Recent zbMATH articles in MSC 33https://zbmath.org/atom/cc/332023-05-31T16:32:50.898670ZWerkzeugProof of a conjecture involving derangements and roots of unityhttps://zbmath.org/1508.050132023-05-31T16:32:50.898670Z"Wang, Han"https://zbmath.org/authors/?q=ai:wang.han"Sun, Zhi-Wei"https://zbmath.org/authors/?q=ai:sun.zhiwei|sun.zhi-wei.1|sun.zhi-weiSummary: Let \(n>1\) be an odd integer, and let \(\zeta\) be a primitive \(n\) th root of unity in the complex field. Via the eigenvector-eigenvalue identity, we show that
\[
\sum_{\tau\in D(n-1)}\,\,\mathrm{sign}(\tau)\prod_{j=1}^{n-1}\frac{1+\zeta^{j-\tau(j)}}{1-\zeta^{j-\tau(j)}}=(-1)^{\frac{n-1}{2}}\frac{((n-2)!!)^2}{n},
\]
where \(D(n-1)\) is the set of all derangements of \(1, \ldots, n-1\). This confirms a previous conjecture of \textit{Z.-W. Sun} [New conjectures in number theory and combinatorics (in Chinese). Harbin: Institute of Technology Press (2021)]. Moreover, for each \(\delta=0,1\) we determine the value of \(\det[x+m_{jk}]_{1\leqslant j,k\leqslant n-1}\) completely, where
\[
m_{jk}=\begin{cases}(1+\zeta^{j-k})/(1-\zeta^{j-k})&\text{ if } j\not=k,\\ \delta&\text{
if }j=k. \end{cases}
\]Generalized Bernoulli numbers, cotangent power sums, and higher-order arctangent numbershttps://zbmath.org/1508.110312023-05-31T16:32:50.898670Z"Isaacson, Brad"https://zbmath.org/authors/?q=ai:isaacson.bradSummary: We explore several interesting consequences of an explicit formula expressing cotangent power sums by generalized Bernoulli numbers. In particular, we (1) examine properties of the generalized Bernoulli numbers used, (2) explicitly express powers of the tangent and cotangent functions as polynomials in their derivatives, (3) obtain an explicit formula for higher-order arctangent numbers, and (4) obtain an explicit formula for certain coefficients of the power series \(\left( \sum_{j=0}^{\infty } \zeta (2j) x^{2j}\right)^n\), where \(\zeta (s)\) denotes the Riemann zeta function.
For the entire collection see [Zbl 1506.11002].A new family of degenerate poly-Bernoulli polynomials of the second kind with its certain related propertieshttps://zbmath.org/1508.110322023-05-31T16:32:50.898670Z"Khan, Waseem A."https://zbmath.org/authors/?q=ai:khan.waseem-ahmad"Muhyi, Abdulghani"https://zbmath.org/authors/?q=ai:muhyi.abdulghani"Ali, Rifaqat"https://zbmath.org/authors/?q=ai:ali.rifaqat"Alzobydi, Khaled Ahmad Hassan"https://zbmath.org/authors/?q=ai:alzobydi.khaled-ahmad-hassan"Singh, Manoj"https://zbmath.org/authors/?q=ai:singh.manoj-kumar"Agarwal, Praveen"https://zbmath.org/authors/?q=ai:agarwal.praveenSummary: The main object of this article is to present type 2 degenerate poly-Bernoulli polynomials of the second kind and numbers by arising from modified degenerate polyexponential function and investigate some properties of them. Thereafter, we treat the type 2 degenerate unipoly-Bernoulli polynomials of the second kind via modified degenerate polyexponential function and derive several properties of these polynomials. Furthermore, some new identities and explicit expressions for degenerate unipoly polynomials related to special numbers and polynomials are obtained. In addition, certain related beautiful zeros and graphical representations are displayed with the help of Mathematica.A \(q\)-analogue of multiple zeta values and its application to number theoryhttps://zbmath.org/1508.110902023-05-31T16:32:50.898670Z"Takeyama, Yoshihiro"https://zbmath.org/authors/?q=ai:takeyama.yoshihiroFrom the text: In this review article, we introduce a \(q\)-analogue of MZVs and discuss its properties
and application to number theory. Roughly speaking, `\(q\)-analogue' of a mathematical
object is a deformation with one parameter, denoted by \(q\), which recovers the original
object in the limit as \(q \to 1\).A study of certain class of strictly positives definite functions and applicationshttps://zbmath.org/1508.110912023-05-31T16:32:50.898670Z"Mehrez, Khaled"https://zbmath.org/authors/?q=ai:mehrez.khaledSummary: In this paper, we present certain class of strictly positive definite functions related to the extended Hurwitz-Lerch Zeta function, digamma function and the modified Bessel function of first kind. The key tools in our proofs are some explicit formulas for the modified Bessel function of the first kind and a new integral representation of the extended Hurwitz-Lerch Zeta function when their terms contain the Fox \(H\)-function. As applications, monotonicity properties and some new functional inequalities of the aforementioned functions are established.On a series of Ramanujan, dilogarithm values, and solitonshttps://zbmath.org/1508.110922023-05-31T16:32:50.898670Z"Boyadzhiev, Khristo N."https://zbmath.org/authors/?q=ai:boyadzhiev.khristo-n"Manns, Steven"https://zbmath.org/authors/?q=ai:manns.stevenIt is well known that the polylogarithm function defined by
\[
\mathrm{Li}_p(x)=\sum_{n=1}^\infty \frac{x^n}{n^p}
\]
has important applications in mathematics, especially in evaluating Euler-type series and logarithmic integrals. In the paper under review, the authors study the extensions of \(\mathrm{Li}_2(x)\) and \(\mathrm{Li}_3(x)\) beyond the unit disk and find various explicit values. They also give a slight improvement of Lewin's formula for the complex dilogarithm \(\mathrm{Li}_2(x)\). Finally, they give a simple proof for the Lewin's formula using the Fourier expansions of the Bernoulli polynomials.
Reviewer: Sami Omar (Sukhair)A brief account of Klein's icosahedral extensionshttps://zbmath.org/1508.120032023-05-31T16:32:50.898670Z"Solanilla, Leonardo"https://zbmath.org/authors/?q=ai:solanilla.leonardo"Barreto, Erick S."https://zbmath.org/authors/?q=ai:barreto.erick-s"Morales, Viviana"https://zbmath.org/authors/?q=ai:morales.vivianaSummary: We present an alternative relatively easy way to understand and determine the zeros of a quintic polynomial whose Galois group is isomorphic to the group of rotational symmetries of a regular icosahedron. The extensive algebraic procedures of Klein in his famous \textit{Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade} are here shortened via Heymann's theory of resolvents. Also, we give a complete explanation of the so-called icosahedral equation and its solution in terms of Gaussian hypergeometric functions. As an innovative element, we construct this solution by using algebraic transformations of hypergeometric series.Power spectrum of the circular unitary ensemblehttps://zbmath.org/1508.150302023-05-31T16:32:50.898670Z"Riser, Roman"https://zbmath.org/authors/?q=ai:riser.roman"Kanzieper, Eugene"https://zbmath.org/authors/?q=ai:kanzieper.eugeneThe sequence of \(N\) ordered energy levels of a given quantum system can naturally be interpreted as a discrete time random process, the times being the indices of the ordered energy levels.
A usual key assumption is then the independence of the mean values on their index (``stationarity''). Without making such a restrictive assumption, the authors first develop a nonperturbative theory to compute the Fourier transform of the covariance matrix (``power spectrum''). Applying it to the circular unitary ensemble (CUE) of random matrices, they express this power spectrum with the tau function of the most general Painlevé function, the sixth one, thus defining a universal probability law.
The additional interest of this paper is the representation of the above result by three different, equivalent methods: a differential equation (the sixth Painlevé), a Fredholm determinant, a Toeplitz determinant.
The paper, which avoids unnecessary esoteric terminology, is clearly written and very pleasant to read.
