Recent zbMATH articles in MSC 33Chttps://zbmath.org/atom/cc/33C2023-11-13T18:48:18.785376ZWerkzeugAutomatic conjecturing and proving of exact values of some infinite families of infinite continued fractionshttps://zbmath.org/1521.110062023-11-13T18:48:18.785376Z"Dougherty-Bliss, Robert"https://zbmath.org/authors/?q=ai:dougherty-bliss.robert"Zeilberger, Doron"https://zbmath.org/authors/?q=ai:zeilberger.doronThe contribution extends a recently developed technique to conjecture values of infinite continued fractions. Instead of conjecturing further values for other fractions, the contribution proves the correct value for three families of continued fractions, which when specialized confirm previously posted conjectures. The authors also provide a Maple implementation of generalized continued fractions that can be used to automatically prove the correct value or conjecture additional values.
The authors are strong proponents of computer-aided discovery and provide suitable tools for the discovery of values for other fractions. Indeed in rare cases the tools might correctly yield a value, which due to the strategy used, even provides a formal proof of the correctness of the value. The implementation relies on a clever representation of continued fractions by means of recurrences. However, the three families of continued fractions that they discuss are proven by traditional means and the authors mention that their implementation currently is unable to handle such cases, but they believe that suitable extensions might be able to handle similar cases automatically soon.
The contribution is excellently written and assumes a fair knowledge of continued fractions and standard techniques involving series. However, all proofs are supplied in sufficient detail and the historical remarks are quite entertaining. Overall, any graduate of mathematics should be able fully appreciate this contribution.
Reviewer: Andreas Maletti (Leipzig)Identities associated to a generalized divisor function and modified Bessel functionhttps://zbmath.org/1521.110542023-11-13T18:48:18.785376Z"Banerjee, Debika"https://zbmath.org/authors/?q=ai:banerjee.debika"Maji, Bibekananda"https://zbmath.org/authors/?q=ai:maji.bibekanandaThe paper investigates, for \(k\in\mathbb{N}\) and \(z\in \mathbb{C}\), the series
\[
\sum_{n=1}^\infty \sigma_z^{(k)}(n)n^{\nu/2}K_\nu(a\sqrt{nx}),
\]
where \(\sigma_z(n):=\sum_{d|n}d^z\), \(\sigma(n):=\sigma_1(n)\), \(d(n):=\sigma_0(n)\), \(\nu\) is a complex number wit \(\mathrm{Re}(\nu)>0\) and the modified \(K\)-Bessel function \(K_\nu(x)\) is defined as one of the solutions of the modified Bessel differential equation \(x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}-(x^2+\nu^2)y=0\).
Out of the five theorems proved, we mention only two of them, the least sophisticated ones.
Theorem 1. Let \(x\) be a positive real number. Then
\[
8\pi^2\sum_{n=1}^\infty \sigma(n)K_0(4\pi\sqrt{nx})=\frac{1}{24x^2}-\frac{1}{2x}-(1+\log (x) +\gamma)\zeta(2)-\zeta'(2)
+\sum_{n=1}^\infty \sigma(n)\left(\frac{1}{n^2}-\frac{1}{(n+x)^2}\right).
\]
Theorem 2. For any positive real number \(x\) such that \(x\not\in\mathbb{Z}_+\), we have
\[
\sum_{n=1}^\infty d(n)K_0(4\pi\sqrt{nx})=\frac{x}{2\pi^2}\sum_{n=1}^\infty d(n)\frac{\log (n/x)}{n^2 - x^2}-\frac{1}{4}\left(\gamma + \frac{\log x}{2} + \frac{\log (2\pi\sqrt{x})}{\pi^2x}\right).
