Recent zbMATH articles in MSC 33https://www.zbmath.org/atom/cc/332021-03-30T15:24:00+00:00WerkzeugExtensions of quadratic transformation identities for hypergeometric functions.https://www.zbmath.org/1455.330032021-03-30T15:24:00+00:00"Qiu, Song-Liang"https://www.zbmath.org/authors/?q=ai:qiu.songliang"Ma, Xiao-Yan"https://www.zbmath.org/authors/?q=ai:ma.xiaoyan"Chu, Yu-Ming"https://www.zbmath.org/authors/?q=ai:chu.yumingThe main purpose of the article is to study the problem of extending
\[\begin{aligned}
F\left(\frac{1}{2}, 1 ; \frac{3}{2} ; \frac{4 r}{(1+r)^{2}}\right) &=(1+r) F\left(\frac{1}{2}, 1 ; \frac{3}{2} ; r\right) \\
F\left(\frac{1}{2}, 1 ; \frac{3}{2} ; \frac{1-r}{1+r}\right) &=\frac{1+r}{2} F\left(\frac{1}{2}, 1 ; \frac{3}{2} ; r^{\prime 2}\right) \\
F\left(\frac{1}{4}, \frac{3}{4} ; 1 ;\left(\frac{1-r}{1+3 r}\right)^{2}\right) &=\frac{\sqrt{1+3 r}}{2} F\left(\frac{1}{4}, \frac{3}{4} ; 1 ; r^{\prime 2}\right) \\
F\left(\frac{1}{4}, \frac{3}{4} ; 1 ; 1-\left(\frac{1-r}{1+3 r}\right)^{2}\right) &=\sqrt{1+3 r} F\left(\frac{1}{4}, \frac{3}{4} ; 1 ; r^{2}\right)
\end{aligned}\]
for \(r\in(0,1)\), to zero-balanced hypergeometric functions. In addition, the authors obtain several new properties of the Ramanujan type constant \(R(a,b)\) and the hypergeometric functions. As examples of applications of these results, several quadratic transformation properties of the generalized Grötzsch ring function, which appears in Ramanujan's modular equations, have been obtained.
Reviewer: Pierluigi Vellucci (Roma)Branching rules for Koornwinder polynomials with one column diagrams and matrix inversions.https://www.zbmath.org/1455.330102021-03-30T15:24:00+00:00"Hoshino, Ayumu"https://www.zbmath.org/authors/?q=ai:hoshino.ayumu"Shiraishi, Jun'ichi"https://www.zbmath.org/authors/?q=ai:shiraishi.junichiSummary: We present an explicit formula for the transition matrix \(\mathcal{C}\) from the type \(BC_n\) Koornwinder polynomials \(P_{(1^r)}(x|a, b, c, d| q, t)\) with one column diagrams, to the type \(BC_n\) monomial symmetric polynomials \(m_{(1^r)}(x)\). The entries of the matrix \(\mathcal{C}\) enjoy a set of four terms recursion relations. These recursions provide us with the branching rules for the Koornwinder polynomials with one column diagrams, namely the restriction rules from \(BC_n\) to \(BC_{n-1}\). To have a good description of the transition matrices involved, we introduce the following degeneration scheme of the Koornwinder polynomials: \(P_{(1^r)}(x|a,b,c,d|q,t) \longleftrightarrow P_{(1^r)}(x |a, -a, c, d| q, t) \longleftrightarrow P_{(1^r)} (x |a, -a, c, -c| q, t) \longleftrightarrow P_{(1^r)} (x |t^{1/2} c, -t^{1/2} c, c, -c| q, t) \longleftrightarrow P_{(1^r)} (x |t^{1/2}, -t^{1/2}, 1, -1| q, t)\). We prove that the transition matrices associated with each of these degeneration steps are given in terms of the matrix inversion formula of Bressoud. As an application, we give an explicit formula for the Kostka polynomials of type \(B_n\), namely the transition matrix from the Schur polynomials \(P^{(B_n, B_n)}_{(1^r)} (x |q; q, q)\) to the Hall-Littlewood polynomials \(P^{(B_n, B_n)}_{(1^r)} (x |t; 0, t)\). We also present a conjecture for the asymptotically free eigenfunctions of the \(B_n q\)-Toda operator, which can be regarded as a branching formula from the \(B_n q\)-Toda eigenfunction restricted to the \(A_{n-1} q\)-Toda eigenfunctions.Electrostatic equilibria on the unit circle via Jacobi polynomials.https://www.zbmath.org/1455.780062021-03-30T15:24:00+00:00"Johnson, K."https://www.zbmath.org/authors/?q=ai:johnson.keith-peter|johnson.kelli-j|johnson.kenneth-o|johnson.kenneth-w|johnson.kerry|johnson.kenneth-c|johnson.k-i|johnson.kenneth-l|johnson.kyle-l|johnson.kenneth-r|johnson.kent|johnson.kenneth-d|johnson.k-g|johnson.keith-a|johnson.ken|johnson.kelly-k|johnson.kristen-e|johnson.kris|johnson.kristina-m|johnson.kurt-n|johnson.kenneth-h|johnson.kathryn-e|johnson.karen-anne|johnson.katie|johnson.kjell"Simanek, B."https://www.zbmath.org/authors/?q=ai:simanek.brian-zSummary: We use classical Jacobi polynomials to identify the equilibrium configurations of charged particles confined to the unit circle. Our main result unifies two theorems from a paper of \textit{P. J. Forrester} and \textit{J. B. Rogers} [SIAM J. Math. Anal. 17, 461--468 (1986; Zbl 0613.33009)].
{\copyright 2020 American Institute of Physics}Evaluation of some non-elementary integrals involving sine, cosine, exponential and logarithmic integrals. II.https://www.zbmath.org/1455.330172021-03-30T15:24:00+00:00"Nijimbere, Victor"https://www.zbmath.org/authors/?q=ai:nijimbere.victorSummary: The non-elementary integrals \({Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx, \beta\ge1,\alpha>\beta+1\) and \({Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx, \beta\ge1,\alpha>2\beta+1\), where \(\{\beta,\alpha\}\in\mathbb{R} \), are evaluated in terms of the hypergeometric function \(_2F_3\). On the other hand, the exponential integral \({Ei}_{\beta,\alpha}=\int (e^{\lambda x^\beta}/x^\alpha) dx, \beta\ge1,\alpha>\beta+1\) is expressed in terms of \(_2F_2\). The method used to evaluate these integrals consists of expanding the integrand as a Taylor series and integrating the series term by term.