Reviewer: Robert Conte (Gif-sur-Yvette)On the annihilator ideal in the \textit{bt}-algebra of tensor spacehttps://zbmath.org/1508.160382023-05-31T16:32:50.898670Z"Ryom-Hansen, Steen"https://zbmath.org/authors/?q=ai:ryom-hansen.steen(no abstract)The rational Sklyanin algebra and the Wilson and para-Racah polynomialshttps://zbmath.org/1508.170162023-05-31T16:32:50.898670Z"Bergeron, Geoffroy"https://zbmath.org/authors/?q=ai:bergeron.geoffroy"Gaboriaud, Julien"https://zbmath.org/authors/?q=ai:gaboriaud.julien"Vinet, Luc"https://zbmath.org/authors/?q=ai:vinet.luc"Zhedanov, Alexei"https://zbmath.org/authors/?q=ai:zhedanov.alexei-sSummary: The relation between Wilson and para-Racah polynomials and representations of the degenerate rational Sklyanin algebra is established. Second-order Heun operators on quadratic grids with no diagonal terms are determined. These special or S-Heun operators lead to the rational degeneration of the Sklyanin algebra; they also entail the contiguity and structure operators of the Wilson polynomials. The finite-dimensional restriction yields a representation that acts on the para-Racah polynomials.
{\copyright 2022 American Institute of Physics}Elliptic double affine Hecke algebrashttps://zbmath.org/1508.200062023-05-31T16:32:50.898670Z"Rains, Eric M."https://zbmath.org/authors/?q=ai:rains.eric-mSummary: We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the \(\tilde{C}_n\) version of the construction to construct a flat noncommutative deformation of the \(n\)th symmetric power of any rational surface with a smooth anticanonical curve, and give a further construction which conjecturally is a corresponding deformation of the Hilbert scheme of points.Series representations for fractional-calculus operators involving generalised Mittag-Leffler functionshttps://zbmath.org/1508.260062023-05-31T16:32:50.898670Z"Fernandez, Arran"https://zbmath.org/authors/?q=ai:fernandez.arran"Baleanu, Dumitru"https://zbmath.org/authors/?q=ai:baleanu.dumitru-i"Srivastava, H. M."https://zbmath.org/authors/?q=ai:srivastava.hari-mohanSummary: We consider an integral transform introduced by \textit{T. R. Prabhakar} [Yokohama Math. J. 19, 7--15 (1971; Zbl 0221.45003)], involving generalised multi-parameter Mittag-Leffler functions, which can be used to introduce and investigate several different models of fractional calculus. We derive a new series expression for this transform, in terms of classical Riemann-Liouville fractional integrals, and use it to obtain or verify series formulae in various specific cases corresponding to different fractional-calculus models. We demonstrate the power of our result by applying the series formula to derive analogues of the product and chain rules in more general fractional contexts. We also discuss how the Prabhakar model can be used to explore the idea of fractional iteration in connection with semigroup properties.Generalized fractional operator with applications in mathematical physicshttps://zbmath.org/1508.260092023-05-31T16:32:50.898670Z"Samraiz, Muhammad"https://zbmath.org/authors/?q=ai:samraiz.muhammad"Mehmood, Ahsan"https://zbmath.org/authors/?q=ai:mehmood.ahsan"Iqbal, Sajid"https://zbmath.org/authors/?q=ai:iqbal.sajid"Naheed, Saima"https://zbmath.org/authors/?q=ai:naheed.saima"Rahman, Gauhar"https://zbmath.org/authors/?q=ai:rahman.gauhar"Chu, Yu-Ming"https://zbmath.org/authors/?q=ai:chu.yuming(no abstract)Necessary and sufficient conditions for a difference defined by four derivatives of a function containing trigamma function to be completely monotonichttps://zbmath.org/1508.260112023-05-31T16:32:50.898670Z"Feng, Qi"https://zbmath.org/authors/?q=ai:qi.fengSummary: In this paper, using convolution theorem for Laplaces transforms, logarithmic convexity of the gamma function, Bernsteins theorem for completely monotonic functions, and other techniques, the author finds necessary and sufficient conditions for a difference defined by four derivatives of a function containing trigamma function to be completely monotonic. Using the Chebyshev integral inequality, the author also presents logarithmic convexity of the sequence of polygamma functions.Weighted Bergman spaces associated with the hyperbolic grouphttps://zbmath.org/1508.301012023-05-31T16:32:50.898670Z"Sánchez-Nungaray, Armando"https://zbmath.org/authors/?q=ai:sanchez-nungaray.armando"Morales-Ramos, Miguel Antonio"https://zbmath.org/authors/?q=ai:morales-ramos.miguel-antonio"Ramírez-Mora, María del Rosario"https://zbmath.org/authors/?q=ai:mora.maria-del-rosario-ramirezSummary: A new way of describing weighted poly-Bergman spaces and true weighted poly-Bergman spaces on the upper half-plane \(\Pi\) emerges using a system of coordinates induced by the \(\mathbb{R}_+\)-action on \(\Pi\), the moment map associated with this action, and the standard symplectic structure. We give an isomorphic description of such spaces using Romanovski polynomials. We also provide a unitary description of Toeplitz operators on these spaces whose symbols are associated with the moment map.Asymptotic formulas for Vasyunin cotangent sumshttps://zbmath.org/1508.330012023-05-31T16:32:50.898670Z"Draziotis, Konstantinos A."https://zbmath.org/authors/?q=ai:draziotis.konstantinos-a"Fikioris, George"https://zbmath.org/authors/?q=ai:fikioris.georgeSummary: We study the Vasyunin-type cotangent sum \(c_0 (h/k) = - \sum_{m = 1}^{k - 1} (m/k) \cot (\pi h m / k)\), where \(h\) and \(k\) are positive coprime integers. This sum is related to Estermann zeta function. By applying the Euler-Maclaurin summation formula to a suitable function, we improve a previous large-\(k\) asymptotic approximation of \(c_0(h/k)\). We also provide a procedure to compute an arbitrary number of terms of the approximation.Decreasing properties of two ratios defined by three and four polygamma functionshttps://zbmath.org/1508.330022023-05-31T16:32:50.898670Z"Qi, Feng"https://zbmath.org/authors/?q=ai:qi.fengAccording to Bernstein's theorem, a real-valued monotonic function on the open interval \((0,\infty)\) is given by a combination of exponential functions. Motivated by a couple of combinations of polygamma functions, i.e., derivatives of Euler's gamma function, the author considers two ratios of polygamma functions and proves his own conjecture from 2021 that these two ratios are decreasing on the open interval. In order to do so, a theorem and six lemma are listed and three lemma are proved. The two conjectures are stated and verified in two theorems. Finally, the author provides an alternative proof of the theorem he started with. Necessary definitions are found in the publication, making it self contained. The publication ends with a long list of remarks, three of which posting new conjectures for further studies, probably by the author himself. As he emphasizes in the last remark, this publication is the eighth in a series of publications devoted to this type of questions.
