2)−(6−4√5)ln(1+√52)−√5ln5,(c)∞∑n=1(18)n(3nn)Hn1+2n=(5√5−3)ln(1+√52)−√5ln5,(d)∞∑n=1(18)n(3nn)On1+2n=2√2ln(3+√101+√2)−(92−3√52)ln(1+√52)−√52ln5,(e)∞∑n=1(18)n(3nn)H3n1+2n=(√5−1)(2+ln2)−(5−3√5)ln(1+√52)−√5ln5. |
Corollary 14 (y→3√3−52 in Theorems 2 and 12).
(a)∞∑n=1(227)n(3nn)ˉH2n1+2n=9+3√32ln(√3−1)+9√6ln(3+2√2√3+√2),(b)∞∑n=1(227)n(3nn)H2n1+2n=92ln(3−√32)+3√32ln((√3+1)312√3)+9√6ln(3+2√2√3+√2),(c)∞∑n=1(227)n(3nn)Hn1+2n=94ln(34)+3√32ln((√3+1)424√3),(d)∞∑n=1(227)n(3nn)On1+2n=34(3−√3)ln(2√3−3)−9ln34√3+9√6ln(3+2√2√3+√2),(e)∞∑n=1(227)n(3nn)H3n1+2n=3√3−3+32ln(2√3−32)+3√32ln(2+√32√3). |
Corollary 15 (y→3√2−4 in Theorems 2 and 12).
(a)∞∑n=1(2+√227)n(3nn)ˉH2n1+2n=3√3ln(√6−2√3−1)+9√2−√2√6ln(6√2−7+√6(2−√2)3),(b)∞∑n=1(2+√227)n(3nn)H2n1+2n=3√3ln(√3−1√2)−(9−9√2)ln(1+√2√3)+9√2−√2√6ln(6√2−7+√6(2−√2)3),(c)∞∑n=1(2+√227)n(3nn)Hn1+2n=3√3ln(√2+√32+√3)−(9−9√2)ln(1+√2√3),(d)∞∑n=1(2+√227)n(3nn)On1+2n=(92−9√2)ln(√6−√3)−3√32ln(√2+√3)+9√2−√2√6ln(6√2−7+√6(2−√2)3),(e)∞∑n=1(2+√227)n(3nn)H3n1+2n=6√2−6+3√3ln(√3−1√2)−(3−3√2)ln(3+2√2√3). |
Corollary 16 (y→15 in Theorems 2 and 12).
(a)∞∑n=1(25216)n(3nn)ˉH2n1+2n=12√65ln(1+√6√5)−12√215ln(1+√212√5),(b)∞∑n=1(25216)n(3nn)H2n1+2n=95ln(53)+12√65ln(1+√6√5)+6√215ln(9−√212√15),(c)∞∑n=1(25216)n(3nn)Hn1+2n=95ln(53)+3√215ln(257−13√21250),(d)∞∑n=1(25216)n(3nn)On1+2n=910ln(53)+12√65ln(1+√6√5)+3√215ln(6−√21√15),(e)∞∑n=1(25216)n(3nn)H3n1+2n=125+35ln(203)+6√215ln(9−√212√15). |
Corollary 17 (y→−110 in Theorems 2 and 12).
(a)∞∑n=1(−100729)n(3nn)ˉH2n1+2n=275arccot3−9√395arccot√39,(b)∞∑n=1(−100729)n(3nn)H2n1+2n=275arccot3−9√3920arccos(6780)−2720ln(65),(c)∞∑n=1(−100729)n(3nn)Hn1+2n=9√3920arccos(79878000)−2720ln(65),(d)∞∑n=1(−100729)n(3nn)On1+2n=275arccot3−9√3920arccos(√2740)−2740ln(65),(e)∞∑n=1(−100729)n(3nn)H3n1+2n=95−9√3920arccos(6780)−920ln(3215). |
Under the parameter replacements "a→6ax,c→2cx", Bailey's cubic transformation (4.1) can be reformulated alternatively as
∞∑n=0(y4(1+y)3)n(6ax)3nn!(1+3ax−cx)n(12+3ax+cx)n=(1+y)6ax×{1+12acx2∞∑k=1(−y4)k(1+6ax)k−1(1+2cx)k−1(1−2cx)kk!(1+3ax−cx)k(12+3ax+cx)k}. |
Both series in the above equation converge subject to the same condition "y∈(ρ,12)", or equivalently, 27y4(1+y)3∈(−1,1) (as before).
Equating the constant terms from both sides yields the algebraic identity below
∞∑n=1(3nn)ynn(1+y)3n=3ln(1+y). |
Instead, the coefficients of x result in the following infinite series identity involving a free variable "y" and two linear parameters a and c:
∞∑n=1yn(1+y)3n(3nn){6aH3n−(3a−c)Hn−2(3a+c)Onn−2an2}=9aln2(1+y)+6c∞∑k=1(k−1)!k(12)k(−y4)k. |
Let U(y) and V(y) stand for the two series
U(y)=∞∑k=1(k−1)!k(12)k(−y4)kandV(y)=∞∑n=1ynn2(1+y)3n(3nn). |
They admit closed formulae as in the lemma below:
Lemma 18.