For Part I, see [the author, ibid. 4, No. 1, 24-42 (2018; Zbl 072556474)].Evaluation of some non-elementary integrals involving sine, cosine, exponential and logarithmic integrals: Part I.https://www.zbmath.org/1455.330162021-03-30T15:24:00+00:00"Nijimbere, Victor"https://www.zbmath.org/authors/?q=ai:nijimbere.victorSummary: The non-elementary integrals \(\text{Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx, \beta\ge1,\alpha\le\beta+1\) and \(\text{Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx, \beta\ge1,\alpha\le2\beta+1\), where \(\{\beta,\alpha\}\in\mathbb{R} \), are evaluated in terms of the hypergeometric functions \(_1F_2\) and \(_2F_3\), and their asymptotic expressions for \(|x|\gg1\) are also derived. The integrals of the form \(\int [\sin^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx\) and \(\int [\cos^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx\), where \(n\) is a positive integer, are expressed in terms \(\text{Si}_{\beta,\alpha}\) and \(\text{Ci}_{\beta,\alpha} \), and then evaluated. \( \text{Si}_{\beta,\alpha}\) and \(\text{Ci}_{\beta,\alpha}\) are also evaluated in terms of the hypergeometric function \(_2F_2\). And so, the hypergeometric functions, \(_1F_2\) and \(_2F_3\), are expressed in terms of \(_2F_2\). The exponential integral \(\text{Ei}_{\beta,\alpha}=\int (e^{\lambda x^\beta}/x^\alpha) dx\) where \(\beta\ge1\) and \(\alpha\le\beta+1\) and the logarithmic integral \(\text{Li}=\int_{\mu}^x dt/\ln{t}, \mu>1\), are also expressed in terms of \(_2F_2\), and their asymptotic expressions are investigated. For instance, it is found that for \(x\gg2, \text{Li}\sim{x}/{\ln{x}}+\ln{\left({\ln{x}}/{\ln{2}}\right)}-2- \ln{2}_2F_2(1,1;2,2;\ln{2})\), where the term \(\ln{\left({\ln{x}}/{\ln{2}}\right)}-2- \ln{2}_2F_2(1,1;2,2;\ln{2})\) is added to the known expression in mathematical literature \(\text{Li}\sim{x}/{\ln{x}} \). The method used in this paper consists of expanding the integrand as a Taylor and integrating the series term by term, and can be used to evaluate the other cases which are not considered here. This work is motivated by the applications of sine, cosine exponential and logarithmic integrals in Science and Engineering, and some applications are given.
For Part II, see [the author, ibid. 4, No. 1, 43-55 (2018; Zbl 072556483)].Asymptotics of some generalized Mathieu series.https://www.zbmath.org/1455.330152021-03-30T15:24:00+00:00"Gerhold, Stefan"https://www.zbmath.org/authors/?q=ai:gerhold.stefan"Hubalek, Friedrich"https://www.zbmath.org/authors/?q=ai:hubalek.friedrich"Tomovski, Živorad"https://www.zbmath.org/authors/?q=ai:tomovski.zivoradLet \(\mu\ge0\) and \(r>0\), and fix two series \(\mathbf{a}=(a_n)_{n\ge0}\), \(\mathbf{b}=(b_n)_{n\ge0}\). The generalized Mathieu series is defined as
\[S_{\mathrm{a}, \mathrm{b}, \mu}(r):=\sum_{n=0}^{\infty} \frac{a_{n}}{\left(b_{n}+r^{2}\right)^{\mu+1}}.\]
Moreover, for \(\alpha, \beta, r>0, \mu \geq 0,\) with \(\alpha-\beta(\mu+1)<-1\) and \(\gamma, \delta \in \mathbb{R}\) the series
\[
S_{\alpha, \beta, \gamma, \delta, \mu}(r):=\sum_{n=2}^{\infty} \frac{n^{\alpha}(\log n)^{\gamma}}{\left(n^{\beta}(\log n)^{\delta}+r^{2}\right)^{\mu+1}}
\]
is defined. Another special case of the generalized Mathieu series is
\[S_{\alpha, \beta, \mu}^{!}(r):=\sum_{n=0}^{\infty} \frac{(n !)^{\alpha}}{\left((n !)^{\beta}+r^{2}\right)^{\mu+1}},\]
for appropriate parameter choice.
The authors then study the asymptotic behavior of these series. Among others, it turns out that
\[S_{\alpha, \beta, \gamma, \delta, \mu}(r) \sim C_{\alpha, \beta, \gamma, \delta, \mu} r^{2(\alpha+1) / \beta-2(\mu+1)}(\log r)^{-\delta(\alpha+1) / \beta+\gamma}, \quad r \uparrow \infty,\]
again, for appropriate parameter choice. Similar asymptotics hold for \(S_{\alpha, \beta, \mu}^{!}(r)\), too. Here \(C_{\alpha, \beta, \gamma, \delta, \mu}\) is an expression involving the Gamma function.
Reviewer: István Mező (Nanjing)On certain transformation formulas involving basic hypergeometric series.https://www.zbmath.org/1455.330092021-03-30T15:24:00+00:00"Singh, Satya Prakash"https://www.zbmath.org/authors/?q=ai:singh.satya-prakash"Yadav, Vijay"https://www.zbmath.org/authors/?q=ai:vijay.yadavSummary: In this paper, making use of Bailey's transform and certain summations of basic hypergeometric series, we have established interesting transformation formulae involving basic hypergeometric series.The heat kernel of sub-Laplace operator on nilpotent Lie groups of step two.https://www.zbmath.org/1455.351292021-03-30T15:24:00+00:00"Chang, Der-Chen"https://www.zbmath.org/authors/?q=ai:chang.der-chen-e"Kang, Qianqian"https://www.zbmath.org/authors/?q=ai:kang.qianqian"Wang, Wei"https://www.zbmath.org/authors/?q=ai:wang.wei.18Summary: The Laguerre calculus is widely used for the inversion of differential operators on the Heisenberg group. Applying the Laguerre calculus established on nilpotent Lie groups of step two in [\textit{D.-C. Chang} et al., ``The Laguerre calculus on the nilpotent Lie group of step two.'' Preprint, \url{arXiv:1901.06513}], we find the explicit formulas for the heat kernel of sub-Laplace operator and the fundamental solution of power of sub-Laplace operator on nilpotent Lie groups of step two. Calin, Chang and Markina [\textit{O. Calin} et al., in: Analysis and mathematical physics. Lectures delivered at the international conference ``New trends in harmonic and complex analysis'', Voss, Norway, May 7--12, 2007. Basel: Birkhäuser. 49--76 (2009; Zbl 1297.53028)] also get the formulas for the heat kernel of sub-Laplace operator on nilpotent Lie groups of step two by using the Hamiltonian and Lagrangian formalisms that are related to geometric mechanics. In this paper, we use a totally different method to prove our main results by using the Laguerre calculus, which is more direct from the point of view of Fourier analysis.Intertwining operators associated with dihedral groups.https://www.zbmath.org/1455.330082021-03-30T15:24:00+00:00"Xu, Yuan"https://www.zbmath.org/authors/?q=ai:xu.yuan.1|xu.yuanThe Fourier analysis associated with reflection groups has attracted
considerable attention.