Reviewer: Stefan Groote (Tartu)A note on the asymptotics for incomplete Betafunctionshttps://zbmath.org/1508.330032023-05-31T16:32:50.898670Z"Schlage-Puchta, Jan-Christoph"https://zbmath.org/authors/?q=ai:schlage-puchta.jan-christophSummary: We determine the asymptotic behaviour of certain incomplete Betafunctions.A power series associated with the generalized hypergeometric functions with the unit argument which are involved in Bell polynomialshttps://zbmath.org/1508.330042023-05-31T16:32:50.898670Z"Choi, Junesang"https://zbmath.org/authors/?q=ai:choi.junesang"Qureshi, Mohd Idris"https://zbmath.org/authors/?q=ai:qureshi.mohd-idris"Majid, Javid"https://zbmath.org/authors/?q=ai:majid.javid"Ara, Jahan"https://zbmath.org/authors/?q=ai:ara.jahanSummary: There have been provided a surprisingly large number of summation formulae for generalized hypergeometric functions and series incorporating a variety of elementary and special functions in their various combinations. In this paper, we aim to consider certain generalized hypergeometric function \(_3F_2\) with particular arguments, through which a number of summation formulas for \(_{p+1}F_p(1)\) are provided. We then establish a power series whose coefficients are involved in generalized hypergeometric functions with unit argument. Also, we demonstrate that the generalized hypergeometric functions with unit argument mentioned before may be expressed in terms of Bell polynomials. Further, we explore several special instances of our primary identities, among numerous others, and raise a problem that naturally emerges throughout the course of this investigation.More indefinite integrals from Riccati equationshttps://zbmath.org/1508.330052023-05-31T16:32:50.898670Z"Conway, John T."https://zbmath.org/authors/?q=ai:conway.john-thomasSummary: Two new methods for obtaining indefinite integrals of a special function using Riccati equations are presented. One method uses quadratic fragments of the Riccati equation, the solutions of which are given by simple quadratic equations. This method is applied to cylinder functions, parabolic cylinder functions and the general solution of the Mathieu equation. No such indefinite integrals for general Mathieu functions seem to have been presented previously. The second method obtains indefinite integrals by assuming simple algebraic forms involving constants for the Riccati variable \(u(x)\) and then choosing the values of these constants to give simple and interesting integrals. This method is illustrated here for cylinder functions and Associated Legendre functions. All integrals obtained have been checked using Mathematica.Relations for a class of terminating \(_4F_3(4)\) hypergeometric serieshttps://zbmath.org/1508.330062023-05-31T16:32:50.898670Z"Mishev, Ilia D."https://zbmath.org/authors/?q=ai:mishev.ilia-dSummary: We derive relations for a certain class of terminating \(_4F_3(4)\) hypergeometric series with three free parameters. The invariance group composed of these relations is shown to be isomorphic to the symmetric group \(S_3\). We further study relations for terminating \(_3F_2(4)\) series that fall under two families. By using a series reversal, we examine the corresponding terminating \(_4F_3(1/4)\) and \(_3F_2(1/4)\) series relations. We additionally derive formulas for the sums of the first \(n + 1\) terms of several nonterminating \(_3F_2(4)\) and \(_3F_2(1/4)\) series. We also show how certain known summation formulas for terminating \(_2F_1(4)\) and \(_3F_2(4)\) series follow as limiting cases of some of our relations.Multi-parameter Mathieu, and alternating Mathieu serieshttps://zbmath.org/1508.330072023-05-31T16:32:50.898670Z"Parmar, Rakesh K."https://zbmath.org/authors/?q=ai:parmar.rakesh-kumar"Milovanović, Gradimir V."https://zbmath.org/authors/?q=ai:milovanovic.gradimir-v"Pogány, Tibor K."https://zbmath.org/authors/?q=ai:pogany.tibor-kSummary: The main purpose of this paper is to present a multi-parameter study of the familiar Mathieu series and the alternating Mathieu series \(\mathcal{S}(\boldsymbol{r})\) and \(\widetilde{\mathcal{S}}(\boldsymbol{r})\). The computable series expansions of the their related integral representations are obtained in terms of higher transcendental hypergeometric functions like Lauricella's hypergeometric function \(F_C^{(m)}[\boldsymbol{x}]\), Fox-Wright \(\Psi\) function, Srivastava-Daoust \(S\) generalized Lauricella function, Riemann Zeta and Dirichlet Eta functions, while the extensions concern products of Bessel and modified Bessel functions of the first kind, hyper-Bessel and Bessel-Clifford functions. Auxiliary Laplace-Mellin transforms, bounding inequalities for the hyper-Bessel and Bessel-Clifford functions are established- which are also of independent but considerable interest. A set of bounding inequalities are presented either for the hyper-Bessel and Bessel-Clifford functions which are to our best knowledge new, or also for all considered extended Mathieu-type series. Next, functional bounding inequalities, log-convexity properties and Turán inequality results are presented for the investigated extensions of multi-parameter Mathieu-type series. We end the exposition by a thorough discussion closes the exposition including important details, bridges to occuring new questions like the similar kind multi-parameter treatment of the complete Butzer-Flocke-Hauss \(\Omega\) function which is intimately connected with the Mathieu series family.The centroid of the zeroes of a polynomial via certain Laguerre-type operatorshttps://zbmath.org/1508.330082023-05-31T16:32:50.898670Z"Aloui, Baghdadi"https://zbmath.org/authors/?q=ai:aloui.baghdadi"Chammam, Wathek"https://zbmath.org/authors/?q=ai:chammam.wathek"Alhussain, Ziyad A."https://zbmath.org/authors/?q=ai:alhussain.ziyad-aliSummary: In this paper, we present several properties of the centroid of the zeroes of a polynomial. As an illustration, we apply these results to the \(d\)-orthogonal polynomials. Finally, we provide the relationship between different centroids of a general monic polynomial and its image under a certain Laguerre-type operator.Characterization of \(q\)-Dunkl-classical symmetric orthogonal \(q\)-polynomialshttps://zbmath.org/1508.330092023-05-31T16:32:50.898670Z"Aloui, Baghdadi"https://zbmath.org/authors/?q=ai:aloui.baghdadi"Souissi, Jihad"https://zbmath.org/authors/?q=ai:souissi.jihadSummary: In this paper, we show that, up to a dilatation, the \(q^2\)-analogue of generalized Hermite and \(q^2\)-analogue of generalized Gegenbauer polynomials are the only \(q\)-Dunkl-classical symmetric orthogonal polynomials.On the Darboux transformations and sequences of \(p\)-orthogonal polynomialshttps://zbmath.org/1508.330102023-05-31T16:32:50.898670Z"Barrios Rolanía, D."https://zbmath.org/authors/?q=ai:barrios-rolania.d"García-Ardila, J. C."https://zbmath.org/authors/?q=ai:garcia-ardila.juan-carlos"Manrique, D."https://zbmath.org/authors/?q=ai:manrique.danielSummary: For a fixed \(p \in \mathbb{N} \), sequences of polynomials \(\{P_n\}\), \(n \in \mathbb{N} \), defined by a \((p + 2)\)-term recurrence relation are related to several topics in Approximation Theory. A \((p + 2)\)-banded matrix \(J\) determines the coefficients of the recurrence relation of any of such sequences of polynomials. The connection between these polynomials and the concept of orthogonality has already been established through a \(p\)-dimension vector of functionals. This work goes further on this topic by analyzing the relation between such vectors for the set of sequences \(\{ P_n^{( j )} \} \), \(n \in N\), associated with the Darboux transformations \(J^{(j)}\), \(j = 1, \ldots, p \), of a given \((p + 2)\)-banded matrix \(J\). This is synthesized in Theorem 1, where, under certain conditions, these relationships are established. Besides, some relationships between the sequences of polynomials \(\{ P_n^{( j )} \}\) are determined in Theorem 2, which will be of interest for future research on \(p\)-orthogonal polynomials. We also provide an example to illustrate the effect of the Darboux transformations of a Hessenberg banded matrix, showing the sequences of \(p\)-orthogonal polynomials and the corresponding vectors of functionals. For the sake of clarity, in this example we have considered the case \(p = 2 \), since the procedure is similar for \(p > 2\).Elementary integral series for Heun functions: application to black-hole perturbation theoryhttps://zbmath.org/1508.330112023-05-31T16:32:50.898670Z"Giscard, P.-L."https://zbmath.org/authors/?q=ai:giscard.pierre-louis"Tamar, A."https://zbmath.org/authors/?q=ai:tamar.avivSummary: Heun differential equations are the most general second order Fuchsian equations with four regular singularities. An explicit integral series representation of Heun functions involving only elementary integrands has hitherto been unknown and noted as an important open problem in a recent review. We provide such representations of the solutions of all equations of the Heun class: general, confluent, bi-confluent, doubly confluent, and triconfluent. All the series are illustrated with concrete examples of use, and Python implementations are available for download. We demonstrate the utility of the integral series by providing the first representation of the solution to the Teukolsky radial equation governing the metric perturbations of rotating black holes that is convergent everywhere from the black hole horizon up to spatial infinity.