\]
Reviewer: Stelian Mihalas (Timişoara)On degenerate gamma matrix functions and related functionshttps://zbmath.org/1521.150072023-11-13T18:48:18.785376Z"Akel, Mohamed"https://zbmath.org/authors/?q=ai:akel.mohamed-s"Bakhet, Ahmed"https://zbmath.org/authors/?q=ai:bakhet.ahmed"Abdalla, Mohamed"https://zbmath.org/authors/?q=ai:abdalla.mohamed"He, Fuli"https://zbmath.org/authors/?q=ai:he.fuliSummary: Recently, the applications of special functions of matrix arguments have received more attention in many fields, such as theoretical physics, number theory, probability theory, engineering and theory of group representations. Gaining enlightenment from these works, in this paper, we introduce the degenerate gamma matrix function, the degenerate zeta matrix function, the degenerate digamma matrix function, the degenerate polygamma matrix function and the degenerate Gauss hypergeometric matrix function. Basic properties of these functions are discussed. In addition, we derive some interesting formulas related to these functions and special cases.Correction to: ``Jacobi ensemble, Hurwitz numbers and Wilson polynomials''https://zbmath.org/1521.150312023-11-13T18:48:18.785376Z"Gisonni, Massimo"https://zbmath.org/authors/?q=ai:gisonni.massimo"Grava, Tamara"https://zbmath.org/authors/?q=ai:grava.tamara"Ruzza, Giulio"https://zbmath.org/authors/?q=ai:ruzza.giulioCorrection to the authors' paper [ibid. 111, No. 3, Paper No. 67, 38 p. (2021; Zbl 1478.15049)].Hardy inequalities for fractional \((k,a)\)-generalized harmonic oscillatorshttps://zbmath.org/1521.220162023-11-13T18:48:18.785376Z"Teng, Wentao"https://zbmath.org/authors/?q=ai:teng.wentaoSummary: We will define a-deformed Laguerre operators \(L_{a,\alpha}\) and a-deformed Laguerre holomorphic semigroups on \(L^2 ((0, \infty), d\mu_{a,\alpha})\). Then we give a spherical harmonic expansion, which reduces to the Bochner-type identity when taking the boundary value \(z = \pi i/2\), of the \((k, a)\)-generalized Laguerre semigroup introduced by Ben Saïd, Kobayashi and Ørsted. We prove a Hardy inequality for fractional powers of the \(a\)-deformed Dunkl harmonic oscillator \(\triangle_{k,a} := |x|^{2-a} \triangle_k - |x|^a\) using this expansion. When \(a = 2\), the fractional Hardy inequality reduces to that of Dunkl-Hermite operators given by Ciaurri, Roncal and Thangavelu. The operators \(L_{a,\alpha}\) also give a tangible characterization of the radial part of the \((k, a)\)-generalized Laguerre semigroup on each \(k\)-spherical component \(\mathcal{H}^m_k (\mathbb{R}^n)\) for
\[
\lambda_{k,a,m} :=\frac{2m + 2 \langle k\rangle + N - 2}{a}\geq -\frac{1}{2}
\]
defined via a decomposition of the unitary representation.Certain formulas for solutions to trinomial and tetranomial algebraic equationshttps://zbmath.org/1521.330032023-11-13T18:48:18.785376Z"Mikhalkin, Evgeniĭ N."https://zbmath.org/authors/?q=ai:mikhalkin.evgenii-nSummary: Algebraic equations with one and two parameters are considered. We prove that solutions to such equations can be represented as linear combination of generalized hypergeometric series. This result allows to express (nonlinearly) solutions to cubic and quartic equations by Gauss hypergeometric series.On the theory of orthogonal polynomials for the weight \(x^\nu \exp (-x-t/x)\). IIhttps://zbmath.org/1521.330042023-11-13T18:48:18.785376Z"Yakubovich, S."https://zbmath.org/authors/?q=ai:yakubovich.semyon-bSummary: Orthogonal polynomials for the weight \(x^\nu \exp (-x-t/x)\), \(x,t>0\), \(\nu \in \mathbb{R}\) are investigated without the use of the Chen-Ismail ladder operators approach. In this part we derive explicit representations, recurrence relations for coefficients, the generating function and Rodrigues-type formula.
For Part I, see [ibid. 33, No. 9, 735--746 (2022; Zbl 07582201)].On the extreme zeros of Jacobi polynomialshttps://zbmath.org/1521.330052023-11-13T18:48:18.785376Z"Nikolov, Geno"https://zbmath.org/authors/?q=ai:nikolov.geno-pSummary: By applying the Euler-Rayleigh method to a specific representation of the Jacobi polynomials as hypergeometric functions, we obtain new bounds for their largest zeros. In particular, we derive upper and lower bound for \(1-x_{nn}^2(\lambda )\), with \(x_{nn}(\lambda )\) being the largest zero of the \(n\)-th ultraspherical polynomial \(P_n^{(\lambda )}\). For every fixed \(\lambda >-1/2\), the limit of the ratio of our upper and lower bounds for \(1-x_{nn}^2(\lambda )\) does not exceed 1.6. This paper is a continuation of [the author, Proc. Am. Math. Soc. 147, No. 4, 1541--1550 (2019; Zbl 1410.33030)].