The purpose of this paper is to study intertwining operators associated with dihedral groups.
The
main result gives an integral representation for the intertwining operator on a class
of functions.
As an application, a closed form formula for the Poisson kernel of
the \(h\)-harmonics associated with the dihedral group \(I_ k\) is obtained.
For one-parameter, the Poisson kernel can be expressed as a hypergeometric
function.
The dihedral groups \(I_{2k}\) and \(I_{2k+1}\) with positive root systems
are considered separately and corresponding Dunkl operators are computed.
The generalized Gegenbauer polynomials can be expressed in terms of
Jacobi polynomials, which leads to formulas for the \(h\)-harmonics.
Finally, an explicit basis of orthogonal polynomials or sieved Gegenbauer polynomials
can be derived.
Reviewer: Thomas Ernst (Uppsala)Densities of bounded primes for hypergeometric series with rational parameters.https://www.zbmath.org/1455.111262021-03-30T15:24:00+00:00"Franc, Cameron"https://www.zbmath.org/authors/?q=ai:franc.cameron"Gill, Brandon"https://www.zbmath.org/authors/?q=ai:gill.brandon"Goertzen, Jason"https://www.zbmath.org/authors/?q=ai:goertzen.jason"Pas, Jarrod"https://www.zbmath.org/authors/?q=ai:pas.jarrod"Tu, Frankie"https://www.zbmath.org/authors/?q=ai:tu.frankieSummary: The set of primes where a hypergeometric series with rational parameters is \(p\)-adically bounded is known by \textit{C. Franc} et al. [J. Number Theory 192, 197--220 (2018; Zbl 1454.11082)] to have a Dirichlet density. We establish a formula for this Dirichlet density and conjecture that it is rare for the density to be large. We provide evidence for this conjecture for hypergeometric series \(_2F_1(x/p,y/p;z/p)\), with \(p\) a prime of the form \(p\equiv 3\pmod{4} \), by establishing an upper bound on the density of bounded primes in this case.Positive solution of nonlinear fractional differential equations with Caputo-like counterpart hyper-Bessel operators.https://www.zbmath.org/1455.340122021-03-30T15:24:00+00:00"Zhang, Kangqun"https://www.zbmath.org/authors/?q=ai:zhang.kangqunIn this paper, the author obtain some Grownwall-type integral inequalities involving Mittag-Leffler functions. Using these inequalities and fixed point theorems the author establish the existence and uniqueness of positive solution of initial value problem to nonlinear fractional differential equation with Caputo-like counterpart hyper-Bessel operators.
Reviewer: Krishnan Balachandran (Coimbatore)Sharp rational bounds for the gamma function.https://www.zbmath.org/1455.330022021-03-30T15:24:00+00:00"Shen, Jian-Mei"https://www.zbmath.org/authors/?q=ai:shen.jian-mei"Yang, Zhen-Hang"https://www.zbmath.org/authors/?q=ai:yang.zhenhang"Qian, Wei-Mao"https://www.zbmath.org/authors/?q=ai:qian.weimao"Zhang, Wen"https://www.zbmath.org/authors/?q=ai:zhang.wen.3"Chu, Yu-Ming"https://www.zbmath.org/authors/?q=ai:chu.yumingThe main result of the paper offers a best upper bound of the type $(x^2+p)/(x+p)$ of the gamma function at $x+1$, where $x$ is in the interval $(0,1)$.
Reviewer's remark: The authors are not aware, that this result has been proved in 2014 by \textit{P. A. Kupán} and \textit{R. Szász} [Integral Transforms Spec. Funct. 25, No. 7, 562--570 (2014; Zbl 1301.26013)].
Reviewer: József Sándor (Cluj-Napoca)Asymptotic expansion and bounds for complete elliptic integrals.https://www.zbmath.org/1455.330132021-03-30T15:24:00+00:00"Wang, Miao-Kun"https://www.zbmath.org/authors/?q=ai:wang.miaokun"Chu, Yu-Ming"https://www.zbmath.org/authors/?q=ai:chu.yuming"Li, Yong-Min"https://www.zbmath.org/authors/?q=ai:li.yongmin"Zhang, Wen"https://www.zbmath.org/authors/?q=ai:zhang.wen.3This paper is concerned with an asymptotic expansion for the complete elliptic integrals of the first kind as \(k\to 1\).
Furthermore, sharp symmetrical bounds for the arithmetic-geometric
mean and a new symmetric mean \(E(a, b)\) are found.
Observe that unusual notations
for the complete elliptic integrals of the first and second kind are used, and
the modulus \(k\) is denoted by \(r\).
The proofs use L'hospital rule and several tedious computations involving the
area tangens hyperbolicus function, which sometimes are not shown.
Reviewer: Thomas Ernst (Uppsala)Algorithmic combinatorics: enumerative combinatorics, special functions and computer algebra. Proceedings of the workshop on combinatorics, special functions and computer algebra (Paule60), Research Institute of Symbolic Computation (RISC), Hagenberg, Austria, May 17--18, 2018. In honour of Peter Paule on his 60th birthday.https://www.zbmath.org/1455.680252021-03-30T15:24:00+00:00"Pillwein, Veronika (ed.)"https://www.zbmath.org/authors/?q=ai:pillwein.veronika"Schneider, Carsten (ed.)"https://www.zbmath.org/authors/?q=ai:schneider.carstenPublisher's description: The book is centered around the research areas of combinatorics, special functions, and computer algebra. What these research fields share is that many of their outstanding results do not only have applications in Mathematics, but also other disciplines, such as computer science, physics, chemistry, etc. A particular charm of these areas is how they interact and influence one another. For instance, combinatorial or special functions' techniques have motivated the development of new symbolic algorithms. In particular, first proofs of challenging problems in combinatorics and special functions were derived by making essential use of computer algebra. This book addresses these interdisciplinary aspects. Algorithmic aspects are emphasized and the corresponding software packages for concrete problem solving are introduced.