{\copyright 2022 American Institute of Physics}Sonine formulas and intertwining operators in Dunkl theoryhttps://zbmath.org/1508.330122023-05-31T16:32:50.898670Z"Rösler, Margit"https://zbmath.org/authors/?q=ai:rosler.margit"Voit, Michael"https://zbmath.org/authors/?q=ai:voit.michaelSummary: Let \(V_k\) denote Dunkl's intertwining operator associated with some root system \(R\) and multiplicity \(k\). For two multiplicities \(k,k'\) on \(R\), we study the intertwiner \(V_{k',k} = V_{k'}\circ V_k^{-1}\) between Dunkl operators with multiplicities \(k\) and \(k'\). It has been a long-standing conjecture that \(V_{k',k}\) is positive if \(k'\geq k\geq 0\). We disprove this conjecture by constructing counterexamples for root system \(B_n\). This matter is closely related to the existence of Sonine-type integral representations between Dunkl kernels and Bessel functions with different multiplicities. In our examples, such Sonine formulas do not exist. As a consequence, we obtain necessary conditions on Sonine formulas for Heckman-Opdam hypergeometric functions of type \(BC_n\) and conditions for positive branching coefficients between multivariable Jacobi polynomials.Connection formulas related with Appell's \(F_2\), Horn's \(H_2\) and Olsson's \(F_P\) functionshttps://zbmath.org/1508.330132023-05-31T16:32:50.898670Z"Mimachi, Katsuhisa"https://zbmath.org/authors/?q=ai:mimachi.katsuhisaSummary: Some of the connection problems associated with the system of differential equations \(E_2\), which is satisfied by Appell's \(F_2\) function, are solved by using integrals of Euler type. The present results give another proof of connection formulas related with Appell's \(F_2\), Horn's \(H_2\) and Olsson's \(F_P\) functions, which are obtained by Olsson.Connection problems for the generalized hypergeometric Appell polynomialshttps://zbmath.org/1508.330142023-05-31T16:32:50.898670Z"Luno, Nataliia"https://zbmath.org/authors/?q=ai:luno.nataliiaSummary: Using a straightforward approach, we derived the solution of the inverse problem for the generalized hypergeometric Appell polynomials. Also, we established the recurrence formulas for the solutions of the connection problem between them and the Bernoulli polynomials, as well as between them and the Gould-Hopper polynomials and between two different generalized hypergeometric Appell polynomial families. In addition, we present one new recurrence identity for the generalized hypergeometric Appell polynomials.A new extension of the (A.2) supercongruence of Van Hammehttps://zbmath.org/1508.330152023-05-31T16:32:50.898670Z"Guo, Victor J. W."https://zbmath.org/authors/?q=ai:guo.victor-j-wSummary: We give a new extension of Van Hamme's (A.2) supercongruence with a parameter \(s\) by establishing a \(q\)-analogue of this result. Our proof uses the `creative microscoping' method, which was developed by the author and Zudilin. We also put forward some related open problems for further study.Bailey and Daum's \(q\)-Kummer theorem and extensionshttps://zbmath.org/1508.330162023-05-31T16:32:50.898670Z"Li, Nadia"https://zbmath.org/authors/?q=ai:li.nadia-na"Chu, Wenchang"https://zbmath.org/authors/?q=ai:chu.wenchangSummary: By means of the linearization method, we establish four analytical formulae for the \(q\)-Kummer sum extended by two integer parameters. Ten closed formulae are presented as examples.On the monotonicity and convexity for generalized elliptic integral of the first kindhttps://zbmath.org/1508.330172023-05-31T16:32:50.898670Z"Chen, Ya-jun"https://zbmath.org/authors/?q=ai:chen.yajun"Zhao, Tie-hong"https://zbmath.org/authors/?q=ai:zhao.tiehongSummary: In this paper, we investigate the monotonicity and convexity of the function \(x\mapsto\mathcal{K}_a(\sqrt{x})/\log (1+c/\sqrt{1-x})\) on \((0, 1)\) for \((a,c)\in (0,1/2]\times (0, \infty)\), and the log-concavity of the function \(x\mapsto (1-x)^\lambda\mathcal{K}_a(\sqrt{x})\) on \((0, 1)\) for \(\lambda\in\mathbb{R}\), where \(\mathcal{K}_a(r)\) is the generalized elliptic integral of the first kind. These results are the generalization of [1, Theorem 2] and [2, Theorems 1.7 and 1.8], also give an affirmative answer of [3, Problem 3.1].The extended multi-index Mittag-Leffler functions and their properties connected with fractional calculus and integral transformshttps://zbmath.org/1508.330182023-05-31T16:32:50.898670Z"Agarwal, Praveen"https://zbmath.org/authors/?q=ai:agarwal.praveen"Suthar, D. L."https://zbmath.org/authors/?q=ai:suthar.daya-l"Jain, Shilpi"https://zbmath.org/authors/?q=ai:jain.shilpi"Momani, Shaher"https://zbmath.org/authors/?q=ai:momani.shaher-mSummary: The aim of this paper is to present the extended multi-index Mittag-Leffer type functions while using the extended beta function and investigate several properties including integral representation, derivatives, beta transform, Mellin transform, Relationships between this function with the Laguerre polynomials and Whittaker functions. Further, several properties of the Riemann-Liouville fractional derivatives and integral operators related to extended multi-index Mittag-Leffer functions are also investigated. Finally, various interesting special cases of these functions are also pointed out.On the fractional calculus of multivariate Mittag-Leffler functionshttps://zbmath.org/1508.330192023-05-31T16:32:50.898670Z"Özarslan, Mehmet Ali"https://zbmath.org/authors/?q=ai:ozarslan.mehmet-ali"Fernandez, Arran"https://zbmath.org/authors/?q=ai:fernandez.arranSummary: Multivariate Mittag-Leffler functions are a strong generalisation of the univariate and bivariate Mittag-Leffler functions which are known to be important in fractional calculus. We consider the general functional operator defined by an integral transform with a multivariate Mittag-Leffler function in the kernel. We prove an expression for this operator as an infinite series of Riemann-Liouville integrals, thereby demonstrating that it fits into the established framework of fractional calculus, and we show the power of this series formula by straightforwardly deducing many facts, some new and some already known but now more quickly proved, about the original integral operator. We illustrate our work here by calculating some examples both analytically and numerically, and comparing the results on graphs. We also define fractional derivative operators corresponding to the established integral operator. As an application, we consider some Cauchy-type problems for fractional integro-differential equations involving this operator, where existence and uniqueness of solutions can be proved using fixed point theory. Finally, we generalise the theory by applying the same operators with respect to arbitrary monotonic functions.Asymptotic expansions of solutions to the second term of the fourth Painlevé hierarchyhttps://zbmath.org/1508.330202023-05-31T16:32:50.898670Z"Anoshin, V. I."https://zbmath.org/authors/?q=ai:anoshin.v-i"Beketova, A. D."https://zbmath.org/authors/?q=ai:beketova.a-d"Parusnikova, A. V."https://zbmath.org/authors/?q=ai:parusnikova.anastasia-v"Romanov, K. V."https://zbmath.org/authors/?q=ai:romanov.k-vSummary: Asymptotic