(a)U(y)=−2ln2(2√y+√4+y),−4<y<4;(b)V(y)=3{π26+lnyln(1+y)−2ln2(1+y)−Li2(11+y)},ρ<y<1/2. |
Proof. The first series U(y) can be reformulated as
U(y)=∞∑k=1(k−1)!k(12)k(−y4)k=∞∑k=1B(k,12)(−y4)k=∞∑k=1(−y/4)kk∫10xk−1dx√1−x=∫10dxx√1−x∞∑k=1(−xy/4)kk=−∫10ln(1+xy4)dxx√1−x=−∫10dTTln(1+(2+y)T+T2(1+T)2)x→4T(1+T)2=∫10dTT{2ln(1+T)−ln(1+4T(√y+√4+y)2)−ln(1+4T(√y−√4+y)2)}=π26+Li2(−4(√y+√4+y)2)+Li2(−4(√y−√4+y)2). |
Then the closed formula for U(y) in the lemma follows by applying the dilogarithmic equation
Li2(−x)+Li2(−x−1)=−π26−ln2x2,wherex>0. |
For the second series, we can rewrite it by reindexing n→n+1 as
V(y)=∞∑n=1ynn2(1+y)3n(3nn)=3y(1+y)3∞∑n=0yn(1+y)3n(3n+2n)1(1+n)2. |
Now specializing Lambert's second series by "α=2 and β=3" and τ→xy/(1+y)3
∞∑n=0xnyn(1+y)3n(3n+2n)=(1+T)31−2T,wherex=T(1+y)3y(1+T)3, |
then integrating twice across this equation, we can proceed with
∞∑n=0yn(1+y)3n(3n+2n)1(1+n)2=∫10dtt∫t0(1+T)31−2Tdx=−∫10(1+T)3lnx1−2Tdx=−(1+y)3y∫y0dT1+Tln(T(1+y)3y(1+T)3)x=T(1+y)3y(1+T)3=−(1+y)3y{∫y0dT1+Tln(T(1+y)3y)−3∫y0ln(1+T)1+TdT}=(1+y)3y{π26+lnyln(1+y)−2ln2(1+y)−Li2(11+y)}. |
By substitution, the closed formula for V(y) in the lemma follows consequently.
Thus, we have the following reduced transformation:
∞∑n=1yn(1+y)3n(3nn){6aH3n−(3a−c)Hn−2(3a+c)Onn}=9aln2(1+y)+6cU(y)+2aV(y). | (5.1) |
By specifying two parameters, a and c, we deduce four independent series
a=0:∞∑n=1yn(1+y)3n(3nn)ˉH2nn=−3U(y),c=0:∞∑n=1yn(1+y)3n(3nn){H3n−H2nn}=32ln2(1+y)+V(y)3,c=3a:∞∑n=1yn(1+y)3n(3nn){H3n−2Onn}=32ln2(1+y)+3U(y)+V(y)3,c=−3a:∞∑n=1yn(1+y)3n(3nn){H3n−Hnn}=32ln2(1+y)−3U(y)+V(y)3. |
For these four equations, replacing U(y) and V(y) first by their algebraic expressions and then relating the resulting equations to that in Theorem 3, we find, after some simplifications, four infinite series identities as below:
Theorem 19 (ρ<y<1/2). Letting Θ(y) be defined as in Theorem 3, the following algebraic identities hold:
(a)∞∑n=1yn(1+y)3n(3nn)ˉH2nn=6ln2(2√y+√4+y),(b)∞∑n=1yn(1+y)3n(3nn)H2nn=π22+3Θ(y)+3lnyln(1+y)−92ln2(1+y)+6ln2(2√y+√4+y),(c)∞∑n=1yn(1+y)3n(3nn)Onn=π24+32Θ(y)+32lnyln(1+y)−94ln2(1+y)+6ln2(2√y+√4+y),(d)∞∑n=1yn(1+y)3n(3nn)H3nn=2π23+3Θ(y)+4lnyln(1+y)−5ln2(1+y)+6ln2(2√y+√4+y)−Li2(11+y). |
By assigning five particular values for y in Theorems 3 and 19, five classes of infinite series identities are displayed in the following corollaries:
Corollary 20 (y→√5−2 in Theorems 3 and 19).
(a) ∞∑n=1(18)n(3nn)ˉH2nn=32ln2(1+√52),(b) ∞∑n=1(18)n(3nn)H2nn=π22+32ln22+3Θ(√5−2)+6ln(1+√5)ln(1+√54),(c) ∞∑n=1(18)n(3nn)Hnn=π22+3Θ(√5−2)+92ln(1+√5)ln(1+√54),(d) ∞∑n=1(18)n(3nn)Onn=π24+32ln22+32Θ(√5−2)+154ln(1+√5)ln(1+√54),(e)∞∑n=1(18)n(3nn)H3nn=2π23+112ln22+3Θ(√5−2)+ln(1+√5)2ln((1+√5)17238)−Li2(1+√54). |
Corollary 21 (y→√5−2√5 in Theorems 3 and 19).