Readers will range from graduate students, researchers to practitioners who are interested in solving concrete problems within mathematics and other research disciplines.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Abramov, Sergei A.}, When the search for solutions can be terminated, 1-7 [Zbl 07293155]
\textit{Alladi, Krishnaswami}, Euler's partition theorem and refinements without appeal to infinite products, 9-23 [Zbl 07293156]
\textit{Andrews, George E.}, Sequences in partitions, double \(q\)-series and the mock theta function \(\rho_3(q)\), 25-45 [Zbl 07293157]
\textit{Berkovich, Alexander; Uncu, Ali Kemal}, Refined \(q\)-trinomial coefficients and two infinite hierarchies of \(q\)-series identities, 47-61 [Zbl 07293158]
\textit{Blümlein, Johannes}, Large scale analytic calculations in quantum field theories, 63-87 [Zbl 07293159]
\textit{Bruder, Andrea; Krattenthaler, Christian; Suslov, Sergei K.}, An eigenvalue problem for the associated Askey-Wilson polynomials, 89-108 [Zbl 07293160]
\textit{Chen, William Y. C.; Hao, Robert X. J.; Yang, Harold R. L.}, Context-free grammars and stable multivariate polynomials over Stirling permutations, 109-135 [Zbl 07293161]
\textit{Cigler, Johann; Tyson, Mike}, An interesting class of Hankel determinants, 137-157 [Zbl 07293162]
\textit{Dominici, Diego; Pillwein, Veronika}, A sequence of polynomials generated by a Kapteyn series of the second kind, 159-179 [Zbl 07293163]
\textit{Dorp, Johannes Vom; Gathen, Joachim von Zur; Loebenberger, Daniel; Lühr, Jan; Schneider, Simon}, Comparative analysis of random generators, 181-196 [Zbl 07293164]
\textit{Gerhold, Stefan; Pinter, Arpad}, Difference equation theory meets mathematical finance, 197-213 [Zbl 07293165]
\textit{Kerber, Adalbert}, Evaluations as \(L\)-subsets, 215-222 [Zbl 07293166]
\textit{Koutschan, Christoph; Wong, Elaine}, Exact lower bounds for monochromatic Schur triples and generalizations, 223-248 [Zbl 07293167]
\textit{Krattenthaler, Christian; Schneider, Carsten}, Evaluation of binomial double sums involving absolute values, 249-295 [Zbl 07293168]
\textit{Prodinger, Helmut; Selkirk, Sarah J.; Wagner, Stephan}, On two subclasses of Motzkin paths and their relation to ternary trees, 297-316 [Zbl 07293169]
\textit{Radu, Cristian-Silviu}, A theorem to reduce certain modular form relations modulo primes, 317-337 [Zbl 07293170]
\textit{Strehl, Volker}, Trying to solve a linear system for strict partitions in `closed form', 339-386 [Zbl 07293171]
\textit{Yao, Yukun; Zeilberger, Doron}, Untying the Gordian knot via experimental mathematics, 387-410 [Zbl 07293172]Additional consideration of the spectral properties of synchrotron light.https://www.zbmath.org/1455.780082021-03-30T15:24:00+00:00"Shishanin, O. E."https://www.zbmath.org/authors/?q=ai:shishanin.o-eSummary: With the help of an identified small parameter, asymptotic of the Bessel function are found here for the first time out to the third order of accuracy. On the basis of these asymptotic, spectral expressions for synchrotron radiation, suitable for physical applications, are calculated in greater detail.Recurrence relations for Mellin transforms of \(G L(n, \mathbb{R})\) Whittaker functions.https://www.zbmath.org/1455.330072021-03-30T15:24:00+00:00"Stade, Eric"https://www.zbmath.org/authors/?q=ai:stade.eric"Trinh, Tien"https://www.zbmath.org/authors/?q=ai:trinh.tienSummary: Using a recursive formula for the Mellin transform \(T_{n , a}(s)\) of a spherical, principal series \(G L(n, \mathbb{R})\) Whittaker function, we develop an explicit recurrence relation for this Mellin transform. This relation, for any \(n \geq 2\), expresses \(T_{n , a}(s)\) in terms of a number of ``shifted'' transforms \(T_{n , a}(s + \Sigma)\), with each coordinate of \(\Sigma\) being a non-negative integer.
We then focus on the case \(n = 4\). In this case, we use the relation referenced above to derive further relations, each of which involves ``strictly positive shifts'' in one of the coordinates of \(s\). More specifically: each of our new relations expresses \(T_{4 , a}(s)\) in terms of \(T_{4 , a}(s + \Sigma)\) and \(T_{4 , a}(s + \Omega)\), where for some \(1 \leq k \leq 3\), the \(k\) th coordinates of both \(\Sigma\) and \(\Omega\) are strictly positive.
Next, we deduce a recurrence relation for \(T_{4 , a}(s)\) involving strictly positive shifts in all three \(s_k\)'s at once. (That is, the condition ``for some \(1 \leq k \leq 3\)'' above becomes ``for all \(1 \leq k \leq 3\).'')
These additional relations on \(G L(4, \mathbb{R})\) may be applied to the explicit understanding of certain poles and residues of \(T_{4 , a}(s)\). This residue information is, as we describe below, in turn relevant to recent results concerning orthogonality of Fourier coefficients of \(S L(4, \mathbb{Z})\) Maass forms, and the \(G L(4)\) Kuznetsov formula.A sharp lower bound for the complete elliptic integrals of the first kind.https://www.zbmath.org/1455.330142021-03-30T15:24:00+00:00"Yang, Zhen-Hang"https://www.zbmath.org/authors/?q=ai:yang.zhenhang"Tian, Jing-Feng"https://www.zbmath.org/authors/?q=ai:tian.jingfeng"Zhu, Ya-Ru"https://www.zbmath.org/authors/?q=ai:zhu.yaruThe complete elliptic integral of the first kind is defined as
\[\mathcal{K}(r)=\int_{0}^{\pi / 2} \frac{1}{\sqrt{1-r^{2} \sin ^{2} t}} d t.\]
It is shown in the paper that
\[\frac{2}{\pi} \mathcal{K}(r)>\left[1-\lambda+\lambda\left(\frac{\operatorname{arth} r}{r}\right)^{q}\right]^{1 / q}\]
holds for \(r\in(0,1)\), and the best constants are
\[\lambda=\frac34,\quad\mbox{and}\quad q=\frac1{10}.\]
Here \(\operatorname{arth} r\) the inverse hyperbolic tangent function.
