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Research article

Construction of blocked designs with multi block variables

  • When experimental units are inhomogeneous, blocking the experimental units into categories is crucial so as to estimate the treatment effects precisely. In practice, the inhomogeneity often comes from different sources known as block variables in design terminology. The paper considers the blocking problems with multi block variables. The construction methods of the optimal blocked regular 2nm designs with multi block variables under the general minimum lower order confounding criterion for 5N16+1nN1 are provided, where N=2nm.

    Citation: Yuna Zhao. Construction of blocked designs with multi block variables[J]. AIMS Mathematics, 2021, 6(6): 6293-6308. doi: 10.3934/math.2021369

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  • When experimental units are inhomogeneous, blocking the experimental units into categories is crucial so as to estimate the treatment effects precisely. In practice, the inhomogeneity often comes from different sources known as block variables in design terminology. The paper considers the blocking problems with multi block variables. The construction methods of the optimal blocked regular 2nm designs with multi block variables under the general minimum lower order confounding criterion for 5N16+1nN1 are provided, where N=2nm.



    In [12,Definition 11.2] and [18,p. 134,Theorem A], the second kind Bell polynomials Bn,k for nk0 are defined by

    Bn,k(x1,x2,,xnk+1)=Nnk+10n!nk+1i=1i!nk+1i=1(xii!)i,

    where N0={0}N, the sum is taken over =(1,2,,nk+1) with iN0 satisfying nk+1i=1i=k and nk+1i=1ii=n. This kind of polynomials are very important in combinatorics, analysis, and the like. See the review and survey article [53] and closely related references therein.

    In [36,pp. 13–15], when studying Grothendieck's inequality and completely correlation-preserving functions, Oertel obtained the interesting identity

    2nk=1(1)k(2n+k)!k!B2n,k(0,16,0,340,0,5112,,1+(1)k+12[(2nk)!!]2(2nk+2)!)=(1)n

    for nN, where

    Bn,k(x1,x2,,xnk+1)=k!n!Bn,k(1!x1,2!x2,,(nk+1)!xnk+1). (1.1)

    In [36,p. 15], Oertel wrote that "However, already in this case we don't know a closed form expression for the numbers

    B2n,k(0,16,0,340,0,5112,,1+(1)k+12[(2nk)!!]2(2nk+2)!). (1.2)

    An even stronger problem appears in the complex case, since already a closed-form formula for the coefficients of the Taylor series of the inverse of the Haagerup function is still unknown''.

    By virtue of the relation (1.1), we see that, to find a closed-form formula for the sequence (1.2), it suffices to discover a closed-form formula for

    B2n,k(0,13,0,95,0,2257,,1+(1)k+12[(2nk)!!]22nk+2). (1.3)

    In this paper, one of our aims is to derive closed-form formulas for the sequence (1.3). The first main result can be stated as the following theorem.

    Theorem 1.1. For k,n0, mN, and xmC, we have

    B2n+1,k(0,x2,0,x4,,1+(1)k2x2nk+2)=0. (1.4)

    For k,nN, we have

    B2n,2k1(0,13,0,95,0,2257,,0,[(2n2k+1)!!]22n2k+3)=22n(2k1)![kp=1(4)p1(2k12p1)(2n+2p12p1)2p2q=0T(n+p1;q,2p2;12)k1p=1(1)p1(2k12p)(2n+2p2p)2p2q=0T(n+p1;q,2p2;1)]

    and

    B2n,2k(0,13,0,95,0,2257,,[(2n2k1)!!]22n2k+1,0)=22n(2k)![kp=1(1)p1(2k2p)(2n+2p2p)2p2q=0T(n+p1;q,2p2;1)kp=1(4)p1(2k2p1)(2n+2p12p1)2p2q=0T(n+p1;q,2p2;12)],

    where s(n,k), which can be generated by

    xn=nm=0s(n,m)xm, (1.5)

    denote the first kind Stirling numbers and

    T(r;q,j;ρ)=(1)q[rm=q(ρ)ms(r,m)(mq)][rm=jq(ρ)ms(r,m)(mjq)]. (1.6)

    In Section 2, for proving Theorem 1.1, we will establish two general expressions for power series expansions of (arcsinx)21 and (arcsinx)2 respectively.

    In Section 3, with the aid of general expressions for power series expansions of the functions (arcsinx)21 and (arcsinx)2 established in Section 2, we will prove Theorem 1.1 in details.

    In Section 4, basing on arguments in [20,p. 308] and [28,Section 2.4] and utilizing general expressions for power series expansions of (arcsinx)21 and (arcsinx)2 established in Section 2, we will derive series representations of generalized logsine functions which were originally introduced in [34] and have been investigating actively, deeply, and systematically by mathematicians [9,10,14,15,16,17,29,30,31,37,38,57] and physicists [3,19,20,28].

    Finally, in Section 5, we will list several remarks on our main results and related stuffs.

    To prove Theorem 1.1, we need to establish the following general expressions of the power series expansions of (arcsinx) for N.

    Theorem 2.1. For N and |x|<1, the functions (arcsinx) can be expanded into power series

    (arcsinx)21=(4)1n=04n(2n)![22q=0T(n+1;q,22;12)]x2n+21(2n+2121) (2.1)

    or

    (arcsinx)2=(1)1n=04n(2n)![22q=0T(n+1;q,22;1)]x2n+2(2n+22), (2.2)

    where s(n,k) denotes the first kind Stirling numbers generated in (1.5) and T(r;q,j;ρ) is defined by (1.6).

    Proof. In [4,pp. 262–263,Proposition 15], [7,p. 3], [20,p. 308], and [28,pp. 49–50], it was stated that the generating expression for the series expansion of (arcsinx)n with nN is

    exp(tarcsinx)==0b(t)x!,

    where b0(t)=1, b1(t)=t, and

    b2(t)=1k=0[t2+(2k)2],b2+1(t)=tk=1[t2+(2k1)2]

    for N. This means that, when writing

    b(t)=k=0β,ktk,0,

    where β0,0=1, β2,0=0, β2,2k+1=0, and β21,2k=0 for k0 and 1, we have

    =0(arcsinx)t!==0x!k=0β,ktk=k=0=kx!β,ktk==0[m=βm,xmm!]t.

    Equating coefficients of t gives

    (arcsinx)=!m=βm,xmm!=!n=0βn+,xn+(n+)!,N. (2.3)

    It is not difficult to see that

    b2(t)=41t2(1it2)1(1+it2)1andb2+1(t)=4t(12it2)(12+it2),

    where i=1 is the imaginary unit and

    (z)n=n1=0(z+)={z(z+1)(z+n1),n11,n=0

    is called the rising factorial of zC, while

    zn=n1=0(z)={z(z1)(zn+1),n11,n=0 (2.4)

    is called the falling factorial of zC. Making use of the relation

    (z)n=(1)nznorzn=(1)n(z)n

    in [52,p. 167], we acquire

    b2(t)=41t2it211it211andb2+1(t)=4tit212it212.

    Utilizing the relation (1.5) in [59,p. 19,(1.26)], we obtain

    b2(t)=41t21m=0s(1,m)2m(it2)m1m=0(1)ms(1,m)2m(it+2)m=41t21m=0s(1,m)2mmk=0(mk)iktk(2)mk1m=0(1)ms(1,m)2mmk=0(mk)iktk2mk=41t21m=0(1)ms(1,m)mk=0(1)k2k(mk)iktk1m=0(1)ms(1,m)mk=012k(mk)iktk=41t21k=0[1m=k(1)m+ks(1,m)2k(mk)]iktk1k=0[1m=k(1)ms(1,m)2k(mk)]iktk=41t22(1)k=0kq=0[1m=q(1)m+qs(1,m)2q(mq)1m=kq(1)ms(1,m)2kq(mkq)]iktk=41t22(1)k=012kkq=0[1m=q(1)m+qs(1,m)(mq)1m=kq(1)ms(1,m)(mkq)]iktk=412(1)k=0ik2k[kq=0(1m=q(1)ms(1,m)(mq))1m=kq(1)ms(1,m)(mkq)]tk+2=412(1)k=0ik2k[kq=0T(1;q,k;1)]tk+2

    and

    b2+1(t)=4tm=0s(,m)2m(it1)mm=0(1)ms(,m)2m(it+1)m=4tm=0s(,m)2mmk=0(1)mk(mk)iktkm=0(1)ms(,m)2mmk=0(mk)iktk=4tk=0[m=k(1)ms(,m)2m(mk)](i)ktkk=0[m=k(1)ms(,m)2m(mk)]iktk=42k=0ik[kq=0(1)q(m=q(1)ms(,m)2m(mq))m=kq(1)ms(,m)2m(mkq)]tk+1=42k=0ik[kq=0T(;q,k;12)]tk+1.

