Recent zbMATH articles in MSC 33Chttps://www.zbmath.org/atom/cc/33C2021-04-16T16:22:00+00:00WerkzeugSelf-replication and Borwein-like algorithms.https://www.zbmath.org/1456.112382021-04-16T16:22:00+00:00"Guillera, Jesús"https://www.zbmath.org/authors/?q=ai:guillera.jesusSummary: Using a self-replicating method, we generalize with a free parameter some Borwein algorithms for the number \(\pi\). This generalization includes values of the gamma function like \(\Gamma (1/3)\), \(\Gamma (1/4)\), and of course \(\Gamma (1/2)=\sqrt{\pi}\). In addition, we give new rapid algorithms for the perimeter of an ellipse.Hyperelliptic integrals modulo \(p\) and Cartier-Manin matrices.https://www.zbmath.org/1456.140362021-04-16T16:22:00+00:00"Varchenko, Alexander"https://www.zbmath.org/authors/?q=ai:varchenko.alexander-nSummary: The hypergeometric solutions of the KZ equations were constructed almost 30 years ago. The polynomial solutions of the KZ equations over the finite field \(\mathbb{F}_p\) with a prime number \(p\) of elements were constructed only recently. In this paper we consider an example of the KZ equations whose hypergeometric solutions are given by hyperelliptic integrals of genus \(g\). It is known that in this case the total \(2g\)-dimensional space of holomorphic (multivalued) solutions is given by the hyperelliptic integrals. We show that the recent construction of the polynomial solutions over the field \(\mathbb{F}_p\) in this case gives only a \(g\)-dimensional space of solutions, that is, a ``half'' of what the complex analytic construction gives. We also show that all the constructed polynomial solutions over the field \(\mathbb{F}_p\) can be obtained by reduction modulo \(p\) of a single distinguished hypergeometric solution. The corresponding formulas involve the entries of the Cartier-Manin matrix of the hyperelliptic curve. That situation is analogous to an example of the elliptic integral considered in the classical paper [\textit{Yu. I. Manin}, Izv. Akad. Nauk SSSR, Ser. Mat. 25, 153--172 (1961; Zbl 0102.27802)].Monotonicity and convexity involving generalized elliptic integral of the first kind.https://www.zbmath.org/1456.330162021-04-16T16:22:00+00:00"Zhao, Tie-Hong"https://www.zbmath.org/authors/?q=ai:zhao.tiehong"Wang, Miao-Kun"https://www.zbmath.org/authors/?q=ai:wang.miaokun"Chu, Yu-Ming"https://www.zbmath.org/authors/?q=ai:chu.yumingThe purpose of this paper is to present monotonicity properties and convexity properties for functions of generalized elliptic integral of the first kind. These generalized elliptic integrals can be expressed in hypergeometric form and the related Grötzsch ring function $\mu_a(r)$ can be written as a quotient of generalized elliptic integrals of the first kind.
The proofs use formulas for hypergeometric functions, Ramanujan constant function $R(a)$, digamma function.
Reviewer: Thomas Ernst (Uppsala)On the zeros of cross-product Bessel functions in oblique derivative boundary-value problems.https://www.zbmath.org/1456.330052021-04-16T16:22:00+00:00"Budzinskiy, S. S."https://www.zbmath.org/authors/?q=ai:budzinskiy.stanislav-sSummary: Combinations of the cross-products of Bessel functions that arise in oblique derivative boundary-value problems for the Laplace operator in a ring are considered. The behavior of zeros of these functions as the ring thickness tends to zero is studied. It is shown that the zeros are divided into two classes, as in the case of Neumann boundary conditions. Some of them remain finite in the limit, while others become infinitely large. Asymptotic expressions for the zeros are found in the case of a fixed inclination angle of the derivative.A small trove of functional equations.https://www.zbmath.org/1456.330032021-04-16T16:22:00+00:00"Glasser, M. Larry"https://www.zbmath.org/authors/?q=ai:glasser.m-larrySummary: A new proof is presented for an old algebraic identity which is then used to produce the general functional relation \[\sum_{k=0}^{n-1} \frac{(m)_k}{k!} g(m,k) + \sum^{m-1}_{k=0} \frac{(n)_k}{k!} g(k,n) = g(0,0), \] where \(g\) is an Euler transform, and a realted integral identity. Several examples are given.Quenched many-body quantum dynamics with \(k\)-body interactions using \(q\)-Hermite polynomials.https://www.zbmath.org/1456.814962021-04-16T16:22:00+00:00"Vyas, Manan"https://www.zbmath.org/authors/?q=ai:vyas.manan"Kota, V. K. B."https://www.zbmath.org/authors/?q=ai:kota.v-k-bError analysis of the Wiener-Askey polynomial chaos with hyperbolic cross approximation and its application to differential equations with random input.https://www.zbmath.org/1456.651732021-04-16T16:22:00+00:00"Luo, Xue"https://www.zbmath.org/authors/?q=ai:luo.xueSummary: It is well-known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the ``curse of dimensionality'' in some degree. In this note, we give the error estimate of hyperbolic cross (HC) approximations with all sorts of Askey polynomials. These polynomials are useful in generalized polynomial chaos (gPC) in the field of uncertainty quantification. The exponential convergences in both regular and optimized HC approximations have been shown under the condition that the random variable depends on the random inputs smoothly in some degree. Moreover, we apply gPC to numerically solve the ordinary differential equations with slightly higher dimensional random inputs. Both regular and optimized HC have been investigated with Laguerre-chaos, Charlier-chaos and Hermite-chaos in the numerical experiment. The discussion of the connection between the standard ANOVA approximation and Galerkin approximation is in the appendix.Blow-up for Strauss type wave equation with damping and potential.https://www.zbmath.org/1456.350452021-04-16T16:22:00+00:00"Dai, Wei"https://www.zbmath.org/authors/?q=ai:dai.wei.3|dai.wei.5|dai.wei.2|dai.wei.6|dai.wei.4|dai.wei.7|dai.wei"Kubo, Hideo"https://www.zbmath.org/authors/?q=ai:kubo.hideo"Sobajima, Motohiro"https://www.zbmath.org/authors/?q=ai:sobajima.motohiroIn this paper, the authors study the blow-up phenomenon for the Cauchy problem of the wave equation with space-dependent critical damping and potential terms in \(\mathbb{R}^n\) (\(n\ge 1\)), \(\partial_t^2 u -\Delta u + Ar^{-1}\partial_t u +Br^{-2}u=|u|^p\), with
\(B>-(n/2-1)^2\), \(0\leq A < n-1+2\rho\) and \(\rho=\sqrt{(n/2-1)^2+B}-(n/2-1)\). Heuristically, the damping coefficient \(A\) and the potential coefficient \(B\) will lead to certain spatial dimensional shift from \(n\) to \(n+A\) and \(n+\rho\) separately. For small, nontrivial, nonnegative data of size \(\varepsilon\), it is shown that the solutions blow up in finite time for \(1+1 / (n+\rho-1) < p< p_c=\max(p_S(n+A), 1+2/(n+\rho-1))\), if \(p_c>1+1/(n+\rho-1)\). Here \(p_S(k)\) is the positive root of \((k-1)p(p-1)=2(p+1)\) (when \(k>1\), otherwise \(p_S(k)=\infty\)), which is also known as the Strauss exponent. In addition, the expected upper bound of the lifespan is also obtained. In order to prove the blow-up result, the authors employ the test function method. The test functions are based on a family of special solutions, inside the light cone \(|x|< t+\lambda\), to the linear dual problem \(\partial_t^2\Psi - Ar^{-1}\partial_t \Psi -\Delta \Psi +Br^{-2}\Psi=0\), which have the form
\(\Psi_{\beta}(t,x)=r^\rho (t+r+\lambda)^{-\beta}\phi(\frac{2r}{t+r+\lambda})\), (\(\beta\in\mathbb{R}\), \(\lambda\geq 0\)), for hypergeometric functions \(\phi\) depending on \(n\), \(A\), \(B\).