(a) ∞∑n=1(564)n(3nn)ˉH2nn=32ln2(5+√510),(b) ∞∑n=1(564)n(3nn)H2nn=π22+3Θ(√5−2√5)+32ln(5+√58)ln(64+128√5)+32ln2(5+√510),(c) ∞∑n=1(564)n(3nn)Hnn=π22+3Θ(√5−2√5)+32ln(5+√58)ln(64+128√5),(d) ∞∑n=1(564)n(3nn)Onn=π24+32Θ(√5−2√5)+34ln(5+√58)ln(64+128√5)+32ln2(5+√510),(e) ∞∑n=1(564)n(3nn)H3nn=2π23+3Θ(√5−2√5)−Li2(5+√58)+32ln2(5+√510)+ln(5+√58)ln(8(5+√5)754). |
Corollary 22 (y→13 in Theorems 3 and 19).
(a)∞∑n=1(964)n(3nn)ˉH2nn=32ln2(7+√136),(b) ∞∑n=1(964)n(3nn)H2nn=π22−32ln(43)ln(643)+3Θ(13)+32ln2(7+√136),(c) ∞∑n=1(964)n(3nn)Hnn=π22−32ln(43)ln(643)+3Θ(13),(d) ∞∑n=1(964)n(3nn)Onn=π24−34ln(43)ln(643)+32Θ(13)+32ln2(7+√136),(e) ∞∑n=1(964)n(3nn)H3nn=2π23−ln(43)ln(10243)+3Θ(13)−Li2(34)+32ln2(7+√136). |
Corollary 23 (y→3√3−52 in Theorems 3 and 19).
(a) ∞∑n=1(227)n(3nn)ˉH2nn=32ln2(√3−1),(b) ∞∑n=1(227)n(3nn)H2nn=π22+32ln(54)ln(23)+6ln(1+√3)ln(1+√32)+3Θ(3√3−53),(c) ∞∑n=1(227)n(3nn)Hnn=π22+32ln(1+√33)ln(27(1+√3)34)+3Θ(3√3−53),(d) ∞∑n=1(227)n(3nn)Onn=π24+34ln(1+√32)ln((1+√3)52)+34ln(54)ln(23)+32Θ(3√3−53),(e) ∞∑n=1(227)n(3nn)H3nn=2π23+32ln2(1+√32)−Li2(1+√33)+3Θ(3√3−53)+ln(1+√33)ln(35(1+√3)724). |
Corollary 24 (y→3√2−4 in Theorems 3 and 19).
(a) ∞∑n=1(2+√227)n(3nn)ˉH2nn=32ln2(1+√32+√6),(b) ∞∑n=1(2+√227)n(3nn)H2nn=π22+3Θ(3√2−4)+32ln2(1+√32+√6)+32ln(1+√23)ln(27(1+√2)2),(c) ∞∑n=1(2+√227)n(3nn)Hnn=π22+3Θ(3√2−4)+32ln(1+√23)ln(27(1+√2)2),(d) ∞∑n=1(2+√227)n(3nn)Onn=π24+32Θ(3√2−4)+32ln2(1+√32+√6)+34ln(1+√23)ln(27(1+√2)2),(e) ∞∑n=1(2+√227)n(3nn)H3nn=2π23+3Θ(3√2−4)+32ln2(1+√32+√6)−Li2(1+√23)+ln(1+√23)ln(35(1+√2)34). |
By integrating Lambert's series and manipulating the cubic transformations for 3F2-series through the "coefficient extraction method", we have shown several algebraic formulae for infinite series containing binomial coefficient (3nn), harmonic-like numbers, and in particular, a free variable "y". As showcases, three classes of infinite series were examined in detail from Sections 3–5. More variants of these series exist and can be treated analogously. For instance, the transformation formula (3.1) under the replacements "a→1+6ax,c→2cx" becomes
3F2[1+2ax,23+2ax,43+2ax1+3ax+cx,32+3ax−cx|27y4(1+y)3]=(1+y)2+6ax1−2y×4F3[43+2ax,1+6ax,2cx,1−2cx13+2ax,1+3ax+cx,32+3ax−cx|−y4]. |
Then by carrying out the same procedure as in the preceding sections, we can establish the algebraic identities for five infinite series, as in the theorem below:
Theorem 25 (ρ<y<1/2). Letting Λ(y) be the function defined by
Λ(y)=3ln(1−2y)+2−y√y2+4yln(2−y(2+y+√y2+4y)2−y(2+y−√y2+4y)), |
we have the closed algebraic formulae for five infinite series:
(a) ∞∑n=1yn(1+y)3n(3n+1n)ˉH2n+1=2(y−2)(1+y)2(1−2y)√y2+4yln√4+y−√y2,(b) ∞∑n=1yn(1+y)3n(3n+1n)H2n+1=(1+y)21−2y{2y−4√y2+4yln√4+y−√y2−Λ(y)2},(c) ∞∑n=1yn(1+y)3n(3n+1n)Hn=(1+y)24y−2Λ(y),(d) ∞∑n=1yn(1+y)3n(3n+1n)On+1=(1+y)21−2y{2y−4√y2+4yln√4+y−√y2−Λ(y)4},(e) ∞∑n=1yn(1+y)3n(3n+1n)H3n+1=(1+y)21−2y{2y−4√y2+4yln√4+y−√y2−Λ(y)2+ln(1+y)}. |
From the algebraic identities exhibited for the series with harmonic-like numbers of the first order, it is natural to ask: what would happen for the corresponding series involving harmonic-like numbers of the second order? Our computations manifested that the resulting expressions are very complex and convoluted, which can be exemplified by the following attempt made by the authors. Extracting the coefficient of x2 across (3.2), we can derive, after a number of reductions and simplifications, the following formulae:
(a)∞∑n=1yn(1+y)3n(3nn){ˉH22n+H⟨2⟩2n}=6y+62y−1{U(y)+W(y)},(b)∞∑n=1yn(1+y)3n(3nn){(H3n−H2n)2+H⟨2⟩2n−H⟨2⟩3n}=1+y1−2yln2(1+y),(c)∞∑n=1yn(1+y)3n(3nn){(2On−H3n)2+4O⟨2⟩n−H⟨2⟩3n}=1+y1−2y{ln2(1+y)+6ln(1+y)W(y)−10U(y)−12W(y)},(d)∞∑n=1yn(1+y)3n(3nn){(Hn−H3n)2+H⟨2⟩n−H⟨2⟩3n}=1+y1−2y{ln2(1+y)−6ln(1+y)W(y)−2U(y)}; |
where W(y)andU(y) are as in Lemmas 4 and 18, respectively, and W(y) is given by
W(y)=∞∑k=1(k−1)!(12)k(−y4)k¯H2k=(1−√y4+y)ln2(√4+y+√y2)−√y4+yLi2(2√y√y+√4+y). |
There are altogether nine harmonic-like numbers appearing in the above four series
{H2n,O2n,H23n;H⟨2⟩n,O⟨2⟩n,H⟨2⟩3n;HnOn,HnH3n,H3nOn}. |