behavior and asymptotic expansions of solutions to the second term of the fourth Painlevé hierarchy are constructed using power geometry methods [\textit{A. Pickering}, Theor. Math. Phys. 137, No. 3, 1733--1742 (2003; Zbl 1178.37088); translation from Teor. Mat. Fiz. 137, No. 3, 445--456 (2003)]. Only results for the case of general position -- for the equation parameters \(\beta ,\delta \ne 0\) -- are provided. For constructing asymptotic expansions, a code written in a computer algebra system is used.On the vanishing of coefficients of the powers of a theta functionhttps://zbmath.org/1508.330212023-05-31T16:32:50.898670Z"Sauloy, Jacques"https://zbmath.org/authors/?q=ai:sauloy.jacques"Zhang, Changgui"https://zbmath.org/authors/?q=ai:zhang.changguiSummary: A result on the Galois theory of \(q\)-difference equations (Sauloy in Théorie analytique locale des équations aux \(q\)-différences de pentes arbitraires. See \url{arXiv:2006.03237v1}, 2020) leads to the following question: If \(q \in{{\mathbf{C}}^*}\), \(\left| q \right| < 1\) and if one sets \({\theta_q}(z) := \sum \nolimits_{m \in{{\mathbf{Z}}}} q^{m(m-1)/2} z^m\), can some coefficients of the Laurent series expansion of \(\theta_q^k(z), k \in{{\mathbf{N}}}^*\), vanish ? We give a partial answer.Winding numbers, unwinding numbers, and the Lambert \(W\) functionhttps://zbmath.org/1508.330222023-05-31T16:32:50.898670Z"Beardon, A. F."https://zbmath.org/authors/?q=ai:beardon.alan-fSummary: The unwinding number of a complex number was introduced to process automatic computations involving complex numbers and multi-valued complex functions, and has been successfully applied to computations involving branches of the Lambert \(W\) function. In this partly expository note we discuss the unwinding number from a purely topological perspective, and link it to the classical winding number of a curve in the complex plane. We also use the unwinding number to give a representation of the branches \(W_k\) of the Lambert \(W\) function as a line integral.Explicit analytical solutions of incommensurate fractional differential equation systemshttps://zbmath.org/1508.340062023-05-31T16:32:50.898670Z"Huseynov, Ismail T."https://zbmath.org/authors/?q=ai:huseynov.ismail-t"Ahmadova, Arzu"https://zbmath.org/authors/?q=ai:ahmadova.arzu"Fernandez, Arran"https://zbmath.org/authors/?q=ai:fernandez.arran"Mahmudov, Nazim I."https://zbmath.org/authors/?q=ai:mahmudov.nazim-idrisogluSummary: Fractional differential equations have been studied due to their applications in modelling, and solved using various mathematical methods. Systems of fractional differential equations are also used, for example in the study of electric circuits, but they are more difficult to analyse mathematically. We present explicit solutions for several families of such systems, both homogeneous and inhomogeneous cases, both commensurate and incommensurate. The results can be written, in several interesting special cases, in terms of a recently defined bivariate Mittag-Leffler function and the associated operators of fractional calculus.The geometry of generalized Lamé equation. III: One-to-one of the Riemann-Hilbert correspondencehttps://zbmath.org/1508.341172023-05-31T16:32:50.898670Z"Chen, Zhijie"https://zbmath.org/authors/?q=ai:chen.zhijie"Kuo, Ting-Jung"https://zbmath.org/authors/?q=ai:kuo.ting-jung"Lin, Chang-Shou"https://zbmath.org/authors/?q=ai:lin.chang-shouSummary: In this paper, the third in a series, we continue to study the generalized Lamé equation \(\mathrm{H}(n_0,n_1,n_2,n_3;B)\) with the Darboux-Treibich-Verdier potential
\[
y''(z)=\bigg[\sum\limits_{k=0}^3n_k(n_k+1)\wp (z+\frac{\omega_k}{2}|\tau)+B\bigg]y(z),\quad n_k\in\mathbb{Z}_{\geq 0}
\]
and a related linear ODE with additional singularities \(\pm P\) from the monodromy aspect. We establish the uniqueness of these ODEs with respect to the global monodromy data. Surprisingly, our result shows that the Riemann-Hilbert correspondence from the set
\[
\{\mathrm{H}(n_0,n_1,n_2,n_3;B)|B\in\mathbb{C}\cup \{\mathrm{H}(n_0+2,n_1,n_2,n_3;B)|B\in\mathbb{C}\}
\]
to the set of group representations \(\rho:\pi_1(E_\tau)\to SL(2,\mathbb{C})\) is one-to-one. We emphasize that this result is not trivial at all. There is an example that for \(\tau=\frac{1}{2}+i\frac{\sqrt{3}}{2}\), there are \(B_1\), \(B_2\) such that the monodromy representations of \(\mathrm{H}(1,0,0,0;B_1)\) and \(\mathrm{H}(4,0,0,0;B_2)\) are \textbf{the same}, namely the Riemann-Hilbert correspondence from the set
\[
\{\mathrm{H}(n_0,n_1,n_2,n_3;B)|B\in\mathbb{C}\}\cup \{\mathrm{H}(n_0+3,n_1,n_2,n_3;B)|B\in\mathbb{C}\}
\]
to the set of group representations is not necessarily one-to-one. This example shows that our result is completely different from the classical one concerning linear ODEs defined on \(\mathbb{CP}^1\) with finite singularities.Asymptotics for a singularly perturbed GUE, Painlevé III, double-confluent Heun equations, and small eigenvalueshttps://zbmath.org/1508.341212023-05-31T16:32:50.898670Z"Yu, Jianduo"https://zbmath.org/authors/?q=ai:yu.jianduo"Li, Chuanzhong"https://zbmath.org/authors/?q=ai:li.chuanzhong|li.chuanzhong.1"Zhu, Mengkun"https://zbmath.org/authors/?q=ai:zhu.mengkun"Chen, Yang"https://zbmath.org/authors/?q=ai:chen.yang.1Summary: We discuss the recurrence coefficients of the three-term recurrence relation for the orthogonal polynomials with a singularly perturbed Gaussian weight \(w(z) = |z|^\alpha\exp\left(-z^2 - t/z^2\right)\), \(z\in\mathbb{R}\), \(t > 0\), \(\alpha > 1\). Based on the ladder operator approach, two auxiliary quantities are defined. We show that the auxiliary quantities and the recurrence coefficients satisfy some equations with the aid of three compatibility conditions, which will be used to derive the Riccati equations and Painlevé III. We show that the Hankel determinant has an integral representation involving a particular \(\sigma\)-form of Painlevé III and to calculate the asymptotics of the Hankel determinant under a suitable double scaling, i.e., \(n\rightarrow\infty\) and \(t\rightarrow0\) such that \(s = (2n + 1 + \lambda)t\) is fixed, where \(\lambda\) is a parameter with \(\lambda := (\alpha\mp 1)/2\). The asymptotic behaviors of the Hankel determinant for large \(s\) and small \(s\) are obtained, and Dyson's constant is recovered here. They have generalized the results in the literature [\textit{C. Min} et al., Nucl. Phys., B 936, 169--188 (2018; Zbl 1400.33020)] where \(\alpha = 0\). By combining the Coulomb fluid method with the orthogonality principle, we obtain the asymptotic expansions of the recurrence coefficients, which are applied to derive the relationship between second order differential equations satisfied by our monic orthogonal polynomials and the double-confluent Heun equations as well as to calculate the smallest eigenvalue of the large Hankel matrices generated by the above weight. In particular, when \(\alpha = t = 0\), the asymptotic behavior of the smallest eigenvalue for the classical Gaussian weight \(\exp(-z^2)\) [\textit{G. Szegö}, Trans. Am. Math. Soc. 40, 450--461 (1936; Zbl 0015.34603; JFM 62.0405.01)] is recovered.
