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A classical problem due to Abel is to determine if a differential equation |$y^{\prime}=\eta y$| admits a non-trivial solution |$y$| algebraic over |$\mathbb C(x)$| when |$\eta $| is a given algebraic function over |$\mathbb C(x)$|⁠. Risch designed an algorithm that, given |$\eta $|⁠ , determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when |$\eta $| admits a Puiseux expansion with rational coefficients at some point in |$\mathbb C\cup \{\infty \}$|⁠ , which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of |$y^{\prime}=\eta y$| if and only if the coefficients of the Puiseux expansion of |$x\eta (x)$| at |$0$| satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations |$y^{\prime}=\eta y$| with an algebraic solution when |$x\eta (x)$| is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present two other applications, namely to diagonals of rational fractions and to directed two-dimensional walks. [ABSTRACT FROM AUTHOR]

In viscoacoustically stratified media where the density is constant, 3-D wave propagation in the frequency–radial wavenumber domain is governed by the Helmholtz equation. In the case that the model is a velocity gradient interface where the squared velocity in depth is represented by a smooth Heaviside function of the Fermi–Dirac distribution type, the Helmholtz equation for a point source at arbitrary location is shown to have analytical solution for the Green's function. The velocity depth profile, which is a modification of the Epstein profile which has been thoroughly studied in different branches of physics, is described by four parameters: the velocities at minus and plus infinity, the reference depth of the gradient interface, and its smoothness. The Helmholtz equation is first solved in a model where the point source is absent. The solution to the source-free equation has four unknown constants that must be determined. The radiation conditions at minus and plus infinity and two conditions on the Green's function at the source depth allow the constants to be found. The Green's function solution can be represented in two mathematically equivalent algebraic forms involving ordinary hypergeometric functions. The first form allows a numerically stable implementation over all wavenumber components. The second form allows a physical, intuitive interpretation and is expressed mathematically as the sum of two terms. Each of the terms contains the product of constant-velocity reference phase shift functions and hypergeometric functions which take the role to adjust the amplitude and phase shift calculated by the reference phase shift functions to account for the depth varying velocity profile. Inverse Fourier transforms take the Green's function from the frequency–wavenumber domain to the frequency–space domain or time–space domain. The Green's function solution is valid for any sharpness of the interface. Selected numerical results are presented for the 1-D and 2-D Helmholtz equation to demonstrate the influence of the velocity gradient zone on the wavefield. The 1-D solution in an acoustic model is compared in time domain to the classical finite-difference wave propagation solution. For the purpose of interpretation of seismograms, we model for comparison the wavefield response in a model of two half-spaces in welded contact. For brevity, the latter model is referenced as the HS model. [ABSTRACT FROM AUTHOR]

A Bayesian prediction problem for the two-dimensional Wishart model is investigated within the framework of decision theory. The loss function is the Kullback–Leibler divergence. We construct a scale-invariant and permutation-invariant prior distribution that shrinks the correlation coefficient. The prior is the geometric mean of the right invariant prior with respect to permutation of the indices, and is characterized by a uniform distribution for Fisher's |$z$| -transformation of the correlation coefficient. The Bayesian predictive density based on the prior is shown to be minimax. [ABSTRACT FROM AUTHOR]

We apply the method of [ 7 ] to compute the Stokes matrices of non-resonant confluent hypergeometric differential equations. We discuss the ambiguity of the presentation of the Stokes matrices regarding different choices. The results rely on an explicit description of the perverse sheaf associated to the non-confluent regular singular hypergeometric system arising via Fourier–Laplace transform. We give assumptions on the parameter such that the Stokes matrices have rational or real values. Under some more restrictive conditions, the Stokes matrices had been computed by Duval–Mitschi before. We compare our results with their formulae in the unramified case. [ABSTRACT FROM AUTHOR]