Reviewer: Chengbo Wang (Hangzhou)Supercongruences arising from hypergeometric series identities.https://www.zbmath.org/1456.110052021-04-16T16:22:00+00:00"Liu, Ji-Cai"https://www.zbmath.org/authors/?q=ai:liu.jicaiThis paper refines a supercongruence of \textit{T. Kilbourn} [Acta Arith. 123, No. 4, 335--348 (2006; Zbl 1170.11008)] about the identity \[ a(p)=p^3-2p^2-7-N(p) \] studied by \textit{S. Ahlgren} and \textit{K. Ono} [J. Reine Angew. Math. 518, 187--212 (2000; Zbl 0940.33002)], by \textit{B. van Geemen} and \textit{N. O. Nygaard} [J. Number Theory 53, No. 1, 45--87 (1995; Zbl 0838.11047)], and by \textit{H. A. Verrill} [CRM Proc. Lecture Notes 19, 333--340. Providence, RI: Amer. Math. Soc. (1999; Zbl 0942.14022] in connection to the modular Calabi-Yau threefold for odd primes \(p\) \[ x+\frac{1}{x}+y+\frac{1}{y}+z+\frac{1}{z}+w+\frac{1}{w}=0 \] associated with truncated hypergeometric series.
Namely, the author establishes that \[a(p) \equiv p \cdot {{_{4}F_3} \left[ \begin{matrix} \frac{1}{2}, & \frac{1}{2}, & \frac{1}{2}, & \frac{1}{2}\;\\ & 1, & \frac{3}{4}, & \frac{5}{4} \end{matrix} \Big| \; 1 \right]}_ \frac{ p-1}{2} \pmod {p^3}\] for any prime \(p \geq 5 \).
In addition, the paper gives a ``human proof'' of a supercongruence already found by the author [J. Math. Anal. Appl. 471, 613--622 (2019; Zbl 1423.11015)], via the Mathematica package \(Sigma\) supplied by \textit{C. Schneider} [Sémin. Lothar. Comb. 56, B56b, 36 p. (2006; Zbl 1188.05001)], as extension of the \(p\)-adic analogue of a Ramanujan's identity conjectured by \textit{L. van Hamme} [Lect. Notes Pure Appl. Math. 192, 223--236 (1997; Zbl 0895.11051)].
Beyond basic properties of the gamma function (both classical and \(p\)-adic), the Taylor's expansion, and the Wolstenholme's congruence, the theorem-proving recalls some results from \textit{W. N. Bailey} [Generalized hypergeometric series. London: Cambridge University Press (1935; Zbl 0011.02303)], from \textit{L. Long} and \textit{R. Ramakrishna} [Adv. Math. 290, 773--808 (2016; Zbl 1336.33018)], and from \textit{F. J. W. Whipple} [Proc. London Math Soc. 24, 247--263 (1926; JFM 51.0283.03)].
Reviewer: Enzo Bonacci (Latina)Some results of fractional integral involving I-function and general class of polynomial.https://www.zbmath.org/1456.330122021-04-16T16:22:00+00:00"Tripathi, Bhupendra"https://www.zbmath.org/authors/?q=ai:tripathi.bhupendra"Sharma, Roshani"https://www.zbmath.org/authors/?q=ai:sharma.roshani"Sharma, C. K."https://www.zbmath.org/authors/?q=ai:sharma.chandra-kSummary: In the present paper, we have derived two multiplication theorems for I-function by using fractional integral formula \(I_{0,x}^{\alpha,\beta,\eta} f(x)\) and \(J_{0,\infty}^{\alpha,\beta,\eta} f(x)\) [\textit{P. K. Banerji} and \textit{S. Choudhary}, Proc. Natl. Acad. Sci. India, Sect. A 66, No. 3, 271--277 (1996; Zbl 1007.26501)]. Further, we established some applications of the theorem in the form of particular case.The algebra of recurrence relations for exceptional Laguerre and Jacobi polynomials.https://www.zbmath.org/1456.420302021-04-16T16:22:00+00:00"Durán, Antonio J."https://www.zbmath.org/authors/?q=ai:duran.antonio-jSummary: Exceptional Laguerre and Jacobi polynomials \(p_n(x)\) are bispectral, in the sense that as functions of the continuous variable \(x\), they are eigenfunctions of a second order differential operator and as functions of the discrete variable \(n\), they are eigenfunctions of a higher order difference operator (the one defined by any of the recurrence relations they satisfy). In this paper, under mild conditions on the sets of parameters, we characterize the algebra of difference operators associated to the higher order recurrence relations satisfied by the exceptional Laguerre and Jacobi polynomials.Precise estimates for the solution of Ramanujan's generalized modular equation.https://www.zbmath.org/1456.110472021-04-16T16:22:00+00:00"Wang, Miao-Kun"https://www.zbmath.org/authors/?q=ai:wang.miaokun"Chu, Yu-Ming"https://www.zbmath.org/authors/?q=ai:chu.yuming"Zhang, Wen"https://www.zbmath.org/authors/?q=ai:zhang.wen.3Summary: In the article, we present several monotonicity theorems and inequalities for the modular equation functions \(m_{a}(r)\) and \(\mu _{a}(r),\) and find the infinite-series formulas for \(m_{1/3}(r)\) and \(m_{1/4}(r)\) which depend only on \(r\). As applications, we find several precise explicit estimates for the solution of Ramanujan's generalized modular equation.Schwarz maps associated with the triangle groups \((2,4,4)\) and \((2,3,6)\).https://www.zbmath.org/1456.330042021-04-16T16:22:00+00:00"Koguchi, Yuto"https://www.zbmath.org/authors/?q=ai:koguchi.yuto"Matsumoto, Keiji"https://www.zbmath.org/authors/?q=ai:matsumoto.keiji.1|matsumoto.keiji"Seto, Fuko"https://www.zbmath.org/authors/?q=ai:seto.fukoSummary: We consider the Schwarz maps with monodromy groups isomorphic to the triangle groups \((2,4,4)\) and \((2,3,6)\) and their inverses. We apply our formulas to studies of mean iterations.The irrationality measure of \(\pi\) is at most 7.103205334137\dots.https://www.zbmath.org/1456.111292021-04-16T16:22:00+00:00"Zeilberger, Doron"https://www.zbmath.org/authors/?q=ai:zeilberger.doron"Zudilin, Wadim"https://www.zbmath.org/authors/?q=ai:zudilin.wadimThe main result of this paper is that the irrationality measure exponent of the number \(\pi\) is less than \(7.103205334138\). The proof uses complex analysis, is based on clever calculating of special integral and is in the spirit of Salikov.