Reviewer: István Mező (Nanjing)Extended Jacobi and Laguerre functions and their applications.https://www.zbmath.org/1455.330042021-03-30T15:24:00+00:00"Eslahchi, M. R."https://www.zbmath.org/authors/?q=ai:eslahchi.mohammad-reza"Abedzadeh, A."https://www.zbmath.org/authors/?q=ai:abedzadeh.aSummary: The aim of this paper is to introduce two new extensions of the Jacobi and Laguerre polynomials as the eigenfunctions of two nonclassical Sturm-Liouville problems. We prove some important properties of these operators such as: These sets of functions are orthogonal with respect to a positive definite inner product defined over the compact intervals \([-1, 1]\) and \([0,\infty)\), respectively and also these sequences form two new orthogonal bases for the corresponding Hilbert spaces. Finally, the spectral and Rayleigh-Ritz methods are carry out using these basis functions to solve some examples. Our numerical results are compared with other existing results to confirm the efficiency and accuracy of our method.On a lattice generalisation of the logarithm and a deformation of the Dedekind eta function.https://www.zbmath.org/1455.110702021-03-30T15:24:00+00:00"Bétermin, Laurent"https://www.zbmath.org/authors/?q=ai:betermin.laurentThe author considers a deformationm the Gannon's deformation of the Dedekind eta function which is related to the two $d$-dimensional simple lattices and two parameters $(m;t)$ where $m$ and $t$ are positive real numbers. The author proves that the minimisers of the lattice theta function are the maximisers of the Gannon's deformation of the Dedekind eta function in the space of lattices with fixed density. The proof is based on the study of a lattice generalisation of the logarithm, called the lattice logarithm, also defined by \textit{T. Gannon} [in: Symmetry in physics. In memory of Robert T. Sharp. Proceedings of the workshop symmetries in physics, Montréal, Canada, September 12--14, 2002. Providence, RI: American Mathematical Society (AMS). 55--66 (2004; Zbl 1087.11025)]. The author also shows that the natural logarithm is characterised by a variational problem over a class of one-dimensional lattice logarithms. The author studies the maximisation of $L 7\to\log L(x)$ for given $x\in (0,1)$ as well as the problem of maximising a lattice energy of type the Gannon's deformation of the Dedekind eta function. In dimension $d=1$, the author shows that the natural logarithm is characterized by a maximality problem. This result allows us to construct functions, in a canonical way, that are smaller and close to $\log(x)$. The author gives many theorems and formulas on deformation of the Dedekind eta function and also lattice generalization of the logarithm.
Reviewer: Yilmaz Simsek (Antalya)The spectral matrices associated with the stochastic Darboux transformations of random walks on the integers.https://www.zbmath.org/1455.600952021-03-30T15:24:00+00:00"de la Iglesia, Manuel D."https://www.zbmath.org/authors/?q=ai:dominguez-de-la-iglesia.manuel"Juarez, Claudia"https://www.zbmath.org/authors/?q=ai:juarez.claudiaSummary: We consider UL and LU stochastic factorizations of the transition probability matrix of a random walk on the integers, which is a doubly infinite tridiagonal stochastic Jacobi matrix. We give conditions on the free parameter of both factorizations in terms of certain continued fractions such that this stochastic factorization is always possible. By inverting the order of the factors (also known as a Darboux transformation) we get new families of random walks on the integers. We identify the spectral matrices associated with these Darboux transformations (in both cases) which are basically conjugations by a matrix polynomial of degree one of a Geronimus transformation of the original spectral matrix. Finally, we apply our results to the random walk with constant transition probabilities with or without an attractive or repulsive force.On linear independence of functions differentiated with respect to parameter.https://www.zbmath.org/1455.111012021-03-30T15:24:00+00:00"Ivankov, Pavel Leonidovich"https://www.zbmath.org/authors/?q=ai:ivankov.pavel-leonidovichSummary: The main difficulty one has to deal with while investigating arithmetic nature of the values of the generalized hypergeometric functions with irrational parameters consists in the fact that the least common denominator of several first coefficients of the corresponding power series increases too fast with the growth of their number. The last circumstance makes it impossible to apply known in the theory of transcendental numbers Siegel's method for carrying out the above mentioned investigation. The application of this method implies usage of pigeon-hole principle for the construction of a functional linear approximating form. This construction is the first step in a long and complicated reasoning that leads ultimately to the required arithmetic result. The attempts to apply pigeon-hole principle in case of functions with irrational parameters encounters insurmountable obstacles because of the aforementioned fast growth of the least common denominator of the coefficients of the corresponding Taylor series. Owing to this difficulty one usually applies effective construction of the linear approximating form (or a system of such forms in case of simultaneous approximations) for the functions with irrational parameters. The effectively constructed form contains polynomials with algebraic coefficients and it is necessary for further reasoning to obtain a satisfactory upper estimate of the modulus of the least common denominator of these coefficients. The known estimates of this type should be in some cases improved. This improvement is carried out by means of the theory of divisibility in quadratic fields. Some facts concerning the distribution of the prime numbers in arithmetic progression are also made use of.
In the present paper we consider one of the versions of effective construction of the simultaneous approximations for the hypergeometric function of the general type and its derivatives. The least common denominator of the coefficients of the polynomials included in these approximations is estimated subsequently by means of the improved variant of the corresponding lemma. All this makes it possible to obtain a new result concerning the arithmetic values of the aforesaid function at a nonzero point of small modulus from some imaginary quadratic field.Monotonicity and sharp inequalities related to complete \((p,q)\)-elliptic integrals of the first kind.https://www.zbmath.org/1455.330122021-03-30T15:24:00+00:00"Wang, Fei"https://www.zbmath.org/authors/?q=ai:wang.fei.1|wang.fei.2"Qi, Feng"https://www.zbmath.org/authors/?q=ai:qi.fengFor \(p,q\in(1,\infty)\), let \(F_{p,q}:[0,1]\to[0,\frac{\pi_{p,q}}{2}]\) be defined by
\[
F_{p,q}(x):=\arcsin_{p,q}(x):=\int_{0}^{x}(1-t^q)^{-1/p}{\operatorname d}t
\]
and \(\pi_{p,q}:=\arcsin_{p,q}(1)\). The corresponding inverse function \(F_{p,q}^{-1}:[0,\frac{\pi_{p,q}}{2}]\to[0,1]\) is the generalized \((p,q)\)-sine function, denoted by \(\sin_{p,q}\). For \(r\in[0,1)\), let
\[
{\mathcal K}_{p,q}(r):=\int_{0}^{\pi_{p,q}/2}(1-r^q\sin_{p,q}^{q}t)^{1/p-1}{\operatorname d}t
\]
be the complete \((p,q)\)-elliptic integral of the first kind, which is a generalization of the classical elliptic integral of the first kind,
\[
{\mathcal K}(r):=\int_{0}^{\pi/2}\frac{{\operatorname d}t}{\sqrt{1-r^2\sin^2t}},
\]
where \(p=q=2\). In this paper, the authors prove some monotonicity properties for \({\mathcal K}_{p,q}(r)\) with respect to \(r\) cumulating in upper and lower bounds for that function.
Reviewer: Klaus Schiefermayr (Wels)Class of integrals involving generalized hypergeometric function.https://www.zbmath.org/1455.330052021-03-30T15:24:00+00:00"Suthar, D. L."https://www.zbmath.org/authors/?q=ai:suthar.daya-l"Hailay, Teklay"https://www.zbmath.org/authors/?q=ai:hailay.teklay"Amsalu, Hafte"https://www.zbmath.org/authors/?q=ai:amsalu.hafte"Singh, Jagdev"https://www.zbmath.org/authors/?q=ai:singh.jagdevIn this paper, the authors evaluate some definite integrals involving generalized hypergeometric functions, a product of algebraic functions, Jacobi functions, Legendre functions and a general class of polynomials.