For the entire collection see [Zbl 1511.65004].Christoffel functions for multiple orthogonal polynomialshttps://zbmath.org/1521.330062023-11-13T18:48:18.785376Z"Świderski, Grzegorz"https://zbmath.org/authors/?q=ai:swiderski.grzegorz"Van Assche, Walter"https://zbmath.org/authors/?q=ai:van-assche.walterSummary: We study weak asymptotic behavior of the Christoffel-Darboux kernel on the main diagonal corresponding to multiple orthogonal polynomials. We show that under some hypotheses the weak limit of \(\frac{1}{n} K_n (x, x) \mathrm{d} \mu\) is the same as the limit of the normalized zero counting measure of type II multiple orthogonal polynomials. We also study an extension of Nevai's operators to our context.Periodic ultradiscrete transformations of the spacehttps://zbmath.org/1521.330072023-11-13T18:48:18.785376Z"Bykovskiĭ, V. A."https://zbmath.org/authors/?q=ai:bykovskii.v-a"Monina, M. D."https://zbmath.org/authors/?q=ai:monina.m-dSummary: In previous works we have constructed five periodic ultradiscrete transformations of the plane. In the present work we construct similar transformation for 3-dimensional space.On some formulas for the Horn function \(H_7(a, b, b^\prime; c; w, z)\)https://zbmath.org/1521.330082023-11-13T18:48:18.785376Z"Brychkov, Yu. A."https://zbmath.org/authors/?q=ai:brychkov.yury-a"Savischenko, N. V."https://zbmath.org/authors/?q=ai:savischenko.nikolay-vSummary: Some new relations for the Horn function \(H_7(a, b, b^\prime; c; w, z)\) are obtained including differentiation, integration, reduction and summation formulas. Some generating functions for various special functions are given in terms of these Horn functions.Semilinear evolution models with scale-invariant friction and visco-elastic dampinghttps://zbmath.org/1521.351202023-11-13T18:48:18.785376Z"Mezadek, Abdelatif Kainane"https://zbmath.org/authors/?q=ai:mezadek.abdelatif-kainane"Reissig, Michael"https://zbmath.org/authors/?q=ai:reissig.michaelSummary: In this paper we study the global (in time) existence of small data Sobolev solutions and blow-up of Sobolev solutions to the Cauchy problem for semilinear evolution models with scale-invariant friction, visco-elastic damping and power nonlinearity. We are interested in critical exponents and the question how higher regularity in the data influences the admissible range of exponents \(p\) in the power nonlinearity to get global (in time) small data Sobolev solutions.Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, circular \(\beta\)-ensemble and double confluent Heun equationhttps://zbmath.org/1521.370882023-11-13T18:48:18.785376Z"Grava, Tamara"https://zbmath.org/authors/?q=ai:grava.tamara"Mazzuca, Guido"https://zbmath.org/authors/?q=ai:mazzuca.guidoSummary: We consider the discrete defocusing nonlinear Schrödinger equation in its integrable version, which is called defocusing Ablowitz-Ladik lattice. We consider periodic boundary conditions with period \(N\) and initial data sampled according to the Generalized Gibbs ensemble. In this setting, the Lax matrix of the Ablowitz-Ladik lattice is a random CMV-periodic matrix and it is related to the Killip-Nenciu Circular \(\beta\)-ensemble at high-temperature. We obtain the generalized free energy of the Ablowitz-Ladik lattice and the density of states of the random Lax matrix by establishing a mapping to the one-dimensional log-gas. For the Gibbs measure related to the Hamiltonian of the Ablowitz-Ladik flow, we obtain the density of states via a particular solution of the double-confluent Heun equation.On approximation of functions on the half-line by Laguerre polynomials and