Reviewer: Jaroslav Hančl (Ostrava)Differential geometry and Lie groups. A second course.https://www.zbmath.org/1456.530012021-04-16T16:22:00+00:00"Gallier, Jean"https://www.zbmath.org/authors/?q=ai:gallier.jean-h"Quaintance, Jocelyn"https://www.zbmath.org/authors/?q=ai:quaintance.jocelynThis book is written as a second course on differential geometry. So the reader is supposed to be familiar with some themes from the first course on differential geometry -- the theory of manifolds and some elements of Riemannian geometry.
In the first two chapters here some topics from linear algebra are provided -- a detailed exposition of tensor algebra and symmetric algebra, exterior tensor products and exterior algebra. These chapters may be useful when studying the material of this book for those students, who did not study these topics in their algebraic course.
Some themes, which are covered in this book, are rather standard for books on differential geometry - they are differential forms, de Rham cohomology, integration on manifolds, connections and curvature in vector bundles, fibre bundles, principal bundles and metrics on bundles. But a number of topics discussed in this book are not always included in courses on differential geometry and are rarely contained in textbooks on differential geometry. The presence of these topics makes this book especially interesting for modern students. Here is a list of some such topics: an introduction to Pontrjagin
classes, Chern classes, and the Euler class, distributions and the Frobenius theorem. Three chapters need to be highlighted separately. Chapter 7 -- spherical harmonics and an introduction to the representations of compact Lie groups. Chapter 8 -- operators on Riemannian manifolds: Hodge Laplacian, Laplace-Beltrami Laplacian, Bochner
Laplacian. Chapter 11 -- Clifford algebras and groups, groups Pin\((n)\), Spin\((n)\).
Not all statements in this book are given with proofs, for some only links to other textbooks are given. But the most important results are given here with complete proofs and accompanied by examples. Each chapter of this book ends with a list of interesting and sometimes very important problems. At the end of the book there is a very detailed list of the notation used (symbol index) and a detailed list (index) of the terms used.
Reviewer: V. V. Gorbatsevich (Moskva)Picard-Vessiot groups of Lauricella's hypergeometric systems \(E_C\) and Calabi-Yau varieties arising integral representations.https://www.zbmath.org/1456.140142021-04-16T16:22:00+00:00"Goto, Yoshiaki"https://www.zbmath.org/authors/?q=ai:goto.yoshiaki"Koike, Kenji"https://www.zbmath.org/authors/?q=ai:koike.kenjiThe authors study the Zariski closure of the monodromy group \(Mon\) of Lauricella's hypergeometric function \(F_C(a,b,c;x)=\sum_{m_1,\ldots ,m_n=0}^{\infty}\frac{(a)_{m_1+\cdots +m_n}(b)_{m_1+\cdots +m_n}}{(c_1)_{m_1}\cdots (c_n)_{m_n}m_1!\cdots m_n!}x_1^{m_1}\cdots x_n^{m_n}\), where \(a,b\in \mathbb{C}\), \(c_i\in \mathbb{C}\setminus \{ 0,-1,-2,\ldots \}\), \((c_i)_{m_i}=\Gamma (c_i+m_i)/\Gamma (c_i)\), and Calabi-Yau varieties arising from its integral representation. When the identity component of \(Mon\) acts irreducibly, then \(\overline{Mon}\cap SL_{2^n}(\mathbb{C})\) is one of the classical groups \(SL_{2^n}(\mathbb{C})\), \(SO_{2^n}(\mathbb{C})\) or \(Sp_{2^n}(\mathbb{C})\).
Reviewer: Vladimir P. Kostov (Nice)Special functions associated with \(K\)-types of degenerate principal series of \(\mathrm{Sp}(n,\mathbb{C})\).https://www.zbmath.org/1456.220052021-04-16T16:22:00+00:00"Mendousse, Grégory"https://www.zbmath.org/authors/?q=ai:mendousse.gregoryThis article is devoted to the study of special vectors contained in various incarnations of generalized principal series representations induced from maximal parabolic subgroups of the complex symplectic group \(\mathrm{Sp}(n,\mathbf{C})\). After reviewing the well-known decomposition of the Hilbert space \(L^2(S^{4n-1})\) under the natural action of \(\mathrm{Sp}(n)\times\mathrm{Sp}(1)\) in terms of spherical harmonics, the author uses quaternionic geometry to establish the existence and uniqueness of bi-invariant spherical harmonics and determines an explicit hypergeometric equation that they satisfy.
The other main result in the paper concerns certain vectors in the so-called \textit{non-standard} picture of the degenerate principal representations. This picture was introduced by \textit{T. Kobayashi} et al. [J. Funct. Anal. 260, No. 6, 1682--1720 (2011; Zbl 1217.22003)] for real symplectic groups and adapted to the complex case by the reviewer in [J. Funct. Anal. 262, No. 9, 4160--4180 (2012; Zbl 1242.22017)]. It is the image of the the classical non-compact picture under a partial Fourier transform afforded by the fact that the unipotent radicals of the inducing parabolic subgroups are Heisenberg groups. The author calculates the image in this picture of particular highest weight vectors, showing that they can be expressed in terms of modified Bessel functions.
Reviewer: Pierre Clare (Williamsburg)A hypergeometric version of the modularity of rigid Calabi-Yau manifolds.https://www.zbmath.org/1456.110732021-04-16T16:22:00+00:00"Zudilin, Wadim"https://www.zbmath.org/authors/?q=ai:zudilin.wadimThis paper considers the fourteen one-parameter families of Calabi-Yau
threefolds whose periods are expressed in terms of hypergeometric functions.