Reviewer: Osman Yürekli (Ithaca)On a family of polynomials related to \(\zeta (2,1)=\zeta (3)\).https://www.zbmath.org/1455.111242021-03-30T15:24:00+00:00"Zudilin, Wadim"https://www.zbmath.org/authors/?q=ai:zudilin.wadimSummary: We give a new proof of the identity \(\zeta (\{2,1\}^l)=\zeta (\{3\}^l)\) of the multiple zeta values, where \(l=1,2,\dots \), using generating functions of the underlying generalized polylogarithms. In the course of study we arrive at (hypergeometric) polynomials satisfying 3-term recurrence relations, whose properties we examine and compare with analogous ones of polynomials originated from an (ex-)conjectural identity of Borwein, Bradley and Broadhurst [\textit{J. M. Borwein} et al., Electron. J. Comb. 4, No. 2, Research paper R5, 19 p. (1997; Zbl 0884.40004)].
For the entire collection see [Zbl 1446.81002].Recursion relations on the power series expansion of the universal Weierstrass sigma function.https://www.zbmath.org/1455.330112021-03-30T15:24:00+00:00"Eilbeck, J. Chris"https://www.zbmath.org/authors/?q=ai:eilbeck.john-chris"Ônishi, Yoshihiro"https://www.zbmath.org/authors/?q=ai:onishi.yoshihiroSummary: The main aim of this paper is an exposition of the theory of Buchstaber and Leykin on the heat equations for the multivariate sigma functions. We treat only the elliptic curve case, but keeping the most general elliptic curve equation, which may be useful for number theoretic applications.New Weierstrass elliptic wave solutions of the Davey-Stewartson equation with power law nonlinearity.https://www.zbmath.org/1455.352192021-03-30T15:24:00+00:00"El Achab, Abdelfattah"https://www.zbmath.org/authors/?q=ai:el-achab.abdelfattahIn this paper the author considers the (2+1)-dimensional
Davey-Stewartson (DS) equations and obtains some previously known
and new solutions through the Weierstrass elliptic function
method. The paper is organized as follows. The first section is an
introduction to the subject. Section 2 deals with the Weierstrass
elliptic function method. In section 3, the author gives some
particular travelling wave solutions of the (2+1)-dimensional (DS)
equations and restates the main points in section 4. The paper is
supported by an appendix containing some properties of Weierstrass
elliptic functions. The paper is well documented.
Reviewer: Ahmed Lesfari (El Jadida)Four identities for third order mock theta functions.https://www.zbmath.org/1455.111362021-03-30T15:24:00+00:00"Andrews, George E."https://www.zbmath.org/authors/?q=ai:andrews.george-eyre"Berndt, Bruce C."https://www.zbmath.org/authors/?q=ai:berndt.bruce-c"Chan, Song Heng"https://www.zbmath.org/authors/?q=ai:chan.song-heng"Kim, Sun"https://www.zbmath.org/authors/?q=ai:kim.sun-mee|kim.sun-myeng|kim.sun-i|kim.sun-hoon|kim.sun-kyoung|kim.sun-myong|kim.sunhee|kim.sun-tsung|kim.sun-tae|kim.sun-ung|kim.sun-jeong|kim.sun-hye|kim.suntak|kim.sun-bin|kim.sun-hyung|kim.sun-kwang|kim.sun-yong|kim.sun-chul|kim.sun-jin|kim.sun-soo"Malik, Amita"https://www.zbmath.org/authors/?q=ai:malik.amitaThe authors provide completely new proofs for the four identities of the third order mock theta functions (using properties of \(q\)-series), which were recorded by \textit{S. Ramanujan} [The Lost Notebook and other unpublished papers. With an introduction by George E. Andrews. New Delhi: Narosa Publishing House; Berlin (FRG): Springer-Verlag (1988; Zbl 0639.01023)] in its page no. 2 [Entries 1.1 and 1.2] and page no. 17 [Entries 1.3 and 1.4]. However, these identities were first proved by \textit{H. Yesilyurt} [Adv. Math. 190, No. 2, 278--299 (2005; Zbl 1106.11013)] with the help of a famous lemma given by [\textit{A. O. L. Atkin} and \textit{P. Swinnerton-Dyer}, Proc. Lond. Math. Soc. (3) 4, 84--106 (1954; Zbl 0055.03805)]. Please note that the third order mock theta functions are intimately connected with ranks of partitions. The proofs of Entries 1.1-1.3 are not difficult but the proof of Entry 1.4 is considerably more difficult, which is based upon a 2-dissection for two special cases of the rank generating function \(G(z,q)\), when \(z=i\) and \(z\) is a primitive eight root of unity. These 2-dissections of the rank, with their immediate consequences, comprise a second major focus of this paper. The complete proof of Entry 1.4 is given in four parts which are spread within sixteen pages. Some immediate consequences of the results on the 2-dissection of the rank function \(G(i,q)\) and a primitive eight root of unity related to ranks and mock theta functions are also discussed. The proofs are nicely presented and clearly understandable.