functionshttps://zbmath.org/1521.410022023-11-13T18:48:18.785376Z"Dautov, R. Z."https://zbmath.org/authors/?q=ai:dautov.rafail-zSummary: In this paper, we define a parametric scale of non-uniformly Laguerre-weighted Sobolev spaces and a scale of spaces isometric to it. The second scale includes the ordinary (non-weighted) Sobolev space. Approximations by generalized Laguerre polynomials and functions in these spaces are studied. Some weighted \(L^2\)-analogues of Bernstein's inequality for these polynomials and functions are obtained.Christoffel-Darboux kernels in several real variableshttps://zbmath.org/1521.420222023-11-13T18:48:18.785376Z"Cichoń, Dariusz"https://zbmath.org/authors/?q=ai:cichon.dariusz"Szafraniec, Franciszek Hugon"https://zbmath.org/authors/?q=ai:szafraniec.franciszek-hugonSummary: The Christoffel-Darboux kernels for orthogonal polynomials in several real variables are investigated within the context of the three-term relation reformulated for this purpose. As suggestive examples of orthogonality, we propose to discuss the two simple algebraic cases: the unit circle and the Bernoulli lemniscate.The two-sided quaternionic Dunkl transform and Hardy's theoremhttps://zbmath.org/1521.440032023-11-13T18:48:18.785376Z"Essenhajy, Mohamed"https://zbmath.org/authors/?q=ai:essenhajy.mohamed"Fahlaoui, Said"https://zbmath.org/authors/?q=ai:fahlaoui.saidSummary: In this paper we study the two-sided quaternionic Dunkl transform. We establish its fundamental properties, such as Plancherel and inversion formula. Finally, we apply the Dunkl transform properties to establish an analogue of Hardy's theorem.Discrete self-similar and ergodic Markov chainshttps://zbmath.org/1521.600422023-11-13T18:48:18.785376Z"Miclo, Laurent"https://zbmath.org/authors/?q=ai:miclo.laurent"Patie, Pierre"https://zbmath.org/authors/?q=ai:patie.pierre"Sarkar, Rohan"https://zbmath.org/authors/?q=ai:sarkar.rohanSummary: The first aim of this paper is to introduce a class of Markov chains on \(\mathbb{Z}_+\) which are discrete self-similar in the sense that their semigroups satisfy an invariance property expressed in terms of a discrete random dilation operator. After showing that this latter property requires the chains to be upward skip-free, we first establish a gateway intertwining relation between the semigroup of such chains and the one of spectrally negative self-similar Markov processes on \(\mathbb{R}_+\). As a by-product, we prove that each of these Markov chains, after an appropriate scaling, converge in the Skorohod metric to the associated self-similar Markov process. By a linear perturbation of the generator of these Markov chains, we obtain a class of ergodic Markov chains which are nonreversible. By means of intertwining relations and their strengthened interweaving versions, we derive several deep analytical properties of such ergodic chains, including the description of the spectrum, the spectral expansion of their semigroups and the study of their convergence to equilibrium in the \(\Phi\)-entropy sense as well as their hypercontractivity property.Some perpetual integral functionals of the three-dimensional Bessel processhttps://zbmath.org/1521.600482023-11-13T18:48:18.785376Z"Tsuzuki, Yukihiro"https://zbmath.org/authors/?q=ai:tsuzuki.yukihiroLaplace transforms of some integral functionals of the three-dimensional Bessel process are computed in terms of modified Bessel functions, Gauss' hypergeometric functions, and confluent hypergeometric functions. For the proof, solutions of the Skorokhod embedding problem are used and the martingale with respect to an enlarged filtration is constructed.