To evaluate the nine corresponding independent "nuclear series", it is necessary to find five similar independent equations. It seems quite a tough problem that the authors failed to arrive at solutions, even though we did succeed in determining the following long and ugly expression involving six terms in dilogarithms (that prevents us from making further simplifications). Perhaps the only value of this formula lies in its existence.
∞∑n=1(3nn)H⟨2⟩nyn(1+y)3n=∫y03(1+y)3ln(T(1+y)3y(1+T)3)dT(1−2y)(1+T)(T2y+Ty2+3Ty−1)=π2(1+y)2(1−2y)+3(1+y)ln2(1+y)1−2y+3(1+y)lnyln(1+y)1−2y−3(1+y)1−2yLi2(11+y)+9(1+y)1−2y√y4+yln(1+y)ln(2+y+√4y+y22)+3(1+y)2(1−2y)(1+√y4+y){3Li2(−2√y(1+y)(√y−√4+y))−3Li2(−2√y√y−√4+y)+Li2(−2y3/2√y(3+y)−(1+y)√4+y)}+3(1+y)2(1−2y)(1−√y4+y){3Li2(−2√y(1+y)(√y+√4+y))−3Li2(−2√y√y+√4+y)+Li2(−2y3/2√y(3+y)+(1+y)√4+y)}. |
Therefore, to evaluate the series containing binomial coefficient (3nn) and harmonic-like numbers of the second and/or higher orders, one needs to find out different approaches. The interested readers are enthusiastically encouraged to make further explorations.
C. Li: Writing, computation and review; W. Chu: Methodology and supervision. Both of authors have read and approved the final version of the manuscript for publication.
The authors declare they have not used Artificial Intelligence tools in the creation of this article.
The authors express their gratitude to anonymous referees for their careful reading, critical comments and valuable suggestions that make the manuscript improved substantially during the revision.
Prof. Wenchang Chu is the Guest Editor of special issue "Combinatorial Analysis and Mathematical Constants" for AIMS Mathematics. Prof. Wenchang Chu was not involved in the editorial review and the decision to publish this article. The authors declare that there are no conflicts of interest regarding the publication of this paper.
[1] | European Commission, Directive of the European Parliament and of the Council on the Reduction of the Impact of Certain Plastic Products on the Environment. The European Parliament and the Council of the European Union, 2019. Available from: https://www.legislation.gov.uk/eudr/2019/904. |
[2] | UK Government, A Green Future: Our 25 Year Plan to Improve the Environment. UK Government, 2018. Available from: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/693158/25-year-environment-plan.pdf. |
[3] | Carrington D, India Will Abolish all Single-use Plastic by 2022, Vows Narendra Modi. The Guardian, 2018. Available from: https://www.theguardian.com/environment/2018/jun/05/india-will-abolish-all-single-use-plastic-by-2022-vows-narendra-modi. |
[4] |
Eze WU, Madufor IC, Onyeagoro GN, et al. (2020) The effect of Kankara zeolite-Y-based catalyst on some physical properties of liquid fuel from mixed waste plastics (MWPs) pyrolysis. Polym Bull 77: 1399–1415. https://doi.org/10.1007/s00289-019-02806-y doi: 10.1007/s00289-019-02806-y
![]() |
[5] | Akter N, Acott RE, Sattar MG, et al. (1997) Medical waste disposal at BRAC health centres: an environmental study. Res Rep 13: 151–179. |
[6] | Asante B, Yanful E, Yaokumah B (2014) Healthcare waste management; its impact: a case study of the Greater Accra Region, Ghana. IJSTR 3: 106–112. |
[7] | WHO, Guidelines for Safe Disposal of Unwanted Pharmaceuticals in and after Emergencies. World Health Organization, 1999. Available from: https://apps.who.int/iris/handle/10665/42238. |
[8] |
Tsakona M, Anagnostopoulou E, Gidarakos E (2007) Hospital waste management and toxicity evaluation: a case study. Waste Manage 27: 912–920. https://doi.org/10.1016/j.wasman.2006.04.019 doi: 10.1016/j.wasman.2006.04.019
![]() |
[9] |
Hantoko D, Li X, Pariatamby A, et al. (2021) Challenges and practices on waste management and disposal during COVID-19 pandemic. J Environ Manage 286: 112140. https://doi.org/10.1016/j.jenvman.2021.112140 doi: 10.1016/j.jenvman.2021.112140
![]() |
[10] |
Vanapalli KR, Sharma HB, Ranjan VP, et al. (2021) Challenges and strategies for effective plastic waste management during and post COVID-19 pandemic. Sci Total Environ 750: 141514. https://doi.org/10.1016/j.scitotenv.2020.141514 doi: 10.1016/j.scitotenv.2020.141514
![]() |
[11] |
Yousefi M, Oskoei V, Jafari AJ, et al. (2021) Municipal solid waste management during COVID-19 pandemic: effects and repercussions. Environ Sci Pollut R 28: 32200–32209. https://doi.org/10.1007/s11356-021-14214-9 doi: 10.1007/s11356-021-14214-9