{\copyright 2022 American Institute of Physics}Use of Atangana-Baleanu fractional derivative in helical flow of a circular pipehttps://zbmath.org/1508.350552023-05-31T16:32:50.898670Z"Abro, Kashif Ali"https://zbmath.org/authors/?q=ai:abro.kashif-ali"Khan, Ilyas"https://zbmath.org/authors/?q=ai:khan.ilyas"Sooppy Nisar, Kottakkaran"https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaranSummary: There is no denying fact that helically moving pipe/cylinder has versatile utilization in industries; as it has multi-purposes, such as foundation helical piers, drilling of rigs, hydraulic simultaneous lift system, foundation helical brackets and many others. This paper incorporates the new analysis based on modern fractional differentiation on infinite helically moving pipe. The mathematical modeling of infinite helically moving pipe results in governing equations involving partial differential equations of integer order. In order to highlight the effects of fractional differentiation, namely, Atangana-Baleanu on the governing partial differential equations, the Laplace and Hankel transforms are invoked for finding the angular and oscillating velocities corresponding to applied shear stresses. Our investigated general solutions involve the gamma functions of linear expressions. For eliminating the gamma functions of linear expressions, the solutions of angular and oscillating velocities corresponding to applied shear stresses are communicated in terms of Fox-\textbf{H} function. At last, various embedded rheological parameters such as friction and viscous factor, curvature diameter of the helical pipe, dynamic analogies of relaxation and retardation time and comparison of viscoelastic fluid models (Burger, Oldroyd-B, Maxwell and Newtonian) have significant discrepancies and semblances based on helically moving pipe.The focusing NLS equation with step-like oscillating background: the genus \(3\) sectorhttps://zbmath.org/1508.351562023-05-31T16:32:50.898670Z"Monvel, Anne Boutet de"https://zbmath.org/authors/?q=ai:boutet-de-monvel.anne-marie"Lenells, Jonatan"https://zbmath.org/authors/?q=ai:lenells.jonatan"Shepelsky, Dmitry"https://zbmath.org/authors/?q=ai:shepelsky.dmitryIn this work, the authors investigate the long-time behavior of the solutions of the focusing nonlinear Schrödinger equation on the real line, generated from a Cauchy data close to plane waves at infinity. Their main goal consists to perform a complete asymptotic analysis according to \(\xi= x/t\) of the solutions of the Cauchy problem
\[
\left\{ \begin{array}{rcl} i q_t + q_{xx} + 2 |q|^ {2} q& = &0\\
{ q}{}_{|t=0} &= &q_0, \end{array} \right.
\]
with \(q_0\) exhibiting the following asymptotic at infinity
\[
q_0(x) \sim \left\{ \begin{array}{rcl} A_1 e^{i \varphi_1} e^{-2i B_1 x}, & x \to + \infty& \\
A_2 e^{i \varphi_2} e^{-2i B_2 x}, & x \to - \infty.& \end{array} \right.
\]
More precisely, the authors provide uniformly in a sector \(\xi_1\leq \xi \leq \xi_2\) the long-time asymptotic of \(q\), up to a remainder term, with a leading term explicitly expressed by means of hyperbolic theta functions and a sub-leading term involving parabolic cylinder and Airy functions.
Reviewer: Hajer Bahouri (Paris)On the motion of billiards in ellipseshttps://zbmath.org/1508.370402023-05-31T16:32:50.898670Z"Stachel, Hellmuth"https://zbmath.org/authors/?q=ai:stachel.hellmuthSummary: For billiards in an ellipse \(e\) with an ellipse as caustic, there exist canonical coordinates on \(e\) such that the billiard transformation from vertex to vertex is equivalent to a shift of coordinates. A kinematic analysis of billiard motions offers a new approach to canonical parametrizations of billiards and associated Poncelet grids. This parametrization uses Jacobian elliptic functions with the modulus equal to the numerical eccentricity of the caustic and is the basis for proving a few invariants of periodic billiards.Box-ball system and the nonautonomous discrete Toda latticehttps://zbmath.org/1508.370992023-05-31T16:32:50.898670Z"Maeda, Kazuki"https://zbmath.org/authors/?q=ai:maeda.kazukiSummary: We discuss the theory of finite orthogonal polynomials based on elementary linear algebra and its connection to the nonautonomous discrete Toda lattice with nonperiodic finite lattice boundary condition. By using the spectral transformation technique for finite orthogonal polynomials, one can give a solution to the initial value problem of the nonautonomous discrete Toda lattice. However, this construction of the solution cannot be ultradiscretized because of so-called ``negative problem''. In this paper, we focus on the rigged configuration technique to solve the initial value problem of the box-ball system and consider a connection between the rigged configuration and orthogonal polynomials.Integrable structures of specialized hypergeometric tau functionshttps://zbmath.org/1508.371002023-05-31T16:32:50.898670Z"Takasaki, Kanehisa"https://zbmath.org/authors/?q=ai:takasaki.kanehisaSummary: Okounkov's generating function of the double Hurwitz numbers of the Riemann sphere is a hypergeometric tau function of the 2D Toda hierarchy in the sense of \textit{A. Yu. Orlov} and \textit{D. M. Scherbin} [Theor. Math. Phys. 128, No. 1, 906--926 (2001; Zbl 0992.37063); translation from Teor. Mat. Fiz. 128, No. 1, 84--108 (2001); Physica D 152--153, 51--65 (2001; Zbl 0988.37091)]. This tau function turns into a tau function of the lattice KP hierarchy by specializing one of the two sets of time variables to constants. When these constants are particular values, the specialized tau functions become solutions of various reductions of the lattice KP hierarchy, such as the lattice Gelfand-Dickey hierarchy, the Bogoyavlensky-Itoh-Narita lattice and the Ablowitz-Ladik hierarchy. These reductions contain previously unknown integrable hierarchies as well.Exactly solvable discrete time birth and death processeshttps://zbmath.org/1508.390112023-05-31T16:32:50.898670Z"Sasaki, Ryu"https://zbmath.org/authors/?q=ai:sasaki.ryuSummary: We present 15 explicit examples of discrete time birth and death processes which are \textit{exactly solvable}. They are related to hypergeometric orthogonal polynomials of the Askey scheme having discrete orthogonality measures. Namely, they are the Krawtchouk, three different kinds of \(q\)-Krawtchouk, (dual, \(q\))-Hahn, \((q)\)-Racah, Al-Salam-Carlitz II, \(q\)-Meixner, \(q\)-Charlier, dual big \(q\)-Jacobi, and dual big \(q\)-Laguerre polynomials. The birth and death rates are determined by using the difference equations governing the polynomials. The stationary distributions are the normalized orthogonality measures of the polynomials. The transition probabilities are neatly expressed by the normalized polynomials and the corresponding eigenvalues. This paper is simply the discrete time versions of the known solutions of the continuous time birth and death processes.