    This means that

    2k=0β2,ktk=2(1)k=2β2,k+2tk+2=2(1)k=0β2,k+2tk+2=412(1)k=0ik2k[kq=0T(1;q,k;1)]tk+2

    and

    2+1k=0β2+1,ktk=2k=1β2+1,k+1tk+1=2k=0β2+1,k+1tk+1=42k=0ik[kq=0T(;q,k;12)]tk+1.

    Further equating coefficients of tk+2 and tk+1 respectively arrives at

    β2,k+2=41ik2kkq=0T(1;q,k;1)andβ2+1,k+1=4ikkq=0T(;q,k;12)

    for k0.

    Replacing by 21 for N in (2.3) leads to

    (arcsinx)21=(21)!n=0βn+21,21xn+21(n+21)!=(21)!n=0β2n+21,21x2n+21(2n+21)!=(21)!n=0[4n+1i2(1)2(1)q=0T(n+1;q,22;12)]x2n+21(2n+21)!=(1)141(21)!n=0[4n2(1)q=0T(n+1;q,22;12)]x2n+21(2n+21)!=(4)1n=04n(2n)![22q=0T(n+1;q,22;12)]x2n+21(2n+2121).

    Replacing by 2 for N in (2.3) leads to

    (arcsinx)2=(2)!n=0βn+2,2xn+2(n+2)!=(2)!n=0β2n+2,2x2n+2(2n+2)!=(1)1(2)!n=0[4n2(1)q=0T(n+1;q,22;1)]x2n+2(2n+2)!=(1)1n=04n(2n)![22q=0T(n+1;q,22;1)]x2n+2(2n+22).

    The proof of Theorem 2.1 is complete.

    We now start out to prove Theorem 1.1.

    In the last line of [18,p. 133], there exists the formula

    1k!(m=1xmtmm!)k=n=kBn,k(x1,x2,,xnk+1)tnn! (3.1)

    for k0. When taking x2m1=0 for mN, the left hand side of the formula (3.1) is even in t(,) for all k0. Therefore, the formula (1.4) is valid.

    Ones know that the power series expansion

    arcsint==0[(21)!!]2(2+1)!t2+1,|t|<1 (3.2)

    is valid, where (1)!!=1. This implies that

    B2n,k(0,13,0,95,0,2257,,1+(1)k+12[(2nk)!!]22nk+2)=B2n,k((arcsint)|t=02,(arcsint)|t=03,(arcsint)(4)|t=04,,(arcsint)(2nk+2)|t=02nk+2).

    Employing the formula

    Bn,k(x22,x33,,xnk+2nk+2)=n!(n+k)!Bn+k,k(0,x2,x3,,xn+1)

    in [18,p. 136], we derive

    B2n,k(0,13,0,95,0,2257,,1+(1)k+12[(2nk)!!]22nk+2)=(2n)!(2n+k)!B2n+k,k(0,(arcsint)|t=0,(arcsint)|t=0,,(arcsint)(2n+1)|t=0).

    Making use of the formula (3.1) yields

    n=0Bn+k,k(x1,x2,,xn+1)k!n!(n+k)!tn+kn!=(m=1xmtmm!)k,n=0Bn+k,k(x1,x2,,xn+1)(n+kk)tn+kn!=(m=1xmtmm!)k,Bn+k,k(x1,x2,,xn+1)=(n+kk)lim

    Setting x_1 = 0 and x_m = (\arcsin t)^{(m)}|_{t = 0} for m\ge2 gives

    \begin{align*} \frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\Biggl[\sum\limits_{m = 0}^\infty x_{m+1}\frac{t^{m}}{(m+1)!}\Biggr]^k & = \frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\Biggl[\frac{1}{t} \sum\limits_{m = 2}^\infty (\arcsin t)^{(m)}|_{t = 0} \frac{t^m}{m!}\Biggr]^k\\ & = \frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\biggl(\frac{\arcsin t-t}{t}\biggr)^k\\ & = \frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\sum\limits_{p = 0}^{k}(-1)^{k-p}\binom{k}{p}\biggl(\frac{\arcsin t}{t}\biggr)^p\\ & = \sum\limits_{p = 1}^{k}(-1)^{k-p}\binom{k}{p}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\biggl(\frac{\arcsin t}{t}\biggr)^p. \end{align*}

    Accordingly, we obtain

    \begin{gather*} \lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\Biggl[\frac{1}{t}\sum\limits_{m = 2}^\infty (\arcsin t)^{(m)}|_{t = 0}\frac{t^m}{m!}\Biggr]^{2k-1} = \sum\limits_{p = 1}^{2k-1}(-1)^{2k-p-1}\binom{2k-1}{p}\lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\biggl(\frac{\arcsin t}{t}\biggr)^p\\ = \sum\limits_{p = 1}^{k}\binom{2k-1}{2p-1}\lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\biggl(\frac{\arcsin t}{t}\biggr)^{2p-1} -\sum\limits_{p = 1}^{k-1}\binom{2k-1}{2p}\lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\biggl(\frac{\arcsin t}{t}\biggr)^{2p} \end{gather*}

    and

    \begin{gather*} \lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\Biggl[\frac{1}{t}\sum\limits_{m = 2}^\infty (\arcsin t)^{(m)}|_{t = 0}\frac{t^m}{m!}\Biggr]^{2k} = \sum\limits_{p = 1}^{2k}(-1)^{2k-p}\binom{2k}{p}\lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\biggl(\frac{\arcsin t}{t}\biggr)^p\\ = \sum\limits_{p = 1}^{k}\binom{2k}{2p}\lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\biggl(\frac{\arcsin t}{t}\biggr)^{2p} -\sum\limits_{p = 1}^{k}\binom{2k}{2p-1}\lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\biggl(\frac{\arcsin t}{t}\biggr)^{2p-1}. \end{gather*}

    From the power series expansions (2.1) and (2.2) in Theorem 2.1, it follows that

    \begin{align*} \lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\biggl(\frac{\arcsin t}{t}\biggr)^{2p-1} & = (-1)^{p-1}4^{p-1}(2p-1)!\\ &\quad\times\lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\sum\limits_{j = 0}^{\infty}\Biggl[4^{j}\sum\limits_{q = 0}^{2p-2} T\biggl(j+p-1;q,2p-2;\frac12\biggr)\Biggr] \frac{t^{2j}}{(2j+2p-1)!}\\ & = (-1)^{p-1}\frac{4^{n+p-1}}{\binom{2n+2p-1}{2n}} \sum\limits_{q = 0}^{2p-2}T\biggl(n+p-1;q,2p-2;\frac12\biggr) \end{align*}

    and

    \begin{align*} \lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\biggl(\frac{\arcsin t}{t}\biggr)^{2p} & = (-1)^{p-1}(2p)!\lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\sum\limits_{j = 0}^{\infty}\Biggl[4^{j} \sum\limits_{q = 0}^{2p-2} T(j+p-1;q,2p-2;1)\Biggr]\frac{t^{2j}}{(2j+2p)!}\\ & = (-1)^{p-1}\frac{4^{n}}{\binom{2n+2p}{2n}} \sum\limits_{q = 0}^{2p-2} T(n+p-1;q,2p-2;1). \end{align*}

    Therefore, we arrive at

    \begin{align*} \lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\Biggl[\frac{1}{t}\sum\limits_{m = 2}^\infty (\arcsin t)^{(m)}|_{t = 0}\frac{t^m}{m!}\Biggr]^{2k-1} & = 4^n\sum\limits_{p = 1}^{k}(-4)^{p-1}\frac{\binom{2k-1}{2p-1}}{\binom{2n+2p-1}{2p-1}}\sum\limits_{q = 0}^{2p-2}T\biggl(n+p-1;q,2p-2;\frac12\biggr)\\ &\quad-4^{n}\sum\limits_{p = 1}^{k-1}(-1)^{p-1}\frac{\binom{2k-1}{2p}}{\binom{2n+2p}{2p}} \sum\limits_{q = 0}^{2p-2} T(n+p-1;q,2p-2;1) \end{align*}

    and

    \begin{align*} \lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\Biggl[\frac{1}{t}\sum\limits_{m = 2}^\infty (\arcsin t)^{(m)}|_{t = 0}\frac{t^m}{m!}\Biggr]^{2k} & = 4^{n}\sum\limits_{p = 1}^{k}(-1)^{p-1}\frac{\binom{2k}{2p}}{\binom{2n+2p}{2p}} \sum\limits_{q = 0}^{2p-2} T(n+p-1;q,2p-2;1)\\ &\quad-4^{n}\sum\limits_{p = 1}^{k}(-4)^{p-1}\frac{\binom{2k}{2p-1}}{\binom{2n+2p-1}{2p-1}} \sum\limits_{q = 0}^{2p-2}T\biggl(n+p-1;q,2p-2;\frac12\biggr). \end{align*}