![]() |
[12] |
Mahmood QK, Jafree SR, Mukhtar S, et al. (2021) Social media use, self-efficacy, perceived threat, and preventive behavior in times of COVID-19: results of a cross-sectional study in Pakistan. Front Psychol 12: 2354. https://doi.org/10.3389/fpsyg.2021.562042 doi: 10.3389/fpsyg.2021.562042
![]() |
[13] |
Van Fan Y, Jiang P, Hemzal M, et al. (2021) An update of COVID-19 influence on waste management. Sci Total Environ 754: 142014. https://doi.org/10.1016/j.scitotenv.2020.142014 doi: 10.1016/j.scitotenv.2020.142014
![]() |
[14] |
Singh N, Tang Y, Ogunseitan OA (2020) Environmentally sustainable management of used personal protective equipment. Environ Sci Technol 54: 8500–8502. https://doi.org/10.1021/acs.est.0c03022 doi: 10.1021/acs.est.0c03022
![]() |
[15] | CDC, Guidelines for Selection and Use of Personal Protective Equipment (PPE) in Health Settings. Centers for Disease Control, 2020. Available from: https://www.cdc.gov/hai/pdfs/ppe/ppeslides6-29-04.pdf. |
[16] | Revoir WH (1997) Respiratory Protection Handbook, New York: Lewis Publisher. |
[17] | CDRH, Guidance for Industry and FDA Staff: Surgical Masks—Premarket Notification (510(k)) Submissions. Center for Devices and Radiological Health, 2004. Available from: https://www.fda.gov/files/medical%20devices/published/Guidance-for-Industry-and-FDA-Staff--Surgical-Masks---Premarket-Notification-%5B510(k)%5D-Submissions--Guidance-for-Industry-and-FDA-(PDF-Version).pdf. |
[18] | Maturaporn T (1995) Disposable face mask with multiple liquid resistant layers. U.S. Patent, US5467765A. |
[19] | Herrick R, Demont J (1994) Industrial hygiene, In: Rosenstock L, Cullen MR, Textbook of Clinical Occupational and Environmental Medicine, 1 Ed., Philadelphia: WB Saunders Company, 169–193. |
[20] |
Mooibroek H, Cornish K (2000) Alternative sources of natural rubber. Appl Microbiol Biot 53: 355–365. https://doi.org/10.1007/s002530051627 doi: 10.1007/s002530051627
![]() |
[21] |
Wei Y, Zhang H, Wu L, et al. (2017) A review on characterization of molecular structure of natural rubber. MOJ Polym Sci 1: 197–199. https://doi.org/10.15406/mojps.2017.01.00032 doi: 10.15406/mojps.2017.01.00032
![]() |
[22] | Barbara J (2002) Single use vs reusable gowns and drapes. Infection Control Today 1: 3234–3237. |
[23] | Leonas KK (2005) Microorganism protection, In: Scott RA, Textiles for Protection, 1 Ed., Boca Raton: Woodhead Publishing-CRC Press, 441–464. https://doi.org/10.1533/9781845690977.2.441 |
[24] |
Whyte W, Carson W, Hambraeus A (1989) Methods for calculating the efficiency of bacterial surface sampling techniques. J Hosp Infect 13: 33–41. https://doi.org/10.1016/0195-6701(89)90093-5 doi: 10.1016/0195-6701(89)90093-5
![]() |
[25] |
Kilinc FS (2015) A review of isolation gowns in healthcare: fabric and gown properties. J Eng Fibers Fabr 10: 180–190. https://doi.org/10.1177/155892501501000313 doi: 10.1177/155892501501000313
![]() |
[26] | Gupta BS (1988) Effect of structural factors on absorbent characteristics of non-wovens. Tappi J 71: 147–152. |
[27] | Africa News of Sunday, Man Carelessly Disposing PPE by Roadside. GhanaWeb, 2020. Available from: https://www.ghanaweb.com/GhanaHomePage/audio/Abba-Kyari-Everyone-at-the-burial-to-be-tested-for-coronavirus-Public-Health-Dept-928402. |
[28] | Isaac K, Africas pressing need for waste management. DW Report, 2017. Available from: https://www.dw.com/en/africas-pressing-need-for-waste-management/a-39623900. |
[29] | SCMP, Coronavirus Leaves China with Mountains of Medical Waste. South China Morning Post, 2020. Available from: https://amp.scmp.com/news/china/society/article/3074722/coronavirus-leaves-china-mountains-medical-waste. |
[30] | James M, Could the U.S., Like China, Face a Medical Waste Crisis? E & E Newsreporter, 2020. Available from: https://www.eenews.net/articles/could-the-u-s-like-china-face-a-medical-waste-crisis/. |
[31] |
Jang YC, Lee C, Yoon OS, et al. (2006) Medical waste management in Korea. J Environ Manage 80: 107–115. https://doi.org/10.1016/j.jenvman.2005.08.018 doi: 10.1016/j.jenvman.2005.08.018