Reviewer: Alexandra Rodkina (College Station)Quantum similarity index and Rényi complexity ratio of Kratzer type potential and compared with that of inverse square and Coulomb type potentialshttps://zbmath.org/1521.810202023-11-13T18:48:18.785376Z"Nath, Debraj"https://zbmath.org/authors/?q=ai:nath.debraj"Carbó-Dorca, Ramon"https://zbmath.org/authors/?q=ai:carbo-dorca.ramonSummary: In this paper, we define three sets of exact solutions for potentials of the type two-parameter Kratzer, inverse square, and Coulomb. The Laguerre polynomials express two sets of solutions, and Bessel functions enter the third one. Also, for these three solutions, we define the exact analytical expressions of the quantum similarities, disequilibria, and entropic moments. In addition, we studied the behavior of quantum dissimilarity, quantum similarity index, and Rényi complexity ratio using every pair of different solutions.Pretty good quantum fractional revival in paths and cycleshttps://zbmath.org/1521.810232023-11-13T18:48:18.785376Z"Chan, Ada"https://zbmath.org/authors/?q=ai:chan.ada"Drazen, Whitney"https://zbmath.org/authors/?q=ai:drazen.whitney"Eisenberg, Or"https://zbmath.org/authors/?q=ai:eisenberg.or"Kempton, Mark"https://zbmath.org/authors/?q=ai:kempton.mark"Lippner, Gabor"https://zbmath.org/authors/?q=ai:lippner.gaborSummary: We initiate the study of pretty good quantum fractional revival in graphs, a generalization of pretty good quantum state transfer in graphs. We give a complete characterization of pretty good fractional revival in a graph in terms of the eigenvalues and eigenvectors of the adjacency matrix of a graph. This characterization follows from a lemma due to Kronecker on Diophantine approximation, and is similar to the spectral characterization of pretty good state transfer in graphs. Using this, we give complete characterizations of when pretty good fractional revival can occur in paths and in cycles.Monotone complexity measures of multidimensional quantum systems with central potentialshttps://zbmath.org/1521.810352023-11-13T18:48:18.785376Z"Dehesa, Jesús S."https://zbmath.org/authors/?q=ai:dehesa.jesus-sSummary: In this work, we explore the (inequality-type) properties of the monotone complexity-like measures of the internal complexity (disorder) of multidimensional non-relativistic electron systems subject to a central potential. Each measure quantifies the combined balance of two spreading facets of the electron density of the system. We show that the hyperspherical symmetry (i.e., the multidimensional spherical symmetry) of the potential allows Cramér-Rao, Fisher-Shannon, and Lopez-Ruiz, Mancini, Calbet-Rényi complexity measures to be expressed in terms of the space dimensionality and the hyperangular quantum numbers of the electron state. Upper bounds, mutual complexity relationships, and complexity-based uncertainty relations of position-momentum type are also found by means of the electronic hyperangular quantum numbers and, at times, the Heisenberg-Kennard relation. We use a methodology that includes a variational approach with a covariance matrix constraint and some algebraic linearization techniques of hyperspherical harmonics and Gegenbauer orthogonal polynomials.
{\copyright 2023 American Institute of Physics}The discrete Chebyshev-Meckler-Mermin-Schwarz polynomials and spin algebrahttps://zbmath.org/1521.811292023-11-13T18:48:18.785376Z"Garg, Anupam"https://zbmath.org/authors/?q=ai:garg.anupamSummary: The polynomials discovered by Chebyshev and subsequently related to spin transition probabilities by \textit{A. Meckler} [``Majorana formula'', Phys. Rev. 111, No. 6, 1447--1449 (1958; \url{doi:10.1103/PhysRev.111.1447})] and \textit{N. D. Mermin} and \textit{G. M. Schwarz} [``Joint distributions and local realism in the higher-spin Einstein-Podolsky-Rosen experiment'', Found. Phys. 12, 101--135 (1982; \url{doi:10.1007/BF00736844})] are studied, and their application to phase space representations of spin states and operators is examined. In particular, a formula relating the end-point value of the polynomials to scale factors relating different phase space representations of spherical harmonic operators is found. This formula is applied to illustrative calculations of Wigner functions for a single spin and the singlet state of a pair of spins.