In this paper, we generalize a result of Cuttler, Greene, Skandera, and Sra that characterizes the majorization order on Young diagrams in terms of nonnegative specializations of Schur polynomials. More precisely, we introduce a generalized notion of majorization associated to an arbitrary crystallographic root system |$\Phi $| and show that it admits a natural characterization in terms of the values of spherical functions on any Riemannian symmetric space with restricted root system |$\Phi.$| We also conjecture a further generalization of this theorem in terms of Heckman–Opdam hypergeometric functions. [ABSTRACT FROM AUTHOR]

We study integral representations of the Gevrey series solutions of irregular hypergeometric systems under certain assumptions. We prove that, for such systems, any Gevrey series solution, along a coordinate hyperplane of its singular support, is the asymptotic expansion of a holomorphic solution given by a carefully chosen integral representation. [ABSTRACT FROM AUTHOR]

Let $V_k$ denote Dunkl's intertwining operator associated with some root system $R$ and multiplicity $k$. For two multiplicities $k, k^{\prime }$ on $R$ , we study the intertwiner $V_{k^{\prime },k} = V_{k^{\prime }}\circ V_k^{-1}$ between Dunkl operators with multiplicities $k$ and $k^{\prime }.$ It has been a long-standing conjecture that $V_{k^{\prime },k}$ is positive if $k^{\prime } \geq k \geq 0.$ 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. [ABSTRACT FROM AUTHOR]

We give a generalization of |$p$| -adic congruences for truncated period functions that were originally discovered for a class of hypergeometric functions by Bernard Dwork. [ABSTRACT FROM AUTHOR]

We introduce the |$q$| -Hahn PushTASEP—an integrable stochastic interacting particle system that is a three-parameter generalization of the PushTASEP, a well-known close relative of the TASEP (totally asymmetric simple exclusion process). The transition probabilities in the |$q$| -Hahn PushTASEP are expressed through the |$_4\phi _3$| basic hypergeometric function. Under suitable limits, the |$q$| -Hahn PushTASEP degenerates to all known integrable (1+1)-dimensional stochastic systems with a pushing mechanism. One can thus view our new system as a pushing counterpart of the |$q$| -Hahn TASEP introduced by Povolotsky [ 37 ]. We establish Markov duality relations and contour integral formulas for the |$q$| -Hahn PushTASEP. In a |$q\to 1$| limit of our process we arrive at a random recursion, which, in a special case, appears to be similar to the inverse-Beta polymer model. However, unlike in recursions for Beta polymer models, the weights (i.e. the coefficients of the recursion) in our model depend on the previous values of the partition function in a nontrivial manner. [ABSTRACT FROM AUTHOR]

We propose a new bilinear Hirota equation for |$\tau$| -functions associated with the |$E_8$| root lattice that provides a "lens" generalisation of the |$\tau$| -functions for the elliptic discrete Painlevé equation. Our equations are characterised by a positive integer |$r$| in addition to the usual elliptic parameters and involve a mixture of continuous variables with additional discrete variables, the latter taking values on the |$E_8$| root lattice. We construct explicit |$W(E_7)$| -invariant hypergeometric solutions of this bilinear Hirota equation, which are given in terms of elliptic hypergeometric sum/integrals. [ABSTRACT FROM AUTHOR]

For any integer |$d\times (n+1)$| matrix |$A$| and parameter |$\beta\in\mathbb{C}^d$| let |$M_A(\beta)$| be the associated |$A$| -hypergeometric (or GKZ) system in the variables |$x_0,\ldots,x_n$|⁠. We describe bounds for the (roots of the) |$b$| -functions of both |$M_A(\beta)$| and its Fourier transform along the hyperplanes |$(x_j=0)$|⁠. We also give an estimate for the |$b$| -function for restricting |$M_A(\beta)$| to a generic point. [ABSTRACT FROM AUTHOR]