![]() |
[32] |
Wu A, Peng Y, Huang B, et al. (2020) Genome composition and divergence of the novel coronavirus (2019-nCoV) originating in China. Cell Host Microbe 27: 325–328. https://doi.org/10.1016/j.chom.2020.02.001 doi: 10.1016/j.chom.2020.02.001
![]() |
[33] | World Health Organization, Preferred Product Characteristics for Personal Protective Equipment for the Health Worker on the Frontline Responding to Viral Hemorrhagic Fevers in Tropical Climates. WHO, 2018. Available from: https://apps.who.int/iris/bitstream/handle/10665/272691/9789241514156-eng.pdf. |
[34] | World Health Organization, WHO Director-General's Opening Remarks at the Media Briefing on Covid-19—19 June 2020. WHO, 2020. Available from: https://www.who.int/director-general/speeches/detail/who-director-general-s-opening-remarks-at-the-media-briefing-on-covid-19---19-june-2020. |
[35] | World Health Organization, Shortage of Personal Protective Equipment Endangering Health Workers Worldwide. WHO, 2020. Available from: https://www.who.int/news/item/03-03-2020-shortage-of-personal-protective-equipment-endangering-health-workers-worldwide. |
[36] | World Health Organization, Health-care Waste. WHO, 2018. Available from: https://www.who.int/news-room/fact-sheets/detail/health-care-waste. |
[37] | Ugom M (2020) Managing medical wastes during the Covid-19 pandemic in Nigeria. Int J Waste Resour 10: 386. |
[38] |
Amasuomo E, Baird J (2016) Solid waste management trends in Nigeria. JMS 6: 35. https://doi.org/10.5539/jms.v6n4p35 doi: 10.5539/jms.v6n4p35
![]() |
[39] | Babs-Shomoye F, Kabir R (2016) Health effects of solid waste disposal at a dumpsite on the surrounding human settlements. JPHDC 2: 268–275. |
[40] |
Eze WU, Madufor IC, Onyeagoro GN, et al. (2021) Study on the effect of Kankara zeolite-Y-based catalyst on the chemical properties of liquid fuel from mixed waste plastics (MWPs) pyrolysis. Polym Bull 78: 377–398. https://doi.org/10.1007/s00289-020-03116-4 doi: 10.1007/s00289-020-03116-4
![]() |
[41] |
Eze WU, Umunakwe R, Obasi HC, et al. (2021) Plastics waste management: A review of pyrolysis technology. Clean Technol Recy 1: 50–69. https://doi.org/10.3934/ctr.2021003 doi: 10.3934/ctr.2021003
![]() |
[42] |
Kaminsky W, Mennerich C, Zhang Z (2009) Feedstock recycling of synthetic and natural rubber by pyrolysis in a fluidized bed. J Anal Appl Pyrol 85: 334–337. https://doi.org/10.1016/j.jaap.2008.11.012 doi: 10.1016/j.jaap.2008.11.012
![]() |
[43] |
Wang J, Jiang J, Wang X, et al. (2019) Catalytic conversion of rubber wastes to produce aromatic hydrocarbons over USY zeolites: Effect of SiO2/Al2O3 mole ratio. Energ Convers Manage 197: 111857. https://doi.org/10.1016/j.enconman.2019.111857 doi: 10.1016/j.enconman.2019.111857
![]() |
[44] |
Abbas-Abadi MS, Haghighi MN, Yeganeh H, et al. (2014) Evaluation of pyrolysis process parameters on polypropylene degradation products. J Anal Appl Pyrol 109: 272–277. https://doi.org/10.1016/j.jaap.2014.05.023 doi: 10.1016/j.jaap.2014.05.023
![]() |
[45] |
Ahmad I, Khan MI, Khan H, et al. (2015) Pyrolysis study of polypropylene and polyethylene into premium oil products. Int J Green Energy 12: 663–671. https://doi.org/10.1080/15435075.2014.880146 doi: 10.1080/15435075.2014.880146
![]() |
[46] |
Fakhrhoseini S, Dastanian M (2013) Pyrolysis of LDPE, PP and PET plastic wastes at different conditions and prediction of products using NRTL activity coefficient model. J Chem 2013: 487676. https://doi.org/10.1155/2013/487676 doi: 10.1155/2013/487676
![]() |
[47] |
Eze WU, Madufor IC, Onyeagoro GN, et al. (2020) The effect of Kankara zeolite-Y-based catalyst on some physical properties of liquid fuel from mixed waste plastics (MWPs) pyrolysis. Polym Bull 77: 1399–1415. https://doi.org/10.1007/s00289-019-02806-y doi: 10.1007/s00289-019-02806-y
![]() |
[48] |
Donaj PJ, Kaminsky W, Buzeto F, et al. (2012) Pyrolysis of polyolefins for increasing the yield of monomers' recovery. Waste Manage 32: 840–846. https://doi.org/10.1016/j.wasman.2011.10.009 doi: 10.1016/j.wasman.2011.10.009
![]() |
[49] |