Reviewer: M. P. Chaudhary (New Delhi)A new class of generalized polynomials associated with Laguerre and Bernoulli polynomials.https://www.zbmath.org/1455.110392021-03-30T15:24:00+00:00"Khan, Nabiullah"https://www.zbmath.org/authors/?q=ai:khan.nabiullah-u"Usman, Talha"https://www.zbmath.org/authors/?q=ai:usman.talha"Choi, Junesang"https://www.zbmath.org/authors/?q=ai:choi.junesangSummary: Motivated by their importance and potential for applications in certain problems in number theory, combinatorics, classical and numerical analysis, and other fields of applied mathematics, a variety of polynomials and numbers with their variants and extensions have recently been introduced and investigated. In this paper, we aim to introduce generalized Laguerre-Bernoulli polynomials and investigate some of their properties such as explicit summation formulas, addition formulas, implicit formulas, and symmetry identities. Relevant connections of the results presented here with those relatively simple numbers and polynomials are considered.Sharp estimates of the Cesàro kernels for weighted orthogonal polynomial expansions in several variables.https://www.zbmath.org/1455.330062021-03-30T15:24:00+00:00"Dai, Feng"https://www.zbmath.org/authors/?q=ai:dai.feng"Ge, Yan"https://www.zbmath.org/authors/?q=ai:ge.yanSummary: We study the Cesàro means of the orthogonal polynomial expansions (OPEs) with respect to the weight function \(\prod_{i = 1}^d | x_i |^{2 \kappa_i}\) on the unit sphere \(\mathbb{S}^{d - 1} \subset \mathbb{R}^d\) for all parameters \(\kappa_1, \cdots, \kappa_d > - \frac{ 1}{ 2} \). We obtain sharp pointwise estimates for the corresponding Cesàro kernels, which were previously known when all parameters are nonnegative. We settle the problem for the case when \(\min_{1 \leq j \leq d} \kappa_j < 0\). Our estimates allow us to establish sharp results on Cesàro summability with less restriction on the parameters. We also establish similar results for the corresponding weighted OPEs on the unit ball and on the simplex.Additive matrix convolutions of Pólya ensembles and polynomial ensembles.https://www.zbmath.org/1455.150092021-03-30T15:24:00+00:00"Kieburg, Mario"https://www.zbmath.org/authors/?q=ai:kieburg.marioDynamic ASEP, duality, and continuous \(q^{-1}\)-Hermite polynomials.https://www.zbmath.org/1455.601292021-03-30T15:24:00+00:00"Borodin, Alexei"https://www.zbmath.org/authors/?q=ai:borodin.alexei"Corwin, Ivan"https://www.zbmath.org/authors/?q=ai:corwin.ivanSummary: We demonstrate a Markov duality between the dynamic asymmetric simple exclusion process (ASEP) and the standard ASEP. We then apply this to step initial data, as well as a half-stationary initial data (which we introduce). While investigating the duality for half-stationary initial data, we uncover and utilize connections to the continuous \(q^{-1}\)-Hermite polynomials. Finally, we introduce a family of stationary initial data which are related to the indeterminate moment problem associated with these \(q^{-1}\)-Hermite polynomials.Some remarks on the generalized Apostol-Bernoulli and Apostol-Euler polynomials.https://www.zbmath.org/1455.110382021-03-30T15:24:00+00:00"Boutiche, Mohamed Amine"https://www.zbmath.org/authors/?q=ai:boutiche.mohamed-amine"Kargin, Levent"https://www.zbmath.org/authors/?q=ai:kargin.levent"Rahmani, Mourad"https://www.zbmath.org/authors/?q=ai:rahmani.mouradSummary: In this paper, we give explicit expressions for generalized Apostol-Bernoulli and Apostol-Euler polynomials. As consequences, we deduce some explicit representations for other Apostol-type polynomials. Moreover, we find an algorithm based on a three-term recurrence for the calculation of generalized Apostol-Euler numbers and polynomials. As an application, we derive a formula for the values of some kinds of Hurwitz-Lerch Zeta functions at negative arguments.A discrete weighted Markov-Bernstein inequality for sequences and polynomials.https://www.zbmath.org/1455.410032021-03-30T15:24:00+00:00"Dimitrov, Dimitar K."https://www.zbmath.org/authors/?q=ai:dimitrov.dimitar-k"Nikolov, Geno P."https://www.zbmath.org/authors/?q=ai:nikolov.geno-pAuthors' abstract: ``For parameters \(c \in(0, 1)\) and \(\beta > 0\), let \(\ell_2(c, \beta)\) be the Hilbert space of real functions defined on \(\mathbb{N} \) (i.e., real sequences), for which
\[
\| f \|_{c , \beta}^2 : = \sum\limits_{k = 0}^\infty \frac{ ( \beta )_k}{ k !} c^k [ f ( k ) ]^2 < \infty .
\]
We study the best (i.e., the smallest possible) constant \(\gamma_n(c, \beta)\) in the discrete Markov-Bernstein inequality
\[
\| {\Delta} P \|_{c , \beta} \leq \gamma_n(c, \beta) \| P \|_{c , \beta}, \;\;\; P \in \mathcal{P}_n,
\]
where \(\mathcal{P}_n\) is the set of real algebraic polynomials of degree at most \(n\) and \({\Delta} f(x) : = f(x + 1) - f(x)\).
We prove that
\begin{itemize}
\item[(i)] \(\gamma_n(c, 1) \leq 1 + \frac{ 1}{ \sqrt{ c}}\) for every \(n \in \mathbb{N}\) and \(\lim\limits_{n \to \infty} \gamma_n(c, 1) = 1 + \frac{ 1}{ \sqrt{ c}} \);
\item[(ii)] For every fixed \(c \in(0, 1), \gamma_n(c, \beta)\) is a monotonically decreasing function of \(\beta\) in \((0, \infty)\);
\item[(iii)] For every fixed \(c \in(0, 1)\) and \(\beta > 0\), the best Markov-Bernstein constants \(\gamma_n(c, \beta)\) are bounded uniformly with respect to \(n\).
\end{itemize}
A similar Markov-Bernstein inequality is proved for sequences, and a relation between the best Markov-Bernstein constants \(\gamma_n(c, \beta)\) and the smallest eigenvalues of certain explicitly given Jacobi matrices is established.''
Added by reviewer: There is an interesting section ``Comments'' where the authors indicate several related unsolved problems and discuss directions for further investigations.
Reviewer: Alexei Lukashov (Saratov)Integer partitions with even parts below odd parts and the mock theta functions.https://www.zbmath.org/1455.111392021-03-30T15:24:00+00:00"Andrews, George E."https://www.zbmath.org/authors/?q=ai:andrews.george-eyreIn this interesting paper, the author investigates a couple of classes of partitions in which each even part is smaller than each odd part and remarks surprising connections with the even-odd crank (which is defined to be the largest even part minus the number of odd parts) and the third order mock theta function \[\nu(q) = \sum_{n=0}^\infty \frac{q^{n^2+n}}{(-q;q^2)_{n+1}}.\]
Reviewer: Mircea Merca (Cornu de Jos)On linear approximating forms.https://www.zbmath.org/1455.111082021-03-30T15:24:00+00:00"Ivankov, P. L."https://www.zbmath.org/authors/?q=ai:ivankov.pavel-leonidovichSummary: Generalized hypergeometric function is defined as a sum of the power series whose coefficients are the products of the values of some fractional rational function. Taken with a minus sign roots of a numerator and denominator of this rational function are called parameters of the corresponding hypergeometric function. For the investigation of the arithmetic nature of the values of hypergeometric functions and their derivatives (including derivatives with respect to parameter) one often makes use of Siegel's method. The corresponding reasoning begins as a rule by the construction of the functional linear approximating form. If parameters of the hypergeometric function are rational one is able to use pigeonhole principle for the construction of this form. In addition the construction is feasible not only for the hypergeometric functions themselves but also for the products of their powers. By this is explained the generality of results obtained by such method. But if there are irrational numbers among the parameters the application of a pigeonhole method is impossible and for carrying out the corresponding investigation it is necessary to employ some additional considerations.
One of the methods of surmounting the difficulty connected with the irrationality of some parameters of a hypergeometric function consists in the application of the effective construction of the linear approximating form from which the reasoning begins. Primarily effective constructions of such approximations appeared for the functions of a special kind (the numerator of the rational function by means of which the coefficients of hypergeometric functions are defined was to be equal to unity). The investigation of the properties of these approximations revealed the fact that they can be useful in case of rational parameters as well for the quantitative results obtained by effective methods turned out to be more precise than their analogs obtained by Siegel's method. Subsequently the methods of effective construction of linear approximating forms were generalized in diverse directions.