{\copyright 2022 American Institute of Physics}Bivariate discrete Mittag-Leffler functions with associated discrete fractional operatorshttps://zbmath.org/1508.390152023-05-31T16:32:50.898670Z"Mohammed, Pshtiwan Othman"https://zbmath.org/authors/?q=ai:mohammed.pshtiwan-othman"Kürt, Cemaliye"https://zbmath.org/authors/?q=ai:kurt.cemaliye"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabet(no abstract)On a multilinear functional equationhttps://zbmath.org/1508.390162023-05-31T16:32:50.898670Z"Illarionov, A. A."https://zbmath.org/authors/?q=ai:illarionov.andrei-aSummary: The following functional equation is solved:
\[f\left( {{x_1} + z} \right) \cdots f\left( {{x_2} + z} \right)f\left( {{x_1} + \cdots + {x_{s - 1}} - z} \right) = {\phi_1}\left( x \right){\psi_1}\left( z \right) + \cdots + {\phi_m}\left( x \right){\psi_m}\left( z \right),\]
where \(x =(x_1,\dots,x_{s -1})\), for the unknowns \(f,{\psi_j}:\mathbb{C} \to \mathbb{C}\) and \({\phi_j}:{\mathbb{C}^{s - 1}} \to \mathbb{C}\) for \(s \geq 3\) and \(m \leq 4s - 5\).On positivity of orthogonal series and its applications in probabilityhttps://zbmath.org/1508.420332023-05-31T16:32:50.898670Z"Szabłowski, Paweł J."https://zbmath.org/authors/?q=ai:szablowski.pawel-jerzySummary: We give necessary and sufficient conditions for an orthogonal series to converge in the mean-squares to a nonnegative function. We present many examples and applications, in analysis and probability. In particular, we give necessary and sufficient conditions for a Lancaster-type of expansion \(\sum_{n\ge 0}c_n\alpha_n(x)\beta_n(y)\) with two sets of orthogonal polynomials \(\left\{ \alpha_n\right\}\) and \(\left\{ \beta_n\right\}\) to converge in means-squares to a nonnegative bivariate function. In particular, we study the properties of the set \(C(\alpha ,\beta )\) of the sequences \(\left\{ c_n\right\},\) for which the above-mentioned series converge to a nonnegative function and give conditions for the membership to it. Further, we show that the class of bivariate distributions for which a Lancaster type expansion can be found, is the same as the class of distributions having all conditional moments in the form of polynomials in the conditioning random variable.Weighted fractional composition operators on certain function spaceshttps://zbmath.org/1508.470352023-05-31T16:32:50.898670Z"Borgohain, D."https://zbmath.org/authors/?q=ai:borgohain.d"Naik, S."https://zbmath.org/authors/?q=ai:naik.suvedha-suresh|naik.shraddha-m|naik.sachin-manjunath|naik.sanjeev-m|naik.shanoja-r|naik.sameer-balkrishna|naik.swatee|naik.satchidananda|naik.shibabrat|naik.siddharth|naik.smita-d|naik.suketu|naik.sunanda|naik.sanket|naik.sunil-l|naik.shankarSummary: In this paper, we give some characterizations for the boundedness of weighted fractional composition operator \(D_{\varphi , u}^\beta\) from \(\alpha \)-Bloch spaces into weighted type spaces by deriving the bounds of its norm. Also, estimates for essential norm are obtained which gives necessary and sufficient conditions for the compactness of the operator \(D_{\varphi , u}^\beta \).Some classes of shapes of the rotating liquid drophttps://zbmath.org/1508.530122023-05-31T16:32:50.898670Z"Pulov, Vladimir I."https://zbmath.org/authors/?q=ai:pulov.vladimir-i"Mladenov, Ivaïlo M."https://zbmath.org/authors/?q=ai:mladenov.ivailo-mSummary: The problem of a fluid body rotating with a constant angular velocity and subjected to uniform external pressure is of real interest in both fluid dynamics and nuclear theory. Besides, from the geometrical viewpoint the sought equilibrium configuration of such system turns out to be equivalent to the problem of determining the surface of revolution with a prescribed mean curvature. In the simply connected case, the equilibrium surface can be parameterized explicitly via elliptic integrals of the first and second kind.Self-associated three-dimensional coneshttps://zbmath.org/1508.530172023-05-31T16:32:50.898670Z"Hildebrand, Roland"https://zbmath.org/authors/?q=ai:hildebrand.rolandSummary: For every proper convex cone \(K \subset{\mathbb{R}}^3\) there exists a unique complete hyperbolic affine 2-sphere with mean curvature \(-1\) which is asymptotic to the boundary of the cone. Two cones are associated if the corresponding affine spheres can be mapped to each other by an orientation-preserving isometry. This equivalence relation is generated by the groups \(SL(3,{\mathbb{R}})\) and \(S^1\), where the former acts by linear transformations of the ambient space, and the latter by multiplication of the cubic holomorphic differential of the affine sphere by unimodular complex constants. The action of \(S^1\) generalizes conic duality, which acts by multiplication of the cubic differential by \(-1\). We call a cone self-associated if it is linearly isomorphic to all its associated cones, in which case the action of \(S^1\) induces (nonlinear) isometries of the corresponding affine sphere. We give a complete classification of the self-associated cones and compute isothermal parameterizations of the corresponding affine spheres. Their metrics can be expressed in terms of degenerate Painlevé III transcendents. The boundaries of generic self-associated cones can be represented as conic hulls of vector-valued solutions of a certain third-order linear ordinary differential equation with periodic coefficients, but there exist also self-associated cones with polyhedral boundary parts. The self-associated cones are the second family of non-trivial 3-dimensional cones for which the affine spheres can be computed explicitly, the first being the semi-homogeneous cones.Probability distributions involving generalized hypergeometric functionshttps://zbmath.org/1508.600172023-05-31T16:32:50.898670Z"Meléndez, Rafael Alfonso"https://zbmath.org/authors/?q=ai:melendez.rafael-alfonso"Castillo, Jaime Antonio"https://zbmath.org/authors/?q=ai:castillo.jaime-antonio"Jiménez, Carlos Jesús"https://zbmath.org/authors/?q=ai:jimenez.carlos-jesusSummary: We define a new function of probability that involves some generalized hypergeometric functions, we found some properties and special cases such as gamma and exponential. We establish some basic functions associated with the new probability distribution like mean, the moments, characteristic function and several graphic representations are obtained for this new function of probability.Dirichlet form analysis of the Jacobi processhttps://zbmath.org/1508.600802023-05-31T16:32:50.898670Z"Grothaus, Martin"https://zbmath.org/authors/?q=ai:grothaus.martin"Sauerbrey, Max"https://zbmath.org/authors/?q=ai:sauerbrey.maxSummary: We construct and analyze the Jacobi process -- in mathematical biology referred to as Wright-Fisher diffusion -- using a Dirichlet form. The corresponding Dirichlet space takes the form of a Sobolev space with different weights for the function itself and its derivative and can be rewritten in a canonical form for strongly local Dirichlet forms in one dimension. Additionally to the statements following from the general theory on these forms, we obtain orthogonal decompositions of the Dirichlet space, derive Sobolev embeddings, verify functional inequalities of Hardy type and analyze the long time behavior of the associated semigroup. We deduce corresponding properties of the Markov process and show that it is up to minor technical modifications a solution to the Jacobi SDE. We also provide uniqueness statements for this SDE, such that properties of general solutions follow.Efficient computation of the Wright function and its applications to fractional diffusion-wave equationshttps://zbmath.org/1508.650142023-05-31T16:32:50.898670Z"Aceto, Lidia"https://zbmath.org/authors/?q=ai:aceto.lidia"Durastante, Fabio"https://zbmath.org/authors/?q=ai:durastante.fabioSummary: In this article, we deal with the efficient computation of the Wright function in the cases of interest for the expression of solutions of some fractional differential equations. The proposed algorithm is based on the inversion of the Laplace transform of a particular expression of the Wright function for which we discuss in detail the error analysis. We also present a code package that implements the algorithm proposed here in different programming languages. The analysis and implementation are accompanied by an extensive set of numerical experiments that validate both the theoretical estimates of the error and the applicability of the proposed method for representing the solutions of fractional differential equations.Novel and accurate Gegenbauer spectral tau algorithms for distributed order nonlinear time-fractional telegraph models in multi-dimensionshttps://zbmath.org/1508.651382023-05-31T16:32:50.898670Z"Ahmed, Hoda F."https://zbmath.org/authors/?q=ai:ahmed.hoda-f"Hashem, W. A."https://zbmath.org/authors/?q=ai:hashem.w-aThis article is concerned with numerical approximations to the multi-dimensional distributed-order nonlinear time-fractional telegraph equations models. The approach relies on the use of shifted Gegenbauer polynomials. The proposed method takes full advantage of the nonlocal nature of distributed order fractional differential operators. Error estimations are obtained. Also, several numerical experiments are included to support the utility of the proposed algorithm.
Reviewer: Marius Ghergu (Dublin)A wavelet collocation method based on Gegenbauer scaling function for solving fourth-order time-fractional integro-differential equations with a weakly singular kernelhttps://zbmath.org/1508.651402023-05-31T16:32:50.898670Z"Faheem, Mo"https://zbmath.org/authors/?q=ai:faheem.mo"Khan, Arshad"https://zbmath.org/authors/?q=ai:khan.arshad-m|khan.arshad-alam|khan.arshad-ahmad|khan.arshad-aliThe paper focuses on the numerical solution of fourth-order time-fractional integro-differential equations defined on a rectangular domain and containing an integral operator with a weakly singular kernel. First, scaling functions are defined as Gegenbauer orthogonal polynomials, and the Riemann-Lieuville fractional integral operator for these scaling functions is expressed using the Laplace transform. Then, the authors use a collocation method with the Gegenbauer scaling functions and uniformly distributed collocation points to convert the problem into a system of linear algebraic equations. Furthermore, the authors establish error estimates for the operators involved and the truncation error estimate. Numerical experiments are presented to confirm the theoretical results.