For these fourteen families, periods are solutions of hypergeometric equations
with parameter \((r, 1-r, t, 1-t)\), where
\begin{multline*}
(r,t)=\Big(\frac{1}{2},\frac{1}{2}\Big),\Big(\frac{1}{2},\frac{1}{3}\Big),\Big(\frac{1}{2},\frac{1}{4}\Big),
\Big(\frac{1}{2},\frac{1}{6}\Big),\Big(\frac{1}{3}\Big),\Big(\frac{1}{3},\frac{1}{4}\Big),\Big(\frac{1}{3},\frac{1}{6}\Big),\\
\Big(\frac{1}{4},\frac{1}{4}\Big),\Big(\frac{1}{4},\frac{1}{6}\Big),\Big(\frac{1}{6},\frac{1}{6}\Big),\Big(\frac{1}{5},\frac{2}{5}\Big),
\Big(\frac{1}{8},\frac{3}{8}\Big),\Big(\frac{1}{10},\frac{3}{10}\Big),\Big(\frac{1}{12},\frac{5}{12}\Big).
\end{multline*}
At a conifold point, any of these Calabi-Yau threefolds becomes rigid,
and the \(p\)-th coefficient \(a(p)\) of the corresponding modular form of weight \(4\)
can be recovered from the truncated partial sums of the corresponding
hypergeometric series modulo a higher power of \(p\), where \(p\) is any good prime \(>5\).
This paper discusses relationships between the critical values of the \(L\)-series of the modular form
and the values of a related basis of solutions to the hypergeometric differential equation.
It is numerically observed that the critical \(L\)-values are \(\mathbb{Q}\)-proportional to the
hypergeometric values \(F_1(1), F_2(1), F_3(1)\), where \(F_j(z)\) are solutions of the hypergeometric
equation for the hypergeometric function \(F_0(z)=_4F_3(z)\) with parameters \((r, 1-r, t, 1-t)\).
This confirms the prediction of Golyshev concerning gamma structures [\textit{V. Golyshev} and \textit{A. Mellit}, J. Geom. Phys. 78, 12--18 (2014; Zbl 1284.33001)].
Reviewer: Noriko Yui (Kingston)An algebraic description of the bispectrality of the biorthogonal rational functions of Hahn type.https://www.zbmath.org/1456.330112021-04-16T16:22:00+00:00"Tsujimoto, Satoshi"https://www.zbmath.org/authors/?q=ai:tsujimoto.satoshi"Vinet, Luc"https://www.zbmath.org/authors/?q=ai:vinet.luc"Zhedanov, Alexei"https://www.zbmath.org/authors/?q=ai:zhedanov.alexei-sBiorthogonal functions appear in the framework of generalized eigenvalue problems (GEVP, in short) of the form $L_{1}U= \lambda L_{2} U,$ where $L_{1}$ and $L_{2}$ are two operators acting on functions of one variable and $\lambda$ denotes the corresponding eigenvalue. Rational functions, as solutions of the GEVP and related to its biorthogonality, are associated with two tridiagonal operators in a certain basis [\textit{A. Zhedanov}, J. Approx. Theory 101, No. 2, 303--329 (1999; Zbl 1058.42502)]. In this framework, the authors of this interesting contribution deal with the analysis of an bispectral problem when three operators $X,Y, Z$ act tridiagonally on a given set of biorthogonal rational functions. Here $X^{(\alpha, \beta)}= (x-\alpha)\mathcal{ I}- x T^{-},$ $Z^{(\alpha, \beta)}= (\alpha-x)^{-1} X^{(\alpha, \beta)},$ $Y^{(\alpha, \beta)}= A_{1}(x) T^{+} + A_{2}(x)T^{-} + A_{3}(x) \mathcal{I},$ where $T^{\pm}f(x)= f(x \pm1),$ and $A_{0}, A_{1}, A_{2}$ are fixed cubic polynomials depending on the parameters $\alpha, \beta.$
The space of real functions defined on the linear grid $\mathbb{Z}$ is restricted to the space of functions defined in a finite set of points $M_{N}= \{0, 1, 2, \cdots, N\}$ where you deal with the standard basis $\{e_{k}\}_{k=0}^{N}.$ In this basis the operators $X, Y, Z$ become matrices of size $(N+1)\times (N+1)$, in such a way $X, Z$ are lower diagonal and $Y$ is tridiagonal. This triplet will play the same role as the Leonard pairs for orthogonal polynomials [\textit{P. Terwilliger}, Linear Algebra Appl. 330, No. 1--3, 149--203 (2001; Zbl 0980.05054)]. These operators can be viewed as elements of the set of operators that satisfy the rational Heun property on the linear grid (see [\textit{S. Tsujimoto} et al., ``The rational Heun operator and Wilson biorthogonal functions', Preprint, \url{arXiv:1912.11571}].
The rational functions $U_{n}(x; \alpha, \beta, N)= \frac{(-1)^{n} (-N)_{n}}{(\beta+1)_{n}}_ {3}F_{2} (-x, -n, \beta- n-N; -N, \alpha-x;1)$ are explicitly obtained as solutions of an GEVP involving $X$ and $Y.$ They appear in the interpolation of the ratio $\frac{\Gamma(\alpha-\beta-x)}{\Gamma(\alpha- x)}$ of two gamma functions by using rational functions with prescribed zeros and poles [\textit{A. Zhedanov},
``Padé interpolation table and biorthogonal rational functions'', Rokko. Lect. Math. 18, 323--363 (2005)]. Their biorthogonal partners $V_{n}(x;\alpha, \beta, N)= U_{n} (N-x; \beta+2-\alpha, \beta, N)$ are identified by using the symmetry properties of the hypergeometric weight distribution. In a next step, the tridiagonal action of the triplet $(X, Y, Z)$ on the functions $U_{n} (x; \alpha, \beta, N)$ is shown and the representation of the operators $X,Y, Z$ in such a basis by square matrices of size $(N+1)\times (N+1)$ is discussed. In such a way, the second order difference equation and the recurrence relation that those rational functions satisfy are deduced from the GEVP associated with $(X, Y, Z).$ Finally, the quadratic algebra realized by such operators is presented. Notice that this algebra encodes the bispectral properties of the above biorthogonal sets as the Askey-Wilson algebra and its specializations do for the hypergeometric orthogonal polynomials.