{\copyright 2022 American Institute of Physics}Two-body Coulomb problem and \(g^{(2)}\) algebra (once again about the hydrogen atom)https://zbmath.org/1521.814942023-11-13T18:48:18.785376Z"Turbiner, Alexander V."https://zbmath.org/authors/?q=ai:turbiner.alexander"Escobar Ruiz, Adrian M."https://zbmath.org/authors/?q=ai:escobar-ruiz.mauricio-aSummary: Taking the Hydrogen atom as an example it is shown that if the symmetry of a three-dimensional system is \(O(2) \oplus Z_2\), the variables \((r, \rho, \varphi)\) allow a separation of the variable \(\varphi\), and the eigenfunctions define a new family of orthogonal polynomials in two variables, \((r, \rho^2)\). These polynomials are related to the finite-dimensional representations of the algebra \(gl(2) \ltimes R^3\in g^{(2)}\) (discovered by S Lie around 1880 which went almost unnoticed), which occurs as the hidden algebra of the \(G_2\) rational integrable system of 3 bodies on the line with 2- and 3-body interactions (the Wolfes model). Namely, those polynomials occur intrinsically in the study of the Zeeman effect on Hydrogen atom. It is shown that in the variables \((r, \rho, \varphi)\) in the quasi-exactly-solvable generalized Coulomb problem new polynomial eigenfunctions in \((r, \rho^2)\)-variables are found.Analysing (cosmological) singularity avoidance in loop quantum gravity using \(\mathrm{U}(1)^3\) coherent states and Kummer's functionshttps://zbmath.org/1521.830442023-11-13T18:48:18.785376Z"Giesel, Kristina"https://zbmath.org/authors/?q=ai:giesel.kristina"Winnekens, David"https://zbmath.org/authors/?q=ai:winnekens.davidSummary: Using a new procedure based on Kummer's Confluent Hypergeometric Functions, we investigate the question of singularity avoidance in loop quantum gravity (LQG) in the context of \(\mathrm{U}(1)^3\) complexifier coherent states and compare obtained results with already existing ones. Our analysis focuses on the dynamical operators, denoted by \(\hat{q}^{i_0}_{I_0}(r)\), whose products are the analogue of the inverse scale factor in LQG and also play a pivotal role for other dynamical operators such as matter Hamiltonians or the Hamiltonian constraint.
For graphs of cubic topology and linear powers in \(\hat{q}^{i_0}_{I_0}(r)\), we obtain the correct classical limit and demonstrate how higher order corrections can be computed with this method. This extends already existing techniques in the way how the involved fractional powers are handled. We also extend already existing formalisms to graphs with higher-valent vertices. For generic graphs and products of \(\hat{q}^{i_0}_{I_0}(r)\), using estimates becomes inevitable and we investigate upper bounds for these semiclassical expectation values. Compared to existing results, our method allows to keep fractional powers involved in \(\hat{q}^{i_0}_{I_0}(r)\) throughout the computations, which have been estimated by integer powers elsewhere. Similar to former results, we find a non-zero upper bound for the inverse scale factor at the initial singularity. Additionally, our findings provide some insights into properties and related implications of the results that arise when using estimates and can be used to look for improved estimates.Vector and scalar spherical harmonic spectral equations for rapidly rotating anisotropic alpha-effect dynamoshttps://zbmath.org/1521.860422023-11-13T18:48:18.785376Z"Phillips, C. G."https://zbmath.org/authors/?q=ai:phillips.christopher-gSummary: Spectral equations are derived for a mean field induction equation, \(\partial\overline{\boldsymbol{B}}/\partial t - \nabla^2\overline{\boldsymbol{B}} = R\nabla\times\boldsymbol{F}\) with an \(\boldsymbol{\alpha}\)-effect, considered appropriate for rapid rotation, given by \(\boldsymbol{F} = \boldsymbol{\alpha}\cdot\overline{\boldsymbol{B}} = a_1\overline{\boldsymbol{B}} + a_3\hat{\boldsymbol{z}}\cdot\overline{\boldsymbol{B}}\hat{\boldsymbol{z}}\), where \((\hat{\boldsymbol{x}}, \hat{\boldsymbol{y}}, \hat{\boldsymbol{z}})\) are Cartesian unit vectors, \(a_1(r, \theta, \phi)\), \(a_3(r, \theta, \phi)\) are scalar functions of position, \((r, \theta, \phi)\) are spherical polar coordinates and \(R\) is the magnetic Reynolds number. The effect of rotation on convection for different boundaries and parameters is discussed. The effect of the flow structure on \(\boldsymbol{\alpha}\) for different geostrophic and near geostrophic models is analysed. The vector spherical harmonics