    Consequently, we acquire

    \begin{align*} &\quad{\rm{B}}_{2n,2k-1}\biggl(0,\frac{1}{3},0,\frac{9}{5},0,\frac{225}{7},\dotsc, 0, \frac{[(2n-2k+1)!!]^2}{2n-2k+3}\biggr)\\ & = \frac{(2n)!}{(2n+2k-1)!}{\rm{B}}_{2n+2k-1,2k-1}\bigl(0,(\arcsin t)''|_{t = 0},(\arcsin t)'''|_{t = 0},\dotsc,(\arcsin t)^{(2n+1)}|_{t = 0}\bigr)\\ & = \frac{(2n)!}{(2n+2k-1)!}\binom{2n+2k-1}{2k-1} \lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\Biggl(\frac{1}{t}\sum\limits_{m = 2}^\infty (\arcsin t)^{(m)}|_{t = 0}\frac{t^m}{m!}\Biggr)^{2k-1}\\ & = \frac{1}{(2k-1)!}\Biggl[4^n\sum\limits_{p = 1}^{k}(-4)^{p-1}\frac{\binom{2k-1}{2p-1}}{\binom{2n+2p-1}{2p-1}} \sum\limits_{q = 0}^{2p-2}T\biggl(n+p-1;q,2p-2;\frac12\biggr)\\ &\quad-4^{n}\sum\limits_{p = 0}^{k-1}(-1)^{p-1}\frac{\binom{2k-1}{2p}}{\binom{2n+2p}{2p}} \sum\limits_{q = 0}^{2p-2} T(n+p-1;q,2p-2;1)\Biggr] \end{align*}

    and

    \begin{align*} &\quad{\rm{B}}_{2n,2k}\biggl(0,\frac{1}{3},0,\frac{9}{5},0,\frac{225}{7},\dotsc,\frac{[(2n-2k-1)!!]^2}{2n-2k+1},0\biggr)\\ & = \frac{(2n)!}{(2n+2k)!}{\rm{B}}_{2n+2k,2k}\bigl(0,(\arcsin t)''|_{t = 0},(\arcsin t)'''|_{t = 0},\dotsc,(\arcsin t)^{(2n+1)}|_{t = 0}\bigr)\\ & = \frac{(2n)!}{(2n+2k)!}\binom{2n+2k}{2k}\lim\limits_{t\to0}\frac{{\rm{d}}^{2n}}{{\rm{d}} t^{2n}}\Biggl(\frac{1}{t}\sum\limits_{m = 2}^\infty (\arcsin t)^{(m)}|_{t = 0}\frac{t^m}{m!}\Biggr)^{2k}\\ & = \frac{1}{(2k)!}\Biggl[4^{n}\sum\limits_{p = 1}^{k}(-1)^{p-1}\frac{\binom{2k}{2p}}{\binom{2n+2p}{2p}} \sum\limits_{q = 0}^{2p-2} T(n+p-1;q,2p-2;1)\\ &\quad-4^{n}\sum\limits_{p = 1}^{k}(-4)^{p-1}\frac{\binom{2k}{2p-1}}{\binom{2n+2p-1}{2p-1}} \sum\limits_{q = 0}^{2p-2} T\biggl(n+p-1;q,2p-2;\frac12\biggr)\Biggr]. \end{align*}

    The proof of Theorem 1.1 is complete.

    The logsine function

    \begin{equation*} {\rm{Ls}}_j(\theta) = -\int_{0}^{\theta}\biggl(\ln\biggl|2\sin\frac{x}{2}\biggr|\biggr)^{j-1}{\rm{d}} x \end{equation*}

    and generalized logsine function

    \begin{equation*} {\rm{Ls}}_j^{(\ell)}(\theta) = -\int_{0}^{\theta}x^\ell\biggl(\ln\biggl|2\sin\frac{x}{2}\biggr|\biggr)^{j-\ell-1}{\rm{d}} x \end{equation*}

    were introduced originally in [34,pp. 191–192], where \ell, j are integers, j\ge\ell+1\ge1 , and \theta is an arbitrary real number. There have been many papers such as [3,9,10,14,15,16,17,19,20,28,29,30,31,37,38,57] devoted to investigation and applications of the (generalized) logsine functions in mathematics, physics, engineering, and other mathematical sciences.

    Theorem 4.1. Let \langle z\rangle_n for z\in\mathbb{C} and n\in\{0\}\cup\mathbb{N} denote the falling factorial defined by (2.4) and let T(r; q, j;\rho) be defined by (1.6). In the region 0 < \theta\le\pi and for j, \ell\in\mathbb{N} , generalized logsine functions {\rm{Ls}}_j^{(\ell)}(\theta) have the following series representations:

    1. for j\ge2\ell+1\ge3 ,

    \begin{equation} \begin{aligned} {\rm{Ls}}_j^{(2\ell-1)}(\theta)& = -\frac{\theta^{2\ell}}{2\ell}\biggl[\ln\biggl(2\sin\frac{\theta}{2}\biggr)\biggr]^{j-2\ell} -(-1)^{\ell}(j-2\ell)(2\ell-1)!(\ln2)^{j-1}\biggl(\frac{2\sin\frac{\theta}{2}}{\ln2}\biggr)^{2\ell}\\ &\quad\times \sum\limits_{n = 0}^{\infty}\frac{\bigl(2\sin\frac{\theta}{2}\bigr)^{2n}}{(2n+2\ell)!} \Biggl[\sum\limits_{q = 0}^{2\ell-2} T(n+\ell-1;q,2\ell-2;1)\Biggr]\\ &\quad\times\Biggl[\sum\limits_{\alpha = 0}^{j-2\ell-1}\biggl(\frac{\ln\sin\frac{\theta}{2}}{\ln2}\biggr)^{\alpha} \binom{j-2\ell-1}{\alpha} \sum\limits_{k = 0}^{\alpha}\frac{(-1)^k\langle\alpha\rangle_{k}} {(2n+2\ell)^{k+1}\bigl(\ln\sin\frac{\theta}{2}\bigr)^{k}}\Biggr]; \end{aligned} \end{equation} (4.1)

    2. for j\ge2\ell+2\ge4 ,

    \begin{equation} \begin{aligned} {\rm{Ls}}_j^{(2\ell)}(\theta)& = -\frac{\theta^{2\ell+1}}{2\ell+1}\biggl[\ln\biggl(2\sin\frac{\theta}{2}\biggr)\biggr]^{j-2\ell-1} +(-1)^{\ell}\frac{(j-2\ell-1)(2\ell)!(\ln2)^{j-1}}{2}\biggl(\frac{4\sin\frac{\theta}{2}}{\ln2}\biggr)^{2\ell+1}\\ &\quad\times\sum\limits_{n = 0}^{\infty}\Biggl[\frac{\bigl(2\sin\frac{\theta}{2}\bigr)^{2n}}{(2n+2\ell+1)!} \sum\limits_{q = 0}^{2\ell}T\biggl(n+\ell;q,2\ell;\frac12\biggr)\Biggr]\\ &\quad\times\Biggl[\sum\limits_{\alpha = 0}^{j-2\ell-2}\binom{j-2\ell-2}{\alpha}\biggl(\frac{\ln\sin\frac{\theta}{2}}{\ln2}\biggr)^\alpha \sum\limits_{k = 0}^{\alpha}\frac{(-1)^k\langle\alpha\rangle_{k}}{(2n+2\ell+1)^{k+1}\bigl(\ln\sin\frac{\theta}{2}\bigr)^{k}}\Biggr]; \end{aligned} \end{equation} (4.2)