Pratama NN, Saptoadi H (2014) Characteristics of waste plastics pyrolytic oil and its applications as alternative fuel on four cylinder diesel engines. Int J Renewable Energy Dev 3: 13–20. https://doi.org/10.14710/ijred.3.1.13-20 doi: 10.14710/ijred.3.1.13-20
![]() |
[50] |
Li H, Jiang X, Cui H, et al. (2015) Investigation on the co-pyrolysis of waste rubber/plastics blended with a stalk additive. J Anal Appl Pyrol 115: 37–42. https://doi.org/10.1016/j.jaap.2015.07.004 doi: 10.1016/j.jaap.2015.07.004
![]() |
[51] |
Hussain Z, Khan A, Naz MY, et al. (2021) Borax-catalyzed valorization of waste rubber and polyethylene using pyrolysis and copyrolysis reactions. Asia-Pac J Chem Eng 16: e2696. https://doi.org/10.1002/apj.2696 doi: 10.1002/apj.2696
![]() |
[52] |
Park J, Díaz-Posada N, Mejía-Dugand S (2018) Challenges in implementing the extended producer responsibility in an emerging economy: The end-of-life tire management in Colombia. J Cleaner Prod 189: 754–762. https://doi.org/10.1016/j.jclepro.2018.04.058 doi: 10.1016/j.jclepro.2018.04.058
![]() |
[53] |
Banguera LA, Sepúlveda JM, Ternero R, et al. (2018) Reverse logistics network design under extended producer responsibility: The case of out-of-use tires in the Gran Santiago city of Chile. Int J Prod Econ 205: 193–200. https://doi.org/10.1016/j.ijpe.2018.09.006 doi: 10.1016/j.ijpe.2018.09.006
![]() |
[54] |
Zarei M, Taghipour H, Hassanzadeh Y (2018) Survey of quantity and management condition of end-of-life tires in Iran: a case study in Tabriz. J Mater Cycles Waste Manage 20: 1099–1105. https://doi.org/10.1007/s10163-017-0674-5 doi: 10.1007/s10163-017-0674-5
![]() |
[55] |
Yagboyaju DA, Akinola AO (2019) Nigerian state and the crisis of governance: A critical exposition. SAGE Open 9: 1–10. https://doi.org/10.1177/2158244019865810 doi: 10.1177/2158244019865810
![]() |
[56] | Leguil-Bayart JF (2009) The State in Africa: the Politics of the Belly, Oxford: Polity Press. |
[57] | Uzodikeo UO (2009) Leadership and governance in Africa. AFFRIKA Journal of Politics, Economics and Society 1: 3–9. |
[58] | Renault V (2022) SWOT analysis: strengths, weaknesses, opportunities, and threats, Community Tool Box: Assessing Community Needs and Resources. Kansas: The University of Kansas. |
[59] | Muniafu M, Kimani NN, Mwangi J (2013) Renewable Energy Governance: Complexities and Challenges, New York: Springer, 397. |
[60] |
Patil DP, Bakthavachalu B, Schoenberg DR (2014) Poly (A) polymerase-based poly (A) length assay. Methods Mol Biol 1125: 13–23. https://doi.org/10.1007/978-1-62703-971-0_2 doi: 10.1007/978-1-62703-971-0_2
![]() |
[61] | Lino FAM, Ismail KAR (2017) Recycling and thermal treatment of MSW in a developing country. IOSRJEN 7: 2278–8719. |
[62] |
Aubert J, Husson B, Saramone N (2006) Utilization of municipal solid waste incineration (MSWI) fly ash in blended cement: Part 1: Processing and characterization of MSWI fly ash. J Hazard Mater 136: 624–631. https://doi.org/10.1016/j.jhazmat.2005.12.041 doi: 10.1016/j.jhazmat.2005.12.041
![]() |
[63] |
Panda AK, Singh RK, Mishra DK (2010) Thermolysis of waste plastics to liquid fuel: A suitable method for plastic waste management and manufacture of value added products—A world prospective. Renewable Sustainable Energy Rev 14: 233–248. https://doi.org/10.1016/j.rser.2009.07.005 doi: 10.1016/j.rser.2009.07.005
![]() |
[64] | Alonso-Torres B, Rodrigez-Martinez A, Domínguez-Patino ML (2010) Design of municipal solid waste incinerator based on hierarchical methodology. Chem Eng Trans 21: 1471–1476. |
[65] | World Health Organization, Findings of an Assessment of Small-scale Incinerators for Healthcare Waste. WHO, 2004. Available from: https://apps.who.int/iris/handle/10665/68775. |
[66] |
Das AK, Islam N, Billah M, et al. (2021) COVID-19 pandemic and healthcare solid waste management strategy—A mini-review. Sci Total Environ 778: 146220. https://doi.org/10.1016/j.scitotenv.2021.146220 doi: 10.1016/j.scitotenv.2021.146220
![]() |
[67] | Tsukiji M, Gamaralalage PJD, Pratomo ISY, et al. (2020) Waste management during the COVID-19 pandemic from response to recovery. United Nations Environment Programme, International Environmental Technology Centre (IETC) IGES Center Collaborating with UNDP on Environmental Technologies (CCET). |