Reviewer: Dana Černá (Liberec)On the inf-sup stability of Crouzeix-Raviart Stokes elements in 3Dhttps://zbmath.org/1508.651602023-05-31T16:32:50.898670Z"Sauter, Stefan"https://zbmath.org/authors/?q=ai:sauter.stefan-a"Torres, Céline"https://zbmath.org/authors/?q=ai:torres.celineSummary: We consider discretizations of the stationary Stokes equation in three spatial dimensions by non-conforming Crouzeix-Raviart elements. The original definition in the seminal paper by \textit{M. Crouzeix} and \textit{P. A. Raviart} [Rev. Franc. Automat. Inform. Rech. Operat., R 7, No. 3, 33--76 (1974; Zbl 0302.65087)] is implicit and also contains substantial freedom for a concrete choice.
In this paper, we introduce \textit{basic} Crouzeix-Raviart spaces in 3D in analogy to the 2D case in a fully explicit way. We prove that this basic Crouzeix-Raviart element for the Stokes equation is inf-sup stable for polynomial degree \(k=2\) (quadratic velocity approximation). We identify spurious pressure modes for the conforming \((k, k-1)\) 3D Stokes element and show that these are eliminated by using the basic Crouzeix-Raviart space.Algebraic structure underlying spherical, parabolic, and prolate spheroidal bases of the nine-dimensional MICZ-Kepler problemhttps://zbmath.org/1508.780052023-05-31T16:32:50.898670Z"Le, Dai-Nam"https://zbmath.org/authors/?q=ai:le.dai-nam"Le, Van-Hoang"https://zbmath.org/authors/?q=ai:le.van-hoangSummary: The nonrelativistic motion of a charged particle around a dyon in \((9 + 1)\) spacetime is known as the nine-dimensional McIntosh-Cisneros-Zwanziger-Kepler problem. This problem has been solved exactly by the variable-separation method in three different coordinate systems: spherical, parabolic, and prolate spheroidal. In the present study, we establish a relationship between the variable separation and the algebraic structure of \(SO(10)\) symmetry. Each of the spherical, parabolic, or prolate spheroidal bases is proved to be a set of eigenfunctions of a corresponding nonuplet of algebraically independent integrals of motion. This finding also helps us establish connections between the bases by the algebraic method. This connection, in turn, allows calculating complicated integrals of confluent Heun, generalized Laguerre, and generalized Jacobi polynomials, which are important in physics and analytics.
{\copyright 2022 American Institute of Physics}Large \(n\) limit for the product of two coupled random matriceshttps://zbmath.org/1508.818292023-05-31T16:32:50.898670Z"Silva, Guilherme L. F."https://zbmath.org/authors/?q=ai:silva.guilherme-l-f"Zhang, Lun"https://zbmath.org/authors/?q=ai:zhang.lunSummary: For a pair of coupled rectangular random matrices we consider the squared singular values of their product, which form a determinantal point process. We show that the limiting mean distribution of these squared singular values is described by the second component of the solution to a vector equilibrium problem. This vector equilibrium problem is defined for three measures with an upper constraint on the first measure and an external field on the second measure. We carry out the steepest descent analysis for a \(4\times 4\) matrix-valued Riemann-Hilbert problem, which characterizes the correlation kernel and is related to mixed type multiple orthogonal polynomials associated with the modified Bessel functions. A careful study of the vector equilibrium problem, combined with this asymptotic analysis, ultimately leads to the aforementioned convergence result for the limiting mean distribution, an explicit form of the associated spectral curve, as well as local Sine, Meijer-G and Airy universality results for the squared singular values considered.New techniques for worldline integrationhttps://zbmath.org/1508.818402023-05-31T16:32:50.898670Z"Edwards, James P."https://zbmath.org/authors/?q=ai:edwards.james-p"Mata, C. Moctezuma"https://zbmath.org/authors/?q=ai:mata.c-moctezuma"Müller, Uwe"https://zbmath.org/authors/?q=ai:muller.uwe"Schubert, Christian"https://zbmath.org/authors/?q=ai:schubert.christianThis article reviews the formulation of quantum field theories like quantum electrodynamics or scalar quantum electrodynamics (scalars and photons) in the worldline formalism. The worldline formalism dates back to Feynman, with two papers from 1950 and 1951, respectively. The article by Edwards, Mata, Müller and Schubert summarises the current state-of-the-art and addresses open problems.
The worldline formalism has the advantage that it offers a nice compact integral representation for the amplitude, the ``Bern-Kosower master formula'' of eq. (3.2). The authors address then the open problem on how to process this formula further without re-expanding it into ordered sectors. They first discuss the case of a polynomial integrand, where a systematic method can be given. As a spin-off this method leads to interesting relations among Bernoulli numbers (the Miki relations and the Faber-Pandharipande-Zagier relations). However, the master formula of Bern-Kosower involves an exponential. In principle one could expand the exponential. Each term of the expansion is then a polynomial and could be treated with the method above. However, it is not obvious if the infinite sum from the expansion can be re-arranged in a closed form, spoiling the initial advantage of a short compact expression. For this reason, the authors advocate a different route: Expansion in inverse derivatives. In order to make this well-defined, they first take out the zero mode. In the last section of this article they report on the current state-of-the-art of the technique of the expansion in inverse derivatives.
Reviewer: Stefan Weinzierl (Mainz)Investigation of the two-cut phase region in the complex cubic ensemble of random matriceshttps://zbmath.org/1508.820172023-05-31T16:32:50.898670Z"Barhoumi, Ahmad"https://zbmath.org/authors/?q=ai:barhoumi.ahmad"Bleher, Pavel"https://zbmath.org/authors/?q=ai:bleher.pavel-m"Deaño, Alfredo"https://zbmath.org/authors/?q=ai:deano.alfredo"Yattselev, Maxim"https://zbmath.org/authors/?q=ai:yattselev.maxim-lSummary: We investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential \(V(M) = - \frac{1}{3}M^3 + tM\), where \(t\) is a complex parameter. As proven in our previous paper [\textit{P. Bleher} et al., J. Stat. Phys. 166, No. 3--4, 784--827 (2017; Zbl 1372.82015)], the whole phase space of the model, \(t\in\mathbb{C}\), is partitioned into two phase regions, \(O_{\mathsf{one}\text{-}\mathsf{cut}}\) and \(O_{\mathsf{two}\text{-}\mathsf{cut}}\), such that in \(O_{\mathsf{one}\text{-}\mathsf{cut}}\) the equilibrium measure is supported by one Jordan arc (cut) and in \(O_{\mathsf{two}\text{-}\mathsf{cut}}\) by two cuts. The regions \(O_{\mathsf{one}\text{-}\mathsf{cut}}\) and \(O_{\mathsf{two}\text{-}\mathsf{cut}}\) are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In [loc. cit.], the one-cut phase region was investigated in detail. In the present paper, we investigate the two-cut region. We prove that in the two-cut region, the endpoints of the cuts are analytic functions of the real and imaginary parts of the parameter \(t\), but not of the parameter \(t\) itself (so that the Cauchy-Riemann equations are violated for the endpoints). We also obtain the semiclassical asymptotics of the orthogonal polynomials associated with the ensemble of random matrices and their recurrence coefficients. The proofs are based on the Riemann-Hilbert approach to semiclassical asymptotics of the orthogonal polynomials and the theory of \(S\)-curves and quadratic differentials.
{\copyright 2022 American Institute of Physics}A high-gain observer with Mittag-Leffler rate of convergence for a class of nonlinear fractional-order systemshttps://zbmath.org/1508.931242023-05-31T16:32:50.898670Z"Martínez-Fuentes, O."https://zbmath.org/authors/?q=ai:martinez-fuentes.oscar"Martínez-Guerra, R."https://zbmath.org/authors/?q=ai:martinez-guerra.rafaelSummary: In this paper, a class of nonlinear fractional systems of commensurate order is analyzed in order to solve the observation problem through a high-gain Mittag-Leffler observer. Some stability conditions are established to design Mittag-Leffler observers, and a high-gain observer is proposed to estimate the unknown states of a fractional-order nonlinear system expressed in the observer canonical form. It is shown that the designed algorithm is a Mittag-Leffler observer. Finally, some numerical simulations validate the proposed methodology.