    3. for j\ge2\ell-1\ge1 ,

    \begin{equation} \begin{aligned} {\rm{Ls}}_j^{(2\ell-2)}(\theta)& = (-1)^{\ell}2^{4\ell-3}(2\ell-2)!(\ln2)^{j}\biggl(\frac{\sin\frac{\theta}{2}}{\ln2}\biggr)^{2\ell-1}\\ &\quad\times\sum\limits_{n = 0}^{\infty}\Biggl[\frac{\bigl(2\sin\frac{\theta}{2}\bigr)^{2n}}{{(2n+2\ell-2)!}}\sum\limits_{q = 0}^{2\ell-2} T\biggl(n+\ell-1;q,2\ell-2;\frac12\biggr)\Biggr]\\ &\quad\times\sum\limits_{\alpha = 0}^{j-2\ell+1}\binom{j-2\ell+1}{\alpha} \biggl(\frac{\ln\sin\frac{\theta}{2}}{\ln2}\biggr)^{\alpha} \sum\limits_{k = 0}^{\alpha}\frac{(-1)^k\langle\alpha\rangle_{k}} {(2n+2\ell-1)^{k+1}\bigl(\ln\sin\frac{\theta}{2}\bigr)^{k}}; \end{aligned} \end{equation} (4.3)

    4. for j\ge2\ell-1\ge1 ,

    \begin{equation} \begin{aligned} {\rm{Ls}}_j^{(2\ell-1)}(\theta)& = (-1)^{\ell}(2\ell-1)!(\ln2)^{j} \biggl(\frac{2\sin\frac{\theta}{2}}{\ln2}\biggr)^{2\ell}\\ &\quad\times\sum\limits_{n = 0}^{\infty}\Biggl[\frac{\bigl(2\sin\frac{\theta}{2}\bigr)^{2n}}{(2n+2\ell-1)!}\sum\limits_{q = 0}^{2\ell-2} T(n+\ell-1;q,2\ell-2;1)\Biggr]\\ &\quad\times\sum\limits_{\alpha = 0}^{j-2\ell}\binom{j-2\ell}{\alpha} \biggl(\frac{\ln\sin\frac{\theta}{2}}{\ln2}\biggr)^{\alpha}\sum\limits_{k = 0}^{\alpha} \frac{(-1)^k\langle\alpha\rangle_{k}}{(2n+2\ell)^{k+1}\bigl(\ln\sin\frac{\theta}{2}\bigr)^{k}}. \end{aligned} \end{equation} (4.4)

    Proof. In [28,p. 49,Section 2.4], it was obtained that

    \begin{equation} {\rm{Ls}}_j^{(k)}(\theta) = -\frac{\theta^{k+1}}{k+1}\biggl[\ln\biggl(2\sin\frac{\theta}{2}\biggr)\biggr]^{j-k-1} +\frac{2^{k+1}(j-k-1)}{k+1}\int_{0}^{\sin(\theta/2)}\frac{(\arcsin x)^{k+1}\ln^{j-k-2}(2x)}{x}{\rm{d}} x \end{equation} (4.5)

    for 0 < \theta\le\pi and j-k-2\ge0 . Making use of Theorem 2.1 and the formula

    \begin{equation} \int x^n\ln^mx{\rm{d}} x = x^{n+1}\sum\limits_{k = 0}^{m}(-1)^k\langle m\rangle_{k}\frac{\ln^{m-k}x}{(n+1)^{k+1}}, \quad m,n\ge0 \end{equation} (4.6)

    in [22,p. 238,2.722], we acquire

    \begin{align*} &\quad\int_{0}^{\sin(\theta/2)}\frac{(\arcsin x)^{2\ell}\ln^{j-2\ell-1}(2x)}{x}{\rm{d}} x\\ & = (-1)^{\ell-1}(2\ell)!\sum\limits_{n = 0}^{\infty}\frac{4^{n}}{(2n+2\ell)!}\Biggl[\sum\limits_{q = 0}^{2\ell-2} T(n+\ell-1;q,2\ell-2;1)\Biggr] \int_{0}^{\sin(\theta/2)}x^{2n+2\ell-1}\ln^{j-2\ell-1}(2x){\rm{d}} x\\ & = (-1)^{\ell-1}(2\ell)!\sum\limits_{n = 0}^{\infty}\frac{4^{n}}{(2n+2\ell)!}\Biggl[\sum\limits_{q = 0}^{2\ell-2} T(n+\ell-1;q,2\ell-2;1)\Biggr]\\ &\quad\times\biggl[\int_{0}^{\sin(\theta/2)}x^{2n+2\ell-1}(\ln2+\ln x)^{j-2\ell-1}{\rm{d}} x\biggr]\\ & = (-1)^{\ell-1}(2\ell)!\sum\limits_{n = 0}^{\infty}\frac{4^{n}}{(2n+2\ell)!}\Biggl[\sum\limits_{q = 0}^{2\ell-2} T(n+\ell-1;q,2\ell-2;1)\Biggr]\\ &\quad\times\Biggl[\sum\limits_{\alpha = 0}^{j-2\ell-1}\binom{j-2\ell-1}{\alpha}(\ln2)^{j-2\ell-\alpha-1} \int_{0}^{\sin(\theta/2)}x^{2n+2\ell-1}(\ln x)^{\alpha}{\rm{d}} x\Biggr]\\ & = (-1)^{\ell-1}(2\ell)!\sum\limits_{n = 0}^{\infty}\frac{4^{n}}{(2n+2\ell)!}\Biggl[\sum\limits_{q = 0}^{2\ell-2} T(n+\ell-1;q,2\ell-2;1)\Biggr]\\ &\quad\times\Biggl[\sum\limits_{\alpha = 0}^{j-2\ell-1}\binom{j-2\ell-1}{\alpha}(\ln2)^{j-2\ell-\alpha-1} \biggl(\sin\frac{\theta}{2}\biggr)^{2n+2\ell}\sum\limits_{k = 0}^{\alpha}\frac{(-1)^k\langle\alpha\rangle_{k}} {(2n+2\ell)^{k+1}}\biggl(\ln\sin\frac{\theta}{2}\biggr)^{\alpha-k}\Biggr]\\ & = (-1)^{\ell-1}(2\ell)!(\ln2)^{j-2\ell-1}\biggl(\sin\frac{\theta}{2}\biggr)^{2\ell} \sum\limits_{n = 0}^{\infty}\frac{4^{n}}{(2n+2\ell)!}\biggl(\sin\frac{\theta}{2}\biggr)^{2n} \Biggl[\sum\limits_{q = 0}^{2\ell-2} T(n+\ell-1;q,2\ell-2;1)\Biggr]\\ &\quad\times\Biggl[\sum\limits_{\alpha = 0}^{j-2\ell-1}\biggl(\frac{\ln\sin\frac{\theta}{2}}{\ln2}\biggr)^{\alpha} \binom{j-2\ell-1}{\alpha} \sum\limits_{k = 0}^{\alpha}\frac{(-1)^k\langle\alpha\rangle_{k}} {(2n+2\ell)^{k+1}\bigl(\ln\sin\frac{\theta}{2}\bigr)^{k}}\Biggr] \end{align*}

    for j\ge2\ell+1\ge3 . Substituting this result into (4.5) for k = 2\ell-1 yields (4.1).