Reviewer: Francisco Marcellán (Leganes)The Virasoro fusion kernel and Ruijsenaars' hypergeometric function.https://www.zbmath.org/1456.814002021-04-16T16:22:00+00:00"Roussillon, Julien"https://www.zbmath.org/authors/?q=ai:roussillon.julienSummary: We show that the Virasoro fusion kernel is equal to Ruijsenaars' hypergeometric function up to normalization. More precisely, we prove that the Virasoro fusion kernel is a joint eigenfunction of four difference operators. We find a renormalized version of this kernel for which the four difference operators are mapped to four versions of the quantum relativistic hyperbolic Calogero-Moser Hamiltonian tied with the root system \(BC_1\). We consequently prove that the renormalized Virasoro fusion kernel and the corresponding quantum eigenfunction, the (renormalized) Ruijsenaars hypergeometric function, are equal.Bessel phase functions: calculation and application.https://www.zbmath.org/1456.330072021-04-16T16:22:00+00:00"Horsley, David E."https://www.zbmath.org/authors/?q=ai:horsley.david-eSummary: The Bessel phase functions are used to represent the Bessel functions as a positive modulus and an oscillating trigonometric term. This decomposition can be used to aid root-finding of certain combinations of Bessel functions. In this article, we give some new properties of the modulus and phase functions and some asymptotic expansions derived from differential equation theory. We find a bound on the error of the first term of this asymptotic expansion and give a simple numerical method for refining this approximation via standard routines for the Bessel functions. We then show an application of the phase functions to the root finding problem for linear and cross-product combinations of Bessel functions. This method improves upon previous methods and allows the roots in ascending order of these functions to be calculated independently. We give some proofs of correctness and global convergence.Pizzetti formula on the Grassmannian of 2-planes.https://www.zbmath.org/1456.430052021-04-16T16:22:00+00:00"Eelbode, D."https://www.zbmath.org/authors/?q=ai:eelbode.david"Homma, Y."https://www.zbmath.org/authors/?q=ai:homma.youkow|homma.yuki|homma.yasushi|homma.yuko|homma.yushiThe authors study the Higgs algebras \(H_3\) in the harmonic analysis of the Grassmann manifolds \(\mathrm{Gr}_0(m,2):=\mathrm{SO}(m)/(\mathrm{SO}(m-2)\times \mathrm{SO}(2))\). These algebras appear in the Howe duality where \(\mathrm{SO}(m)\) acts on
\(\mathrm{Gr}_0(m,2)\). In particular, an orthogonal decomposition of the vector space of all poynomials under the joint action of \(\mathrm{SO}(m)\times H_3\) is given. This decomposition is then used to derive a Pizetti formula for integrals of the form \(\int_{\mathrm{Gr}_0(m,2)} f\> d\mu\) with the uniform distribution \(\mu\) on \(\mathrm{Gr}_0(m,2)\). The authors also present a connection to a Pizetti formula for Stiefel manifolds by K.~Coulembier and M.~Kieburg.
Reviewer: Michael Voit (Dortmund)Multi-integral representations for associated Legendre and Ferrers functions.https://www.zbmath.org/1456.330092021-04-16T16:22:00+00:00"Cohl, Howard S."https://www.zbmath.org/authors/?q=ai:cohl.howard-s"Costas-Santos, Roberto S."https://www.zbmath.org/authors/?q=ai:costas-santos.roberto-sSummary: For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order, including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis, we obtain some new results for the associated Legendre function of the second kind, including parameter values for which this function is identically zero.Barnes-Ismagilov integrals and hypergeometric functions of the complex field.https://www.zbmath.org/1456.330082021-04-16T16:22:00+00:00"Neretin, Yury A."https://www.zbmath.org/authors/?q=ai:neretin.yuri-aThe purpose of this paper is first to extend Eulerian integrals for generalized hypergeometric functions \(\ _{p }\textup{F}_{q}\) to complex integrals. Several special cases are indicated and there are many similarities with the Meijer G-function. It seems that there are more than one definition available and the usual stringent structure ``definition, theorem, proof'' is unfortunately missing. It would be better not to have so many references, the paper has kind of a physics style. The concept Gamma-function of the complex field on page 4 is not properly defined.
Reviewer: Thomas Ernst (Uppsala)Multicritical continuous random trees.https://www.zbmath.org/1456.824312021-04-16T16:22:00+00:00"Bouttier, J."https://www.zbmath.org/authors/?q=ai:bouttier.jeremie"Di Francesco, P."https://www.zbmath.org/authors/?q=ai:di-francesco.philippe"Guitter, E."https://www.zbmath.org/authors/?q=ai:guitter.emmanuelOn mixed \(AR(1)\) time series model with approximated beta marginal.https://www.zbmath.org/1456.622132021-04-16T16:22:00+00:00"Popović, Božidar V."https://www.zbmath.org/authors/?q=ai:popovic.bozidar-v"Pogány, Tibor K."https://www.zbmath.org/authors/?q=ai:pogany.tibor-k"Nadarajah, Saralees"https://www.zbmath.org/authors/?q=ai:nadarajah.saralees(no abstract)Precise error estimate of the Brent-McMillan algorithm for Euler's constant.https://www.zbmath.org/1456.330062021-04-16T16:22:00+00:00"Demailly, Jean-Pierre"https://www.zbmath.org/authors/?q=ai:demailly.jean-pierreSummary: \textit{R. P. Brent} and \textit{E. M. McMillan} [Math. Comput. 34, 305--312 (1980; Zbl 0442.10002)] introduced in 1980 a new algorithm for the computation of Euler's constant \(\gamma\), based on the use of the Bessel functions \(I_0(x)\) and \(K_0(x)\). It is the fastest known algorithm for the computation of \(\gamma\). The time complexity can still be improved by evaluating a certain divergent asymptotic expansion up to its minimal term. Brent-McMillan conjectured in 1980 that the error is of the same magnitude as the last computed term, and \textit{R. P. Brent} and \textit{F. Johansson} [Math. Comput. 84, No. 295, 2351--2359 (2015; Zbl 1320.33007)] partially proved it in 2015. They also gave some numerical evidence for a more precise estimate of the error term. We find here an explicit expression of that optimal estimate, along with a complete self-contained formal proof and an even more precise error bound.Condition numbers for real eigenvalues in the real elliptic Gaussian ensemble.https://www.zbmath.org/1456.600192021-04-16T16:22:00+00:00"Fyodorov, Yan V."https://www.zbmath.org/authors/?q=ai:fyodorov.yan-v"Tarnowski, Wojciech"https://www.zbmath.org/authors/?q=ai:tarnowski.wojciechSummary: We study the distribution of the eigenvalue condition numbers \(\kappa_i = \sqrt{(\mathbf{l}_i^* \mathbf{l}_i) (\mathbf{r}_i^* \mathbf{r}_i)}\) associated with real eigenvalues \(\lambda_i\) of partially asymmetric \(N \times N\) random matrices from the real Elliptic Gaussian ensemble. The large values of \(\kappa_i\) signal the non-orthogonality of the (bi-orthogonal) set of left \(\mathbf{l}_i\) and right \(\mathbf{r}_i\) eigenvectors and enhanced sensitivity of the associated eigenvalues against perturbations of the matrix entries. We derive the general finite \(N\) expression for the joint density function (JDF) \(\mathcal{P}_N (z, t)\) of \(t = \kappa_i^2 - 1\) and \(\lambda_i\) taking value \(z\), and investigate its several scaling regimes in the limit \(N \rightarrow \infty\). When the degree of asymmetry is fixed as \(N \rightarrow \infty\), the number of real eigenvalues is \(\mathcal{O} (\sqrt{N})\), and in the bulk of the real spectrum \(t_i = \mathcal{O}(N)\), while on approaching the spectral edges the non-orthogonality is weaker: \(t_i = \mathcal{O} (\sqrt{N})\). In both cases the corresponding JDFs, after appropriate rescaling, coincide with those found in the earlier studied case of fully asymmetric (Ginibre) matrices. A different regime of weak asymmetry arises when a finite fraction of \(N\) eigenvalues remain real as \(N \rightarrow \infty\). In such a regime eigenvectors are weakly non-orthogonal, \(t = \mathcal{O}(1)\), and we derive the associated JDF, finding that the characteristic tail \(\mathcal{P} (z, t) \sim t^{-2}\) survives for arbitrary weak asymmetry. As such, it is the most robust feature of the condition number density for real eigenvalues of asymmetric matrices.Computing special \(L\)-values of certain modular forms with complex multiplication.https://www.zbmath.org/1456.110572021-04-16T16:22:00+00:00"Li, Wen-Ching Winnie"https://www.zbmath.org/authors/?q=ai:li.wen-ching-winnie"Long, Ling"https://www.zbmath.org/authors/?q=ai:long.ling"Tu, Fang-Ting"https://www.zbmath.org/authors/?q=ai:tu.fang-tingThis is an expository article in which the authors present two explicit examples
of computing special \(L\)-values of modular forms admitting complex multiplication.