In this paper we propose a new effective construction of approximating form in case when for the hypergeometric functions derivatives with respect to parameter are also considered. This construction is made use of for the sharpening of the lower estimates of the linear independence measure of the values of corresponding functions.New Pieri type formulas for Jack polynomials and their applications to interpolation Jack polynomials.https://www.zbmath.org/1455.050782021-03-30T15:24:00+00:00"Shibukawa, Genki"https://www.zbmath.org/authors/?q=ai:shibukawa.genkiSummary: We present new Pieri type formulas for Jack polynomials. As an application, we give a new derivation of higher order difference equations for interpolation Jack polynomials originally found by \textit{F. Knop} and \textit{S. Sahi} [Int. Math. Res. Not. 1996, No. 10, 473--486 (1996; Zbl 0880.43014)]. We also propose Pieri formulas for interpolation Jack polynomials and intertwining relations for a kernel function for Jack polynomials.The affine VW supercategory.https://www.zbmath.org/1455.180102021-03-30T15:24:00+00:00"Balagović, M."https://www.zbmath.org/authors/?q=ai:balagovic.martina"Daugherty, Z."https://www.zbmath.org/authors/?q=ai:daugherty.zajj"Entova-Aizenbud, I."https://www.zbmath.org/authors/?q=ai:aizenbud.inna-entova"Halacheva, I."https://www.zbmath.org/authors/?q=ai:halacheva.iva"Hennig, J."https://www.zbmath.org/authors/?q=ai:hennig.johanna-m|hennig.jorg-dieter"Im, M. S."https://www.zbmath.org/authors/?q=ai:im.mee-seong"Letzter, G."https://www.zbmath.org/authors/?q=ai:letzter.gail"Norton, E."https://www.zbmath.org/authors/?q=ai:norton.emily"Serganova, V."https://www.zbmath.org/authors/?q=ai:serganova.vera-v"Stroppel, C."https://www.zbmath.org/authors/?q=ai:stroppel.catharina-hIn the article under review, the authors introduce two linear monoidal supercategories: the Brauer supercategory \(s\mathcal{B}r\) and the affine VW (or Nazarov-Wenzl) supercategory. The main result of this work is to provide explicit bases for their morphism spaces (see Theorems 1 and 2). This is proved in two steps. The first one consists in showing that the given sets span the corresponding morphism spaces, by a topological argument (see Proposition 11 and 12). Secondly, they prove that the previous sets are linearly independent (see Sections 4.10 and 4.11). Finally, as an application, in Section 5 they describe the center of the associated endomorphism algebras of the affine VW supercategory (see Theorem 53).
Reviewer: Estanislao Herscovich (Gières)A complete monotonicity property of the multiple gamma function.https://www.zbmath.org/1455.330012021-03-30T15:24:00+00:00"Das, Sourav"https://www.zbmath.org/authors/?q=ai:das.souravThis paper is concerned with the complete monotonicity property of
some complicated, implicitly defined functions.
Rather than mention these functions, I note that since logarithms and
exponentials occur on many places, it would be advisable to define
a branch of the logarithm. The multiple gamma function, as well as the
gamma function and the (multiple) \(q\)-gamma function, which occurs in
another paper of the author [C. R., Math., Acad. Sci. Paris 358, No. 3, 327--332 (2020; Zbl 1450.33004)], are by definition complex functions.
Graphs of the functions are given in the end.
Reviewers remarks:
\begin{itemize}\item[1.]
The complex logarithm is denoted by log, not ln. Both notations are used in the paper.
\item[2.]
There is a misprint in the definition on line 5 on page 919. It should say:
Similarly, one can define multiple \(\Phi\) function \(\Phi_n=\frac{G_n'}{G_n}\).
\item[3.]
The application of Leibnitz' rule in the proof of theorem 5 is not clear,
since it is not possible to distinguish the factors \(u\) and \(v\) in the proof.
\end{itemize}
Reviewer: Thomas Ernst (Uppsala)Numerical approach to the controllability of fractional order impulsive differential equations.https://www.zbmath.org/1455.340652021-03-30T15:24:00+00:00"Kumar, Avadhesh"https://www.zbmath.org/authors/?q=ai:kumar.avadhesh"Vats, Ramesh K."https://www.zbmath.org/authors/?q=ai:vats.ramesh-kumar"Kumar, Ankit"https://www.zbmath.org/authors/?q=ai:kumar.ankit"Chalishajar, Dimplekumar N."https://www.zbmath.org/authors/?q=ai:chalishajar.dimplekumar-nSummary: In this manuscript, a numerical approach for the stronger concept of exact controllability (\textit{total controllability}) is provided. The proposed control problem is a nonlinear fractional differential equation of order \(\alpha \in (1,2]\) with non-instantaneous impulses in finite-dimensional spaces. Furthermore, the numerical controllability of an integro-differential equation is briefly discussed. The tool for studying includes the Laplace transform, the Mittag-Leffler matrix function and the iterative scheme. Finally, a few numerical illustrations are provided through MATLAB graphs.Analytical and number-theoretical properties of the two-dimensional sigma function.https://www.zbmath.org/1455.110372021-03-30T15:24:00+00:00"Ayano, Takanori"https://www.zbmath.org/authors/?q=ai:ayano.takanori"Bukhshtaber, Viktor Matveevich"https://www.zbmath.org/authors/?q=ai:bukhshtaber.viktor-matveevichSummary: This survey is devoted to the classical and modern problems related to the entire function \({\sigma({\mathbf{u}};\lambda)} \), defined by a family of nonsingular algebraic curves of genus 2, where \({\mathbf{u}} = (u_1,u_3)\) and \(\lambda = (\lambda_4, \lambda_6,\lambda_8,\lambda_{10})\). It is an analogue of the Weierstrass sigma function \(\sigma({{u}};g_2,g_3)\) of a family of elliptic curves. Logarithmic derivatives of order \(2\) and higher of the function \({\sigma({\mathbf{u}};\lambda)}\) generate fields of hyperelliptic functions of \({\mathbf{u}} = (u_1,u_3)\) on the Jacobians of curves with a fixed parameter vector \(\lambda \). We consider three Hurwitz series \(\sigma({\mathbf{u}};\lambda)=\sum_{m,n\ge0}a_{m,n}(\lambda)\frac{u_1^mu_3^n}{m!n!}, \sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\xi_k(u_1;\lambda)\frac{u_3^k}{k!}\) and \(\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\mu_k(u_3;\lambda)\frac{u_1^k}{k!} \). The survey is devoted to the number-theoretic properties of the functions \(a_{m,n}(\lambda), \xi_k(u_1;\lambda)\) and \(\mu_k(u_3;\lambda)\). It includes the latest results, which proofs use the fundamental fact that the function \({\sigma ({\mathbf{u}};\lambda)}\) is determined by the system of four heat equations in a nonholonomic frame of six-dimensional space.