    Similarly, by virtue of Theorem 2.1 and the formula (4.6), we also have

    \begin{align*} &\quad\int_{0}^{\sin(\theta/2)}\frac{(\arcsin x)^{2\ell+1}\ln^{j-2\ell-2}(2x)}{x}{\rm{d}} x\\ & = (-1)^{\ell}4^{\ell}(2\ell+1)!\sum\limits_{n = 0}^{\infty}\Biggl[\frac{4^{n}}{(2n+2\ell+1)!}\sum\limits_{q = 0}^{2\ell} T\biggl(n+\ell;q,2\ell;\frac{1}{2}\biggr)\Biggr] \int_{0}^{\sin(\theta/2)}x^{2n+2\ell}\ln^{j-2\ell-2}(2x){\rm{d}} x\\ & = (-1)^{\ell}4^{\ell}(2\ell+1)!\sum\limits_{n = 0}^{\infty}\Biggl[\frac{4^{n}}{(2n+2\ell+1)!}\sum\limits_{q = 0}^{2\ell} T\biggl(n+\ell;q,2\ell;\frac{1}{2}\biggr)\Biggr]\\ &\quad\times\sum\limits_{\alpha = 0}^{j-2\ell-2}\binom{j-2\ell-2}{\alpha}(\ln2)^{j-2\ell-\alpha-2}\int_{0}^{\sin(\theta/2)}x^{2n+2\ell}(\ln x)^\alpha{\rm{d}} x\\ & = (-1)^{\ell}4^{\ell}(2\ell+1)!\sum\limits_{n = 0}^{\infty}\Biggl[\frac{4^{n}}{(2n+2\ell+1)!}\sum\limits_{q = 0}^{2\ell} T\biggl(n+\ell;q,2\ell;\frac{1}{2}\biggr)\Biggr]\\ &\quad\times\sum\limits_{\alpha = 0}^{j-2\ell-2}\binom{j-2\ell-2}{\alpha}(\ln2)^{j-2\ell-\alpha-2} \biggl(\sin\frac{\theta}{2}\biggr)^{2n+2\ell+1} \sum\limits_{k = 0}^{\alpha}(-1)^k\langle\alpha\rangle_{k}\frac{\bigl(\ln\sin\frac{\theta}{2}\bigr)^{\alpha-k}}{(2n+2\ell+1)^{k+1}}\\ & = (-1)^{\ell}4^{\ell}(2\ell+1)!\biggl(\sin\frac{\theta}{2}\biggr)^{2\ell+1}(\ln2)^{j-2\ell-2} \sum\limits_{n = 0}^{\infty}\Biggl[\frac{4^{n}}{(2n+2\ell+1)!}\biggl(\sin\frac{\theta}{2}\biggr)^{2n} \sum\limits_{q = 0}^{2\ell}T\biggl(n+\ell;q,2\ell;\frac{1}{2}\biggr)\Biggr]\\ &\quad\times \Biggl[\sum\limits_{\alpha = 0}^{j-2\ell-2}\binom{j-2\ell-2}{\alpha}\biggl(\frac{\ln\sin\frac{\theta}{2}}{\ln2}\biggr)^\alpha \sum\limits_{k = 0}^{\alpha}\frac{(-1)^k\langle\alpha\rangle_{k}}{(2n+2\ell+1)^{k+1}\bigl(\ln\sin\frac{\theta}{2}\bigr)^{k}}\Biggr] \end{align*}

    for \ell\in\mathbb{N} and j\ge2(\ell+1)\ge4 . Substituting this result into (4.5) for k = 2\ell yields (4.2).

    In [20,p. 308], it was derived that

    \begin{equation} {\rm{Ls}}_j^{(k)}(\theta) = -2^{k+1}\int_{0}^{\sin(\theta/2)}\frac{(\arcsin x)^k}{\sqrt{1-x^2}\,}\ln^{j-k-1}(2x){\rm{d}} x \end{equation} (4.7)

    for 0 < \theta\le\pi and j\ge k+1\ge1 . Differentiating with respect to x on both sides of the formulas (2.1) and (2.2) in Theorem 2.1 results in

    \begin{equation} \frac{(\arcsin x)^{2\ell-2}}{\sqrt{1-x^2}\,} = (-1)^{\ell-1}4^{\ell-1}(2\ell-2)!\sum\limits_{n = 0}^{\infty}\Biggl[4^{n}\sum\limits_{q = 0}^{2\ell-2}T\biggl(n+\ell-1;q,2\ell-2;\frac{1}{2}\biggr)\Biggr] \frac{x^{2n+2\ell-2}}{(2n+2\ell-2)!} \end{equation} (4.8)

    and

    \begin{equation} \frac{(\arcsin x)^{2\ell-1}}{\sqrt{1-x^2}\,} = (-1)^{\ell-1}(2\ell-1)!\sum\limits_{n = 0}^{\infty}\Biggl[4^{n} \sum\limits_{q = 0}^{2\ell-2} T(n+\ell;q,2\ell;1)\Biggr]\frac{x^{2n+2\ell-1}}{(2n+2\ell-1)!} \end{equation} (4.9)

    for \ell\in\mathbb{N} . Substituting the power series expansions (4.8) and (4.9) into (4.7) and employing the indefinite integral (4.6) respectively reveal

    \begin{align*} {\rm{Ls}}_j^{(2\ell-2)}(\theta)& = -2^{2\ell-1}\int_{0}^{\sin(\theta/2)}\frac{(\arcsin x)^{2\ell-2}}{\sqrt{1-x^2}\,}\ln^{j-2\ell+1}(2x){\rm{d}} x\\ & = (-1)^{\ell}2^{4\ell-3}(2\ell-2)!\sum\limits_{n = 0}^{\infty}\Biggl[\frac{4^{n}}{{(2n+2\ell-2)!}}\sum\limits_{q = 0}^{2\ell-2} T\biggl(n+\ell-1;q,2\ell-2;\frac{1}{2}\biggr)\Biggr]\\ &\quad\times \int_{0}^{\sin(\theta/2)} x^{2n+2\ell-2}(\ln2+\ln x)^{j-2\ell+1}{\rm{d}} x\\ & = (-1)^{\ell}2^{4\ell-3}(2\ell-2)!\sum\limits_{n = 0}^{\infty}\Biggl[\frac{4^{n}}{{(2n+2\ell-2)!}}\sum\limits_{q = 0}^{2\ell-2} T\biggl(n+\ell-1;q,2\ell-2;\frac{1}{2}\biggr)\Biggr]\\ &\quad\times\sum\limits_{\alpha = 0}^{j-2\ell+1}\binom{j-2\ell+1}{\alpha} (\ln2)^{j-2\ell-\alpha+1}\int_{0}^{\sin(\theta/2)} x^{2n+2\ell-2}(\ln x)^{\alpha}{\rm{d}} x\\ & = (-1)^{\ell}2^{4\ell-3}(2\ell-2)!(\ln2)^{j}\biggl(\frac{\sin\frac{\theta}{2}}{\ln2}\biggr)^{2\ell-1} \sum\limits_{n = 0}^{\infty}\Biggl[\frac{4^{n}}{{(2n+2\ell-2)!}}\biggl(\sin\frac{\theta}{2}\biggr)^{2n}\\ &\quad\times\sum\limits_{q = 0}^{2\ell-2}T\biggl(n+\ell-1;q,2\ell-2;\frac{1}{2}\biggr)\Biggr]\\ &\quad\times\sum\limits_{\alpha = 0}^{j-2\ell+1}\binom{j-2\ell+1}{\alpha} \biggl(\frac{\ln\sin\frac{\theta}{2}}{\ln2}\biggr)^{\alpha} \sum\limits_{k = 0}^{\alpha}\frac{(-1)^k\langle\alpha\rangle_{k}} {(2n+2\ell-1)^{k+1}\bigl(\ln\sin\frac{\theta}{2}\bigr)^{k}} \end{align*}

    for j\ge2\ell-1\ge1 and

    \begin{align*} {\rm{Ls}}_j^{(2\ell-1)}(\theta)& = -2^{2\ell}\int_{0}^{\sin(\theta/2)}\frac{(\arcsin x)^{2\ell-1}}{\sqrt{1-x^2}\,}\ln^{j-2\ell}(2x){\rm{d}} x\\ & = (-1)^{\ell}2^{2\ell}(2\ell-1)!\sum\limits_{n = 0}^{\infty}\Biggl[\frac{4^{n}}{(2n+2\ell-1)!} \sum\limits_{q = 0}^{2\ell-2} T(n+\ell-1;q,2\ell-2;1)\Biggr]\\ &\quad\times \int_{0}^{\sin(\theta/2)}x^{2n+2\ell-1}(\ln2+\ln x)^{j-2\ell}{\rm{d}} x\\ & = (-1)^{\ell}2^{2\ell}(2\ell-1)!\sum\limits_{n = 0}^{\infty}\Biggl[\frac{4^{n}}{(2n+2\ell-1)!} \sum\limits_{q = 0}^{2\ell-2} T(n+\ell-1;q,2\ell-2;1)\Biggr]\\ &\quad\times\sum\limits_{\alpha = 0}^{j-2\ell}\binom{j-2\ell}{\alpha}(\ln2)^{j-2\ell-\alpha}\int_{0}^{\sin(\theta/2)}x^{2n+2\ell-1} (\ln x)^{\alpha}{\rm{d}} x\\ & = (-1)^{\ell}(2\ell-1)!(\ln2)^{j} \biggl(\frac{2\sin\frac{\theta}{2}}{\ln2}\biggr)^{2\ell} \sum\limits_{n = 0}^{\infty}\Biggl[\frac{\bigl(2\sin\frac{\theta}{2}\bigr)^{2n}}{(2n+2\ell-1)!}\sum\limits_{q = 0}^{2\ell-2} T(n+\ell-1;q,2\ell-2;1)\Biggr]\\ &\quad\times\sum\limits_{\alpha = 0}^{j-2\ell}\binom{j-2\ell}{\alpha} \biggl(\frac{\ln\sin\frac{\theta}{2}}{\ln2}\biggr)^{\alpha}\sum\limits_{k = 0}^{\alpha} \frac{(-1)^k\langle\alpha\rangle_{k}}{(2n+2\ell)^{k+1}\bigl(\ln\sin\frac{\theta}{2}\bigr)^{k}} \end{align*}

    for j\ge2\ell\ge1 . The series representations (4.3) and (4.4) are thus proved. The proof of Theorem 4.1 is complete.