Two methods for this task are discussed. One uses hypergeometric functions,
and the other one Eisenstein series, and approaches are rather computational.
Two main examples are given in the following theorems. Here \(\eta(\tau)\)
denotes the Dedekind eta-function.
Theorem 1: Let \(\psi\) be the idéle class character of \({\mathbb{Q}}(\sqrt{-1})\)
such that \(L(\psi, s-\frac{1}{2})\) is the Hasse-Weil \(L\)-function of the CM elliptic
curve \(E_1:y^4+x^2=1\) of conductor \(32\). Then
\[2L\Big(\psi,\frac{1}{2}\Big)^2=L(\psi^2,1).\]
In terms of cusp forms with CM by \({\mathbb{Q}}(\sqrt{-1})\), the above identity is reformulated as
\[2L(\eta(4\tau)^2\eta(8\tau)^2,1)^2=L(\eta(4\tau)^6,2),\]
where \(\eta(4\tau)^2\eta(8\tau)^2\) is the weight \(2\) level \(32\) cuspidal eigenform corresponding
to \(\psi\), and \(\eta(4\tau)^6\) is the weight \(3\) level \(16\) cuspidal eigenform corresponding
to \(\psi^2\).
Theorem 2: Let \(\chi\) be the idéle class character of \({\mathbb{Q}}(\sqrt{-3})\) such
that \(L(\chi,s-\frac{1}{2})\) is the Hasse-Weil \(L\)-function of the CM elliptic curve
\(E_2: x^3+y^3=\frac{1}{4}\) of conductor \(36\). Then
\[\frac{3}{2}L\Big(\chi,\frac{1}{2}\Big)^2=L(\chi^2,1)\quad\text{and}\quad
\frac{8}{3}L\Big(\chi,\frac{1}{2}\Big)^3=L\Big(\chi^3,\frac{3}{2}\Big).\]
In terms of cusp forms with CM by \({\mathbb{Q}}(\sqrt{-3})\), these identities are reformulated
respectively as follows:
\[\frac{3}{2}L(\eta(6\tau)^4,1)^2=L(\eta(2\tau)^3\eta(6\tau)^3,2)\]
and
\[\frac{8}{3}L(\eta(6\tau)^4,1)^3=L(\eta(3\tau)^8,3).\]
Here \(\eta(6\tau)^4\) is the level \(36\) weight \(2\) cuspidal Hecke eigenform corresponding
to \(\chi\), \(\eta(2\tau)^3\eta(6\tau)^3\) is the level \(12\) weight \(3\) Hecke eigenform
corresponding to \(\chi^2\), and \(\eta(3\tau)^8\) is the level \(9\) weight \(4\) Hecke eigenform
corresponding to \(\chi^3\).
Geometrically, weight \(2, 3\) and \(4\) cusp forms come from elliptic curves \(E_1\) and \(E_2\),
K3 surfaces, and a Calabi-Yau threefold.
Reviewer: Noriko Yui (Kingston)Free-fermion entanglement and orthogonal polynomials.https://www.zbmath.org/1456.810582021-04-16T16:22:00+00:00"Crampé, Nicolas"https://www.zbmath.org/authors/?q=ai:crampe.nicolas"Nepomechie, Rafael I."https://www.zbmath.org/authors/?q=ai:nepomechie.rafael-i"Vinet, Luc"https://www.zbmath.org/authors/?q=ai:vinet.lucUncertainty, ghosts, and resolution in Radon problems.https://www.zbmath.org/1456.440032021-04-16T16:22:00+00:00"Louis, Alfred K."https://www.zbmath.org/authors/?q=ai:louis.alfred-karlSummary: We study the nonuniqueness problem for Radon transforms for finitely many directions. In the early days of the application of computed tomography, they caused some confusion about the possible information content in the reconstructions from tomographic data. The existence of nontrivial functions in the null space started the analysis of these then so-called ghosts. A result of \textit{B. F. Logan} [Duke Math. J. 42, 661--706 (1975; Zbl 0354.46015)] described properties of the spectrum of those functions. Only with the description of those functions in terms of special functions by \textit{A. K. Louis} [Math. Methods Appl. Sci. 3, 1--10 (1981; Zbl 0459.44004)] a more detailed study and an improvement of earlier results was possible. Here, we describe the essential steps to find those characterizations and the analysis of the spectral properties allowing for resolution results.