[68] | Chu LM (2008) Landfills, In: Jorgensen SE, Fath B, Encyclopedia of Ecology, Netherlands: Elsevier, 2099–2103. https://doi.org/10.1016/B978-008045405-4.00345-1 |
[69] | Stauffer B, Landfills, SSWM—Find Tools for Sustainable Sanitation and Water Management. International Solid Waste Association Report, 2020. Available from: https://sswm.info/water-nutrient-cycle/wastewater-treatment/hardwares/solid-waste/landfills. |
[70] | Waste Management Bioreactor Program Report, The Bioreactor Landfill—Next Generation Landfill Technology. EPA, 2004. Available from: https://www.epa.gov/landfills/bioreactor-landfills. |
[71] | UNEP, A Directory of Environmentally Sound Technologies for the Integrated Management of Solid, Liquid and Hazardous Waste for Small Island Developing States (SIDS) in the Pacific Region. International Waters Learning Exchange & Resource Network Report, 2021. Available from: https://iwlearn.net/documents/3901. |
[72] |
Fereja WM, Chemeda DD (2021) Status, characterization, and quantification of municipal solid waste as a measure towards effective solid waste management: The case of Dilla Town, Southern Ethiopia. J Air Waste Manage 72: 187–201. https://doi.org/10.1080/10962247.2021.1923585 doi: 10.1080/10962247.2021.1923585
![]() |
[73] |
Okwesili J, Iroko C (2016) Urban solid waste management and environmental sustainability in Abakaliki Urban, Nigeria. Eur Sci J 12: 160. https://doi.org/10.19044/esj.2016.v12n23p155 doi: 10.19044/esj.2016.v12n23p155
![]() |
[74] |
Ojuri OO, Ajijola TO, Akinwumi II (2018) Design of an engineered landfill as possible replacement for an existing dump at Akure, Nigeria. African J Sci Technol Innov Dev 10: 835–843. https://doi.org/10.1080/20421338.2018.1523827 doi: 10.1080/20421338.2018.1523827
![]() |
[75] | Sonia A, Many in Northern Syria Live off Rubbish Dumps. The Pulse of the Middle East, Al-Monitor, 2020. Available from: https://www.al-monitor.com/originals/2020/03/syria-north-chidlren-women-begging-garbage-collect-poverty.html. |
[76] |
De Feo G, De Gisi S, Williams ID (2013) Public perception of odour and environmental pollution attributed to MSW treatment and disposal facilities: A case study. Waste Manage 33: 974–987. https://doi.org/10.1016/j.wasman.2012.12.016 doi: 10.1016/j.wasman.2012.12.016
![]() |
[77] |
Kumar S, Gaikwad SA, Shekdar AV, et al. (2004) Estimation method for national methane emission from solid waste landfills. Atmos Environ 38: 3481–3487. https://doi.org/10.1016/j.atmosenv.2004.02.057 doi: 10.1016/j.atmosenv.2004.02.057
![]() |
[78] |
Njoku PO, Edokpayi JN, Odiyo JO (2019) Health and environmental risks of residents living close to a landfill: A case study of Thohoyandou Landfill, Limpopo Province, South Africa. Int J Environ Res Public Health 16: 2125. https://doi.org/10.3390/ijerph16122125 doi: 10.3390/ijerph16122125
![]() |
[79] | Toyese O, Ademola O, Olusanya JJ (2021) Preliminary investigation on the screening of selected metallic oxides, M2O3 (M = Fe, La, and Gd) for the capture of carbon monoxide using a computational approach. JESC 3: 1–14. |
[80] | Pakistan Today, WB to Assist in Making Landfill Site for Karachi. Pakistan Today, 2019. Available from: https://archive.pakistantoday.com.pk/2019/10/09/wb-to-assist-in-making-landfill-site-for-karachi/. |
[81] | WOIMAD, Rowning in Waste—Case Accra, Ghana. WOIMA Corporation, 2021. Available from: https://woimacorporation.com/drowning-in-waste-case-accra-ghana/. |
[82] |
Galadima A, Garba ZN, Ibrahim BM, et al. (2011) Biofuels production in Nigeria: The policy and public opinions. J Sustain Dev 4: 22–31. https://doi.org/10.5539/jsd.v4n4p22 doi: 10.5539/jsd.v4n4p22
![]() |
[83] | Toyese O, Jibiril BEY (2016) Design and feasibility study of a 5MW bio-power plant in Nigeria. Int J Renew Energy Res 6: 1496–1505. |
1. | Chunli Li, Wenchang Chu, Generating Functions for Binomial Series Involving Harmonic-like Numbers, 2024, 12, 2227-7390, 2685, 10.3390/math12172685 | |
2. | Artūras Dubickas, On the Range of Arithmetic Means of the Fractional Parts of Harmonic Numbers, 2024, 12, 2227-7390, 3731, 10.3390/math12233731 |