    Finally, we list several remarks on our main results and related stuffs.

    Remark 5.1. For n\ge k\ge1 , the first kind Stirling numbers s(n, k) can be explicitly computed by

    \begin{equation} |s(n+1,k+1)| = n!\sum\limits_{\ell_1 = k}^{n} \frac1{\ell_1}\sum\limits_{\ell_2 = k-1}^{\ell_1-1}\frac1{\ell_2}\dotsm \sum\limits_{\ell_{k-1} = 2}^{\ell_{k-2}-1} \frac1{\ell_{k-1}} \sum\limits_{\ell_{k} = 1}^{\ell_{k-1}-1}\frac1{\ell_{k}}. \end{equation} (5.1)

    The formula (5.1) was derived in [41,Corollary 2.3] and can be reformulated as

    \begin{equation*} \frac{|s(n+1,k+1)|}{n!} = \sum\limits _{m = k}^{n}\frac{|s(m,k)|}{m!} \end{equation*}

    for n\ge k\ge1 . From the equation (1.5), by convention, we assume s(n, k) = 0 for n < k and k, n < 0 . In recent years, the first kind Stirling numbers s(n, k) have been investigated in [39,40,41,42,45] and closely related references therein.

    Remark 5.2. For |x| < 1 , we have the following series expansions of \arcsin x and its powers.

    1. The series expansion (3.2) of \arcsin x can be rewritten as

    \begin{equation} \frac{\arcsin x}{x} = 1!\sum\limits_{n = 0}^{\infty}[(2n-1)!!]^2\frac{x^{2n}}{(2n+1)!}, \end{equation} (5.2)

    where (-1)!! = 1 . Various forms of (5.2) can be found in [1,4.4.40] and [2,p. 121,6.41.1].

    2. The series expansion of (\arcsin x)^2 can be rearranged as

    \begin{equation} \biggl(\frac{\arcsin x}{x}\biggr)^2 = 2!\sum\limits_{n = 0}^{\infty} [(2n)!!]^2 \frac{x^{2n}}{(2n+2)!}. \end{equation} (5.3)

    The variants of (5.3) can be found in [2,p. 122,6.42.1], [4,pp. 262–263,Proposition 15], [5,pp. 50–51 and p. 287], [6,p. 384], [7,p. 2,(2.1)], [13,Lemma 2], [20,p. 308], [21,pp. 88-90], [22,p. 61,1.645], [32,p. 1011], [33,p. 453], [47,Section 6.3], [58], [60,p. 59,(2.56)], or [62,p. 676,(2.2)]. It is clear that the series expansion (5.3) and its equivalent forms have been rediscovered repeatedly. For more information on the history, dated back to 1899 or earlier, of the series expansion (5.3) and its equivalent forms, see [7,p. 2] and [32,p. 1011].

    3. The series expansion of (\arcsin x)^3 can be reformulated as

    \begin{equation} \biggl(\frac{\arcsin x}{x}\biggr)^3 = 3!\sum\limits_{n = 0}^{\infty}[(2n+1)!!]^2 \Biggl[\sum\limits_{k = 0}^{n}\frac{1}{(2k+1)^2}\Biggr]\frac{x^{2n}}{(2n+3)!}. \end{equation} (5.4)

    Different variants of (5.4) can be found in [2,p. 122,6.42.2], [4,pp. 262–263,Proposition 15], [11,p. 188,Example 1], [20,p. 308], [21,pp. 88–90], [22,p. 61,1.645], or [27,pp. 154–155,(832)].

    4. The series expansion of (\arcsin x)^4 can be restated as

    \begin{equation} \biggl(\frac{\arcsin x}{x}\biggr)^4 = 4!\sum\limits_{n = 0}^{\infty}[(2n+2)!!]^2\Biggl[\sum\limits_{k = 0}^{n}\frac{1}{(2k+2)^2}\Biggr] \frac{x^{2n}}{(2n+4)!}. \end{equation} (5.5)

    There exist three variants of (5.5) in [4,pp. 262–263,Proposition 15], [7,p. 3,(2.2)], and [20,p. 309].

    5. Basing on the formula (2.21) in [28,p. 50], we concretely obtain

    \begin{equation} \biggl(\frac{\arcsin x}{x}\biggr)^5 = \frac{5!}{2}\sum\limits_{n = 0}^{\infty}[(2n+3)!!]^2 \Biggl[\Biggl(\sum\limits_{k = 0}^{n+1}\frac{1}{(2k+1)^2}\Biggr)^2 -\sum\limits_{k = 0}^{n+1}\frac{1}{(2k+1)^4}\Biggr] \frac{x^{2n}}{(2n+5)!}. \end{equation} (5.6)

    6. In [7], the special series expansions

    \begin{align*} \biggl(\arcsin\frac{x}{2}\biggr)^2& = \frac{1}{2}\sum\limits_{n = 1}^{\infty}\frac{x^{2n}}{\binom{2n}{n}n^2},\\ \biggl(\arcsin\frac{x}{2}\biggr)^4& = \frac{3}{2} \sum\limits_{n = 1}^{\infty} \Biggl(\sum\limits_{m = 1}^{n-1} \frac{1}{m^2}\Biggr) \frac{x^{2n}}{\binom{2n}{n}n^2},\\ \biggl(\arcsin\frac{x}{2}\biggr)^6& = \frac{45}{4}\sum\limits_{n = 1}^{\infty}\Biggl(\sum\limits_{m = 1}^{n-1} \frac{1}{m^2}\sum\limits_{\ell = 1}^{m-1}\frac{1}{\ell^2}\Biggr)\frac{x^{2n}}{\binom{2n}{n}n^2},\\ \biggl(\arcsin\frac{x}{2}\biggr)^8& = \frac{315}{2}\sum\limits_{n = 1}^{\infty}\Biggl(\sum\limits_{m = 1}^{n-1} \frac{1}{m^2}\sum\limits_{\ell = 1}^{m-1}\frac{1}{\ell^2}\sum\limits_{p = 1}^{\ell-1}\frac{1}{p^2}\Biggr) \frac{x^{2n}}{\binom{2n}{n}n^2} \end{align*}

    were listed. In general, it was obtained in [7,pp. 1–2] that

    \begin{equation} \biggl(\arcsin\frac{x}{2}\biggr)^{2\ell} = (2\ell)!\sum\limits_{n = 1}^{\infty}H_\ell(n)\frac{x^{2n}}{\binom{2n}{n}n^2}, \quad \ell\in\mathbb{N} \end{equation} (5.7)

    and

    \begin{equation} \biggl(\arcsin\frac{x}{2}\biggr)^{2\ell+1} = (2\ell+1)! \sum\limits_{n = 1}^{\infty}G_\ell(n)\frac{\binom{2n}{n}}{2^{4n+1}}\frac{x^{2n+1}}{2n+1}, \quad \ell\in\{0\}\cup\mathbb{N}, \end{equation} (5.8)

    where H_1(n) = \frac{1}{4} , G_0(n) = 1 ,

    \begin{equation*} H_{\ell+1}(n) = \frac{1}{4}\sum\limits_{m_1 = 1}^{n-1}\frac{1}{(2m_1)^2} \sum\limits_{m_2 = 1}^{m_1-1}\frac{1}{(2m_2)^2} \dotsm\sum\limits_{m_\ell = 1}^{m_{\ell-1}-1}\frac{1}{(2m_\ell)^2}, \end{equation*}

    and

    \begin{equation*} G_\ell(n) = \sum\limits_{m_1 = 0}^{n-1}\frac{1}{(2m_1+1)^2} \sum\limits_{m_2 = 0}^{m_1-1}\frac{1}{(2\ell_2+1)^2} \dotsm\sum\limits_{m_\ell = 0}^{m_{\ell-1}-1}\frac{1}{(2m_\ell+1)^2}. \end{equation*}

    The convention is that the sum is zero if the starting index exceeds the finishing index.