\[
\boldsymbol{Y}^m_{n, n_1}(\theta, \phi) = (-1)^{n-m}[2n+1]^{1/2}\sum\limits_{\mu = -1, 0, 1}\begin{pmatrix} n & n_1 & 1 \\ m & \mu-m & -\mu \end{pmatrix}\mathrm{Y}_{n_1}^{m-\mu}\boldsymbol{e}_\mu,
\]
where \(\boldsymbol{e}_{-1} = (\hat{\boldsymbol{x}}-\mathrm{i}\hat{\boldsymbol{y}})/2^{1/2}\), \(\boldsymbol{e}_0 = \hat{\boldsymbol{z}}\), \(\boldsymbol{e}_1 = -(\hat{\boldsymbol{x}}-\mathrm{i}\hat{\boldsymbol{y}})/2^{1/2}\), the \(2 \times 3\) matrix is a Wigner 3J coefficient and \(\mathrm{Y}^m_n = \mathrm{Y}^m_n(\theta, \phi)\) are scalar spherical harmonics, are used to derive the vector \(\boldsymbol{Y}^m_{n, n_1}\) forms of the induction equation for this \(\boldsymbol{\alpha}\)-effect. The solenoidal condition \(\nabla\cdot\overline{\boldsymbol{B}} = 0\) is imposed by relating the \(\boldsymbol{Y}^m_{n, n_1}\) formalism to the toroidal-poloidal harmonic formalism, \(\boldsymbol{T}^m_n = \nabla\times(\boldsymbol{r}T^m_n\mathrm{Y}^m_n)\) and \(\boldsymbol{S}^m_n = \nabla\times\nabla\times(\boldsymbol{r}S^m_n\mathrm{Y}^m_n)\). The \(T^m_n\) and \(S^m_n\) components of the induction equation are thus derived in terms of \(F^m_{n, n_1}\), the \(\boldsymbol{Y}^m_{n, n_1}\) components of \(\boldsymbol{F}\); \(\boldsymbol{F} = \sum_{n_1=n-1}^{n+1}\sum_{m=-n}^n\sum^\infty_{n=0}F^m_{n, n_1}\boldsymbol{Y}^m_{n. n_1}\). These combined \(T^m_n/\boldsymbol{Y}^m_{n, n_1}\), \(S^m_n/\boldsymbol{Y}^m_{n, n_1}\) vector spectral equations are then transformed into interaction type \((a_{n_a}S_nS_N)_{\mathrm{I}}\), \((a_{n_a}T_nT_N)_{\mathrm{I}}\), \((a_{n_a}S_nT_N)_{\mathrm{I}}\), \((a_{n_a}T_nS_N)_{\mathrm{I}}\) and \((a_{n_a}S_nS_N)_{\mathrm{A}}\), \((a_{n_a}T_nT_N)_{\mathrm{A}}\), \((a_{n_a}S_nT_N)_{\mathrm{A}}\), \((a_{n_a}T_nS_N)_{\mathrm{A}}\) equations for the isotropic and anisotropic components of \(\boldsymbol{\alpha}\). As an application of the general spectral equations derived herein, the interaction equations can be specialised by restricting \(a_1\) and \(a_3\) to be proportional to \(r\cos\theta\) or \(\cos\theta\), or restricting \(\overline{\boldsymbol{B}}\) and \(\boldsymbol{\alpha}\) to be axisymmetric. These equations are then compared to those of previous works. The differences between the equations derived herein and those of past works provide corrections and account for, at least in part, the differences in numerical solutions of the past works.Kramer-type sampling theorems associated with higher-order differential equationshttps://zbmath.org/1521.940172023-11-13T18:48:18.785376Z"Markett, Clemens"https://zbmath.org/authors/?q=ai:markett.clemensSummary: For many decades, \textit{H. P. Kramer}'s sampling theorem [J. Math. Phys., Mass. Inst. Techn. 38, 68--72 (1959; Zbl 0196.31702)] has been attracting enormous interest in view of its important applications in various branches. In this paper we present a new approach to a Kramer-type theory based on spectral differential equations of higher order on an interval of the real line. Its novelty relies partly on the fact that the corresponding eigenfunctions are orthogonal with respect to a scalar product involving a classical measure together with a point mass at a finite endpoint of the domain. In particular, a new sampling theorem is established, which is associated with a self-adjoint Bessel-type boundary value problem of fourth-order on the interval \([0,1]\). Moreover, we consider the Laguerre and Jacobi differential equations and their higher-order generalizations and establish the Green-type formulas of the differential operators as an essential key towards a corresponding sampling theory.