For the entire collection see [Zbl 1420.44001].\(q\)-supercongruences modulo the fourth power of a cyclotomic polynomial via creative microscoping.https://www.zbmath.org/1456.110242021-04-16T16:22:00+00:00"Guo, Victor J. W."https://www.zbmath.org/authors/?q=ai:guo.victor-j-wSummary: By applying the Chinese remainder theorem for coprime polynomials and the ``creative microscoping'' method recently introduced by the author and \textit{W. Zudilin} [Adv. Math. 346, 329--358 (2019; Zbl 07035902)], we establish parametric generalizations of three \(q\)-supercongruences modulo the fourth power of a cyclotomic polynomial. The original \(q\)-supercongruences then follow from these parametric generalizations by taking the limits as the parameter tends to 1 (l'Hôpital's rule is utilized here). In particular, we prove a complete \(q\)-analogue of the (J.2) supercongruence of Van Hamme and a complete \(q\)-analogue of a ``divergent'' Ramanujan-type supercongruence, thus confirming two recent conjectures of the author. We also put forward some related conjectures, including a \(q\)-supercongruence modulo the fifth power of a cyclotomic polynomial.Combinatorics of some fifth and sixth order mock theta functions.https://www.zbmath.org/1456.050132021-04-16T16:22:00+00:00"Rana, Meenakshi"https://www.zbmath.org/authors/?q=ai:rana.meenakshi"Sharma, Shruti"https://www.zbmath.org/authors/?q=ai:sharma.shrutiSummary: The goal of this paper is to provide a new combinatorial meaning to two fifth order and four sixth order mock theta functions. Lattice paths of \textit{A. K. Agarwal} and \textit{D. M. Bressoud} [Pac. J. Math. 136, No. 2, 209--228 (1989; Zbl 0674.33002)] with certain modifications are used as a tool to study these functions.Fractional supersymmetric quantum mechanics and lacunary Hermite polynomials.https://www.zbmath.org/1456.812062021-04-16T16:22:00+00:00"Bouzeffour, F."https://www.zbmath.org/authors/?q=ai:bouzeffour.fethi"Garayev, M."https://www.zbmath.org/authors/?q=ai:garayev.mubariz-tapdigogluSummary: We consider a realization of fractional supersymmetric of quantum mechanics of order \(r\), where the Hamiltonian and supercharges involve reflection operators. It is shown that the Hamiltonian has \(r\)-fold degenerate spectrum and the eigenvalues of hermitian supercharges are zeros of the associated Hermite polynomials of Askey and Wimp. Also it is shown that the associated eigenfunctions involve lacunary Hermite polynomials.Bounds for the perimeter of an ellipse in terms of power means.https://www.zbmath.org/1456.260232021-04-16T16:22:00+00:00"He, Zai-Yin"https://www.zbmath.org/authors/?q=ai:he.zaiyin"Wang, Miao-Kun"https://www.zbmath.org/authors/?q=ai:wang.miaokun"Jiang, Yue-Ping"https://www.zbmath.org/authors/?q=ai:jiang.yueping"Chu, Yu-Ming"https://www.zbmath.org/authors/?q=ai:chu.yumingIt is well known that the perimeter \(L\) of an ellipse is strongly related to the Gauss hypergeometric function. Using properties of this function, the authors improve certain results from 2012 and 2014, by proving lower and upper bounds for \(L\) in terms of particular power means.
Reviewer: József Sándor (Cluj-Napoca)Basic properties of incomplete Macdonald function with applications.https://www.zbmath.org/1456.330012021-04-16T16:22:00+00:00"Shu, Jian-Jun"https://www.zbmath.org/authors/?q=ai:shu.jianjun"Shastri, Kunal Krishnaraj"https://www.zbmath.org/authors/?q=ai:shastri.kunal-krishnarajThe purpose of this paper is to derive key properties of the Shu function \(S_{\nu}\), such as recurrence and differential relations and series and asymptotic expansions. A parabolic partial differential equation for \(S_{\nu}\) with a modified Bessel operator is derived. The proofs use integration by parts, incomplete gamma function, approximation of Macdonald function. Graphs for leading term approximations are shown.
Reviewer: Thomas Ernst (Uppsala)Transitions of generalised Bessel kernels related to biorthogonal ensembles.https://www.zbmath.org/1456.150372021-04-16T16:22:00+00:00"Kawamoto, Yosuke"https://www.zbmath.org/authors/?q=ai:kawamoto.yosukeSummary: Biorthogonal ensembles are generalisations of classical orthogonal ensembles such as the Laguerre or the Hermite ensembles. Local fluctuation of these ensembles at the origin has been studied, and determinantal kernels in the limit are described by the Wright generalised Bessel functions. The limit kernels are one parameter deformations of the Bessel kernel and the sine kernel for the Laguerre weight and the Hermite weight, respectively. We study transitions from these generalised Bessel kernels to the sine kernel under appropriate scaling limits in common with classical kernels.Analytical approach to Ramanujan type and Ramanujan's modular equations of degree 7.https://www.zbmath.org/1456.110612021-04-16T16:22:00+00:00"Mahadevaswamy"https://www.zbmath.org/authors/?q=ai:mahadevaswamy.Some Mellin transforms for the Riemann zeta function in the critical strip.https://www.zbmath.org/1456.111582021-04-16T16:22:00+00:00"Patkowski, Alexander E."https://www.zbmath.org/authors/?q=ai:patkowski.alexander-ericOn generalized Lagrange-Hermite-Bernoulli and related polynomials.https://www.zbmath.org/1456.110262021-04-16T16:22:00+00:00"Khan, Waseem A."https://www.zbmath.org/authors/?q=ai:khan.waseem-ahmed|khan.waseem-ahmad|khan.waseem-asghar"Pathan, M. A."https://www.zbmath.org/authors/?q=ai:pathan.mahmood-ahmadThe purpose of this paper is to introduce a new class of polynomials, generalizing Hermite, Lagrange, Bernoulli, Miller-Lee, and Laguerre polynomials. Several relations between special cases of these polynomials are proved by using generating functions. Then 'implicit summation formulas' for several polynomials are proved. Finally, a relation between the Lagrange-Hermite-Bernoulli polynomials and Laguerre polynomials is found. It is assumed that the exponents \(\alpha\in\mathbb{C}\) and a branch of the logarithm is chosen.