    7. In [7,(2.9) and (4.3)], [25,p. 480,(88.2.2)], and [56,p. 124], there exist the formulas

    \begin{equation} \begin{aligned} \biggl(\frac{\arcsin x}{x}\biggr)^\ell& = \sum\limits_{n = 0}^{\infty}\Biggl[\Biggl(\prod\limits_{k = 1}^{\ell-1} \Biggl\{\sum\limits_{n_k = 0}^{n_{k-1}}\frac{(2n_{k-1}-2n_k)!}{[(n_{k-1}-n_k)!]^2(2n_{k-1}-2n_k+1)}\frac{1}{2^{2n_{k-1}-2n_k}}\Biggr\}\Biggr)\\ &\quad\times\frac{(2n_{\ell-1})!}{(n_{\ell-1}!)^2(2n_{\ell-1}+1)}\frac{1}{2^{2n_{\ell-1}}}\Biggr]x^{2n} \end{aligned} \end{equation} (5.9)

    and

    \begin{equation} \biggl(\frac{\arcsin x}{x}\biggr)^\ell = \ell!\sum\limits_{n = 0}^{\infty}\Biggl[\sum\limits_{n_1 = 0}^{n}\frac{\binom{2n_1}{n_1}}{2n_1+1} \sum\limits_{n_2 = n_1}^{n}\frac{\binom{2n_2-2n_1}{n_2-n_1}}{2n_2+2}\dotsm \sum\limits_{n_\ell = n_{\ell-1}}^{n}\frac{\binom{2n_\ell-2n_{\ell-1}}{n_\ell-n_{\ell-1}}}{2n_\ell+\ell}\frac{1}{4^{n_\ell}}\Biggr]x^n. \end{equation} (5.10)

    All the power series expansions from (5.2) to (5.6) can also be deduced from Theorem 2.1.

    By the way, we notice that the quantity in the pair of bigger brackets, the coefficient of x^{2n} , in the formula (5.9) has no explicit relation with n . This means that there must be some misprints and typos somewhere in the formula (5.9). On 30 January 2021, Christophe Vignat (Tulane University) pointed out that n_0 = n is the missing information in the formula (5.9).

    In [28,pp. 49–50,Section 2.4], the power series expansions of (\arcsin x)^k for 2\le k\le 13 were concretely and explicitly written down in alternative forms. The main idea in the study of the power series expansions of (\arcsin x)^k for 2\le k\le 13 was related with series representations for generalized logsine functions in [28,p. 50,(2.24) and (2.25)]. The special interest is special values of generalized logsine functions defined by [28,p. 50,(2.26) and (2.27)].

    In [54,Theorem 1.4] and [55,Theorem 2.1], the n th derivative of \arcsin x was explicitly computed.

    In [43,44], three series expansions (5.2), (5.3), (5.4) and their first derivatives were used to derive known and new combinatorial identities and others.

    Because coefficients of x^{2n+2\ell-1} and x^{2n+2\ell} in (2.1) and (2.2) contain three times sums, coefficients of x^{2n} and x^{2n+1} in (5.7) and (5.8) contain \ell times sums, coefficients of x^{2n} in (5.9) contain \ell-1 times sums, and coefficients of x^n in (5.10) contain \ell times sums, we conclude that the series expansions (2.1) and (2.2) are more elegant, more operable, more computable, and more applicable.

    Remark 5.3. Two expressions (2.1) and (2.2) in Theorem 2.1 for series expansions of (\arcsin x)^{2\ell-1} and (\arcsin x)^{2\ell} are very close and similar to, but different from, each other. Is there a unified expression for series expansions of (\arcsin x)^{2\ell-1} and (\arcsin x)^{2\ell} ? If yes, two closed-form formulas for {\rm{B}}_{2n, k} in Theorem 1.1 would also be unified. We believe that the formula

    \begin{equation} \exp\biggl(2a\arcsin\frac{x}{2}\biggr) = \sum\limits_{n = 0}^{\infty}\frac{(ia)_{n/2}}{(ia+1)_{-n/2}}\frac{(-ix)^n}{n!} \end{equation} (5.11)

    mentioned in [7,p. 3,(2.7)] and collected in [25,p. 210,(10.49.33)] would be useful for unifying two expressions (2.1) and (2.2) in Theorem 2.1, where extended Pochhammer symbols

    \begin{equation} (ia)_{n/2} = \frac{\Gamma\bigl(ia+\frac{n}{2}\bigr)}{\Gamma(ia)}\quad{\rm{and}}\quad (ia+1)_{-n/2} = \frac{\Gamma\bigl(ia+1-\frac{n}{2}\bigr)}{\Gamma(ia+1)} \end{equation} (5.12)

    were defined in [25,p. 5,Section 2.2.3], and the Euler gamma function \Gamma(z) is defined [59,Chapter 3] by

    \begin{equation*} \Gamma(z) = \lim\limits_{n\to\infty}\frac{n!n^z}{\prod\limits_{k = 0}^n(z+k)}, \quad z\in\mathbb{C}\setminus\{0,-1,-2,\dotsc\}. \end{equation*}

    What are closed forms and why do we care closed forms? Please read the paper [8].

    Remark 5.4. In [2,p. 122,6.42], [27,pp. 154–155,(834)], [33,p. 452,Theorem], and [47,Section 6.3,Theorem 21,Sections 8 and 9], it was proved or collected that

    \begin{equation} \frac{\arcsin x}{\sqrt{1-x^2}\,} = \sum\limits_{n = 0}^{\infty}2^{2n}(n!)^2\frac{x^{2n+1}}{(2n+1)!}, \quad |x|\le1. \end{equation} (5.13)

    In [6,p. 385], [47,Theorem 24], and [61,p. 174,(10)], it was proved that

    \begin{equation} \sum\limits_{n = 1}^{\infty}\frac{(2x)^{2n}}{\binom{2n}{n}} = \frac{x^2}{1-x^2}+\frac{x\arcsin x}{\bigl(1-x^2\bigr)^{3/2}}, \quad |x| < 1. \end{equation} (5.14)

    These series expansions (5.13) and (5.14) can be derived directly from the series expansion for (\arcsin x)^2 and are a special case of (4.9) for \ell = 1 .

    Remark 5.5. The series expansion of the function \sqrt{1-x^2}\, \arcsin x was listed in [2,p. 122,6.42.4] which can be corrected and reformulated as

    \begin{equation} \sqrt{1-x^2}\,\arcsin x = x-1!\sum\limits_{n = 1}^{\infty}[(2n-2)!!]^2(2n)\frac{x^{2n+1}}{(2n+1)!}, \quad |x|\le1. \end{equation} (5.15)

    Basing on the relation

    \begin{equation*} \bigl(1-x^2\bigr)\bigl[(\arcsin x)^\ell\bigr]' = \ell\sqrt{1-x^2}\,(\arcsin x)^{\ell-1} \end{equation*}

    and utilizing series expansions of (\arcsin x)^3 and (\arcsin x)^4 , after simple operations, we can readily derive

    \begin{equation} \sqrt{1-x^2}\,(\arcsin x)^2 = x^2-2!\sum\limits_{n = 1}^{\infty}[(2n-1)!!]^2\Biggl[(2n+1) \sum\limits_{k = 0}^{n-1}\frac{1}{(2k+1)^2}-1\Biggr]\frac{x^{2n+2}}{(2n+2)!} \end{equation} (5.16)

    and

    \begin{equation} \sqrt{1-x^2}\,(\arcsin x)^3 = x^3-3!\sum\limits_{n = 1}^{\infty}[(2n)!!]^2 \Biggl[(2n+2)\sum\limits_{k = 0}^{n-1} \frac{1}{(2k+2)^2}-1\Biggr]\frac{x^{2n+3}}{(2n+3)!}. \end{equation} (5.17)

    From (4.8) and (4.9), we can generalize the series expansions (5.15), (5.16), and (5.17) as

    \begin{equation} \begin{aligned} \sqrt{1-x^2}\,(\arcsin x)^{2\ell-2} & = x^{2\ell-2}+(-1)^{\ell-1}4^{\ell-1}(2\ell-2)! \\ &\quad\times\sum\limits_{n = 1}^{\infty}[A(\ell,n)-(2n+2\ell-2)(2n+2\ell-3)A(\ell,n-1)] \frac{x^{2n+2\ell-2}}{(2n+2\ell-2)!} \end{aligned} \end{equation} (5.18)

    and

    \begin{equation} \begin{aligned} \sqrt{1-x^2}\,(\arcsin x)^{2\ell-1} & = x^{2\ell-1}+(-1)^{\ell-1}(2\ell-1)!\\ &\quad\times\sum\limits_{n = 1}^{\infty}[B(\ell,n)-(2n+2\ell-1)(2n+2\ell-2)B(\ell,n-1)]\frac{x^{2n+2\ell-1}}{(2n+2\ell-1)!} \end{aligned} \end{equation} (5.19)

    for \ell\in\mathbb{N} , where

    \begin{align*} A(\ell,n)& = 4^{n}\sum\limits_{q = 0}^{2\ell-2}T\biggl(n+\ell-1;q,2\ell-2;\frac12\biggr),\\ B(\ell,n)& = 4^{n} \sum\limits_{q = 0}^{2\ell-2} T(n+\ell-1;q,2\ell-2;1), \end{align*}

    and T(r; q, j;\rho) is defined by (1.6). Considering both coefficients of x^{2\ell-2} and x^{2\ell-1} in the power series expansions (5.18) and (5.19) must be 1 , we acquire two combinatorial identities

    \begin{equation*} \sum\limits_{q = 0}^{2\ell}T\biggl(\ell;q,2\ell;\frac12\biggr) = \frac{(-1)^{\ell}}{4^{\ell}} \quad{\rm{and}}\quad \sum\limits_{q = 0}^{2\ell} T(\ell;q,2\ell;1) = (-1)^{\ell} \end{equation*}

    for \ell\in\{0\}\cup\mathbb{N} , where T(r; q, j;\rho) is defined by (1.6).

    Remark 5.6. Making use of Theorem 1.1, we readily obtain the first several values of the sequence (1.3) in Tables 1 and 2.

    Table 1.  The sequence {\rm{B}}_{2n, 2k-1} in (1.3) for 1\le n, k\le8 .
    {\rm{B}}_{2n, 2k-1} k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
    n=1 \frac{1}{3} 0 0 0 0 0 0 0
    n=2 \frac{9}{5} 0 0 0 0 0 0 0
    n=3 \frac{225}{7} \frac{5}{9} 0 0 0 0 0 0
    n=4 1225 42 0 0 0 0 0 0
    n=5 \frac{893025}{11} 3951 \frac{35}{9} 0 0 0 0 0
    n=6 \frac{108056025}{13} \frac{2515524}{5} 1155 0 0 0 0 0
    n=7 1217431215 85621185 314314 \frac{5005}{81} 0 0 0 0
    n=8 \frac{4108830350625}{17} 18974980350 \frac{284770486}{3} \frac{140140}{3} 0 0 0 0

     | Show Table
    DownLoad: CSV
    Table 2.  The sequence {\rm{B}}_{2n, 2k} in (1.3) for 1\le n, k \le 8 .
    {\rm{B}}_{2n, 2k} k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
    n=1 0 0 0 0 0 0 0 0
    n=2 \frac{1}{3} 0 0 0 0 0 0 0
    n=3 9 0 0 0 0 0 0 0
    n=4 \frac{2067}{5} \frac{35}{27} 0 0 0 0 0 0
    n=5 30525 210 0 0 0 0 0 0
    n=6 \frac{23483925}{7} 35211 \frac{385}{27} 0 0 0 0 0
    n=7 516651345 \frac{106790684}{15} 7007 0 0 0 0 0
    n=8 106480673775 \frac{8891683281}{5} 2892890 \frac{25025}{81} 0 0 0 0

     | Show Table
    DownLoad: CSV

    In the papers [46,48,49,50,51,52,53,54,55] and closely related references therein, the authors and their coauthors discovered and applied closed form expressions for many special values of the second kind Bell polynomials {\rm{B}}_{n, k}(x_1, x_2, \dotsc, x_{n-k+1}) for n\ge k\ge0 .

    Remark 5.7. Taking \theta = \frac{\pi}{3} in (4.3) and (4.4) give

    \begin{align*} {\rm{Ls}}_j^{(2\ell-2)}\biggl(\frac{\pi}{3}\biggr) & = (-1)^{\ell}(4\ell-4)!!(\ln2)^{j-2\ell+1} \sum\limits_{n = 0}^{\infty}\Biggl[\frac{1}{{(2n+2\ell-2)!}}\sum\limits_{q = 0}^{2\ell-2} T\biggl(n+\ell-1;q,2\ell-2;\frac12\biggr)\Biggr] \\ &\quad\times\sum\limits_{\alpha = 0}^{j-2\ell+1}(-1)^{\alpha} \binom{j-2\ell+1}{\alpha} \sum\limits_{k = 0}^{\alpha}\frac{\langle\alpha\rangle_{k}} {(2n+2\ell-1)^{k+1}(\ln2)^{k}} \end{align*}

    and

    \begin{align*} {\rm{Ls}}_j^{(2\ell-1)}\biggl(\frac{\pi}{3}\biggr) & = (-1)^{\ell}(2\ell-1)!(\ln2)^{j-2\ell}\sum\limits_{n = 0}^{\infty}\Biggl[\frac{1}{(2n+2\ell-1)!} \sum\limits_{q = 0}^{2\ell-2} T(n+\ell-1;q,2\ell-2;1)\Biggr]\\ &\quad\times\sum\limits_{\alpha = 0}^{j-2\ell}(-1)^{\alpha}\binom{j-2\ell}{\alpha} \sum\limits_{k = 0}^{\alpha} \frac{\langle\alpha\rangle_{k}}{(2n+2\ell)^{k+1}(\ln2)^{k}} \end{align*}

    for \ell\in\mathbb{N} , where \langle z\rangle_n for z\in\mathbb{C} and n\in\{0\}\cup\mathbb{N} denotes the falling factorial defined by (2.4) and T(r; q, j;\rho) is defined by (1.6). In [28,p. 50], it was stated that the values {\rm{Ls}}_j^{(\ell)}\bigl(\frac{\pi}{3}\bigr) have been related to special interest in the calculation of the multiloop Feynman diagrams [19,20].

    Similarly, we can also deduce series representations for special values of the logsine function {\rm{Ls}}_j^{(\ell)}(\theta) at \theta = \frac{\pi}{2} , \frac{\pi}{4} , \frac{\pi}{6} and \theta = \pi . These special values were originally derived in [30,31,34] and also considered in [3,9,10,14,15,16,17,19,20,28,29,37,38,57] and closely related references therein.

    Remark 5.8. This paper is a revised version of electronic arXiv preprints [23,24].

    The authors thank

    1. Frank Oertel (Philosophy, Logic & Scientific Method Centre for Philosophy of Natural and Social Sciences, London School of Economics and Political Science, UK; f.oertel@email.de) for his citing the paper [53] in his electronic preprint [35]. On 10 October 2020, this citation and the Google Scholar Alerts leaded the authors to notice the numbers (1.2) in [35]. On 26 January 2021, he sent the important paper [7] to the authors and others. We communicated and discussed with each other many times.

    2. Chao-Ping Chen (Henan Polytechnic University, China; chenchaoping@sohu.com) for his asking the combinatorial identity in [43,Theorem 2.2], or the one in [44,Theorem 2.1], via Tencent QQ on 18 December 2020. Since then, we communicated and discussed with each other many times.

    3. Mikhail Yu. Kalmykov (Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Russia; kalmykov.mikhail@googlemail.com) for his noticing [43,Remark 4.2] and providing the references [19,20,28,30,31,34] on 9 and 27 January 2021. We communicated and discussed with each other many times.

    4. Li Yin (Binzhou University, China; yinli7979@163.com) for his frequent communications and helpful discussions with the authors via Tencent QQ online.

    5. Christophe Vignat (Department of Physics, Universite d'Orsay, France; Department of Mathematics, Tulane University, USA; cvignat@tulane.edu) for his sending electronic version of those pages containing the formulas (5.9), (5.11), and (5.12) in [25,56] on 30 January 2021 and for his sending electronic version of the monograph [27] on 8 February 2021.

    6. Frédéric Ouimet (California Institute of Technology, USA; ouimetfr@caltech.edu) for his photocopying by Caltech Library Services and transferring via ResearchGate those two pages containing the formulas (5.9) and (5.11) on 2 February 2021.

    7. anonymous referees for their careful corrections to and valuable comments on the original version of this paper.

    The author Dongkyu Lim was partially supported by the National Research Foundation of Korea under Grant NRF-2021R1C1C1010902, Republic of Korea.

    All authors contributed equally to the manuscript and read and approved the final manuscript.

    The authors declare that they have no conflict of interest.



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