Reviewer: Thomas Ernst (Uppsala)Schoenberg coefficients and curvature at the origin of continuous isotropic positive definite kernels on spheres.https://www.zbmath.org/1456.420072021-04-16T16:22:00+00:00"Arafat, Ahmed"https://www.zbmath.org/authors/?q=ai:arafat.ahmed"Gregori, Pablo"https://www.zbmath.org/authors/?q=ai:gregori.pablo"Porcu, Emilio"https://www.zbmath.org/authors/?q=ai:porcu.emilioSummary: We consider the class \(\Psi_d\) of continuous functions that define isotropic covariance functions in the \(d\)-dimensional sphere \(\mathbb{S}^d\). We provide a new recurrence formula for the solution of Problem 1 in \textit{T. Gneiting} [``Strictly and non-strictly positive definite functions on spheres: online supplement'' (2013), \url{https://projecteuclid.org/download/suppdf_1/euclid.bj/1377612854}], solved by \textit{J. Fiedler} [``From Fourier to Gegenbauer: dimension walks on spheres'' (2013), Preprint, \url{arXiv:1303.6856}]. In addition, we have improved the current bounds for the curvature at the origin of locally supported covariances (Problem 3 in T. Gneiting [loc. cit.]), which is of applied interest at least for \(d=2\).Single Bessel tractor-beam tweezers.https://www.zbmath.org/1456.761182021-04-16T16:22:00+00:00"Mitri, F. G."https://www.zbmath.org/authors/?q=ai:mitri.f-gSummary: The tractor behavior of a zero-order Bessel acoustic beam acting on a fluid sphere, and emanating from a finite circular aperture (as opposed to waves of infinite extent) is demonstrated theoretically. Conditions for an attractive force acting in opposite direction of the radiating waves, determined by the choice of the beam's half-cone angle, the size of the radiator, and its distance from a fluid sphere, are established and discussed. Numerical predictions for the radiation force function, which is the radiation force per unit energy density and cross-sectional surface, are provided using a partial-wave expansion method stemming from the acoustic scattering. The results suggest a simple and reliable analysis for the design of Bessel beam acoustical tweezers and tractor beam devices.Vertex operators, solvable lattice models and metaplectic Whittaker functions.https://www.zbmath.org/1456.820972021-04-16T16:22:00+00:00"Brubaker, Ben"https://www.zbmath.org/authors/?q=ai:brubaker.ben"Buciumas, Valentin"https://www.zbmath.org/authors/?q=ai:buciumas.valentin"Bump, Daniel"https://www.zbmath.org/authors/?q=ai:bump.daniel"Gustafsson, Henrik P. A."https://www.zbmath.org/authors/?q=ai:gustafsson.henrik-p-aThis paper discusses two mechanisms by which the quantum groups \(U_q (\hat{\mathfrak{g}})\), for a simple Lie algebra or superalgebra \(\mathfrak{g}\), produce families of special functions with a number of interesting properties related to functional equations, branching rules and unexpected algebraic relations. The first mechanism uses solvable lattice models associated to finite-dimensional modules of \(U_q (\hat{\mathfrak{g}})\). The second mechanism uses actions of Heisenberg and Clifford algebras on a fermionic Fock space, exploiting the boson-fermion correspondence arising in connection with soliton theory, dating back to [\textit{M. Jimbo} and \textit{T. Miwa}, Publ. Res. Inst. Math. Sci. 19, 943--1001 (1983; Zbl 0557.35091)] and pushed forward by \textit{T. Lam} [Math. Res. Lett. 13, No. 2--3, 377--392 (2006; Zbl 1160.05056)] and especially by [\textit{M. Kashiwara} et al., Sel. Math., New Ser. 1, No. 4, 787--805 (1995; Zbl 0857.17013)]. These two points of view provide new insight into the theory of metaplectic Whittaker functions for the general linear group and relate them to LLT polynomials (known also as ribbon symmetric functions). The main theorem of the paper considers two solvable lattice models, named Gamma ice and Delta, and details in Section 4 their row transfer matrices. In this study, metaplectic ice models are exploited, whose partition functions are metaplectic Whittaker functions. In the process, the authors introduce new symmetric functions termed metaplectic symmetric functions and explain how they are related to Whittaker functions. It is explained that half vertex operators agree with Lam's construction, and this interpretation allows for many new identities for metaplectic symmetric and Whittaker functions, including Cauchy identities. While both metaplectic symmetric functions and LLT polynomials [\textit{A. Lascoux} et al., J. Math. Phys. 38, No. 2, 1041--1068 (1997; Zbl 0869.05068)] can be related to vertex operators on the quantum Fock space, only metaplectic symmetric functions are connected to solvable lattice models. A number of links with the existing literature is identified as well.
Reviewer: Piotr Garbaczewski (Opole)The adelic Grassmannian and exceptional Hermite polynomials.https://www.zbmath.org/1456.330102021-04-16T16:22:00+00:00"Kasman, Alex"https://www.zbmath.org/authors/?q=ai:kasman.alex"Milson, Robert"https://www.zbmath.org/authors/?q=ai:milson.robertLet \(h_n\) denote the non-Appell Hermite polynomial.
The bivariate Hermite polynomial \(H_ n (x, y)\)
is defined by
\[H_ n (x, y):=(-y)^{n/2}h_n\left(\frac{x}{\sqrt{-4y}}\right).
\]
An equivalent definition is the generating function
\[\sum_{n=0}^{\infty}
H_ n (x, y)\frac{z^n}{n!}=\exp(xz+yz^2),
\] which means that \(H_ n (x, y)\) is an Appell polynomial
in the variable \(x\).
Exceptional Hermite polynomials satisfy a second-order eigenvalue equation
and are defined as the
Wronskian of classical Hermite polynomials.
Furthermore, exceptional Hermite polynomials associated to a partition
can be expressed simply in terms of the Schur functions produced from that partition
via insertion.
The purpose of this paper is first to prove that
the exceptional Hermite polynomials are generated by the wave functions of certain points in
George Wilson's adelic
Grassmannian \(Gr^{ad}\).
The second part of the paper is devoted to bispectrality,
stabilizer algebras.
The Cauchy-Euler operator preserves the Wronskian of the partition
and together with the Grassmannian generate an operator algebra.
Then, lowering and recurrence relations for the exceptional Hermites
are proved.
Finally, algorithms and examples for special partitions are given
and Maya diagrams are drawn.
The proofs use bivariate Hermite polynomials,
Maya diagrams, Taylor series, Cauchy-Euler operators, exceptional Jacobi operators.
Reviewer: Thomas Ernst (Uppsala)Discrete orthogonality relations for multi-indexed Laguerre and Jacobi polynomials.https://www.zbmath.org/1456.420312021-04-16T16:22:00+00:00"Ho, Choon-Lin"https://www.zbmath.org/authors/?q=ai:ho.choon-lin"Sasaki, Ryu"https://www.zbmath.org/authors/?q=ai:sasaki.ryuSummary: The discrete orthogonality relations hold for all the orthogonal polynomials obeying three term recurrence relations. We show that they also hold for multi-indexed Laguerre and Jacobi polynomials, which are new orthogonal polynomials obtained by deforming these classical orthogonal polynomials. The discrete orthogonality relations could be considered as a more encompassing characterization of orthogonal polynomials than the three term recurrence relations. As the multi-indexed orthogonal polynomials start at a positive degree \(\ell_{\mathcal{D}} \geq 1\), the three term recurrence relations are broken. The extra \(\ell_{\mathcal{D}}\) ``lower degree polynomials,'' which are necessary for the discrete orthogonality relations, are identified. The corresponding Christoffel numbers are determined. The main results are obtained by the blow-up analysis of the second order differential operators governing the multi-indexed orthogonal polynomials around the zeros of these polynomials at a degree \(\ell_{\mathcal{D}} + \mathcal{N}\). The discrete orthogonality relations are shown to hold for another group of ``new'' orthogonal polynomials called Krein-Adler polynomials based on the Hermite, Laguerre, and Jacobi polynomials.Certain fractional integral and fractional derivative formulae with their image formulae involving generalized multi-index Mittag-Leffler function.https://www.zbmath.org/1456.260072021-04-16T16:22:00+00:00"Chand, Mehar"https://www.zbmath.org/authors/?q=ai:chand.mehar"Kasmaei, Hamed Daei"https://www.zbmath.org/authors/?q=ai:kasmaei.hamed-daei"Senol, Mehmet"https://www.zbmath.org/authors/?q=ai:senol.mehmetSummary: The main objective of this paper is to establish some image formulas by applying the Riemann-Liouville fractional derivative and integral operators to the product of generalized multiindex Mittag-Leffler function \(E^{\gamma,q}_{(\alpha_j,\beta_j)_m}(.)\). Some more image formulas are derived by applying integral transforms. The results obtained here are quite general in nature and capable of yielding a very large number of known and (presumably) new results.