Motivation Transcriptional regulators play a major role in most biological processes. Alterations in their activities are associated with a variety of diseases and in particular with tumor development and progression. Hence, it is important to assess the effects of deregulated regulators on pathological processes. Results Here, we present REGulator-Gene Association Enrichment (REGGAE), a novel method for the identification of key transcriptional regulators that have a significant effect on the expression of a given set of genes, e.g. genes that are differentially expressed between two sample groups. REGGAE uses a Kolmogorov–Smirnov-like test statistic that implicitly combines associations between regulators and their target genes with an enrichment approach to prioritize the influence of transcriptional regulators. We evaluated our method in two different application scenarios, which demonstrate that REGGAE is well suited for uncovering the influence of transcriptional regulators and is a valuable tool for the elucidation of complex regulatory mechanisms. Availability and implementation REGGAE is freely available at https://regulatortrail.bioinf.uni-sb.de. Supplementary information Supplementary data are available at Bioinformatics online. [ABSTRACT FROM AUTHOR]

After an extended period of declining health, Adrienne Winifred (Freda) Kemp, died on 6 March 2022, aged 91, leaving husband David, four children and six grandchildren. In 1970, she and David moved to live in Ilkley and to work at the University of Bradford, David as Professor of Statistics and Freda as an Honorary Research Fellow (lecturer 4 years later). In 1983, Freda and David both took early retirement and moved to St Andrews where both were given honorary research positions within the School of Mathematics and Statistics. [Extracted from the article]

We give a complete description of finite braid group orbits in Aff(C)-character varieties of the punctured Riemann sphere. This is performed, thanks to a coalescence procedure and to the theory of finite complex reflection groups. We then derive consequences in the theory of differential equations. These concern algebraicity of isomonodromic deformations for reducible rank two logarithmic connections on the sphere, the Riemann-Hilbert problem and FD-type Lauricella hypergeometric functions. [ABSTRACT FROM AUTHOR]

Accurate and efficient methods to evaluate cosmological distances are an important tool in modern precision cosmology. In a flat ∧ cold dark matter (CDM) cosmology, the luminosity distance can be expressed in terms of elliptic integrals. We derive an alternative and simple expression for the luminosity distance in a flat ∧CDM based on hypergeometric functions. Using a timing experiment, we compare the computation time for the numerical evaluation of the various exact formulae, as well as for two approximate fitting formulae available in the literature. We find that our novel expression is the most efficient exact expression in the redshift range of z ≳ 1. Ideally, it can be combined with the expression based on Carlson's elliptic integrals in the range of z ≲ 1 for high-precision cosmology distance calculations over the entire redshift range. On the other hand, for practical work where relative errors of about 0.1 per cent are acceptable, the analytical approximation proposed by Adachi & Kasai (2012) is a suitable alternative. [ABSTRACT FROM AUTHOR]

The focus of this article is on the nature of the likelihood associated with N-mixture models for repeated count data. It is shown that the infinite sum embedded in the likelihood associated with the Poisson mixing distribution can be expressed in terms of a hypergeometric function and, thence, in closed form. The resultant expression for the likelihood can be readily computed to a high degree of accuracy and is algebraically tractable. Specifically, the likelihood equations can be simplified to some advantage, the concentrated likelihood in the probability of detection formulated and problematic cases identified. The results are illustrated by means of a simulation study and a real world example. The study is extended to N-mixture models with a negative binomial mixing distribution and results similar to those for the Poisson case obtained. N-mixture models with mixing distributions which accommodate excess zeros and, separately, with a beta-binomial distribution rather than a binomial used to model the intra-site counts are also investigated. However the results for these settings, while computationally attractive, do not provide insight into the nature of the maximum likelihood estimates. [ABSTRACT FROM AUTHOR]

We give a lower bound for the sup-norm of an L2-normalized newform in an irreducible, unitary, cuspidal representation π of GL2 over a number field. When the central character of π is sufficiently ramified, this bound improves upon the trivial bound by a positive power of N, where N is the norm of the conductor of π. This generalizes a result of Templier, who dealt with the special case when the conductor of the central character equals the conductor of the representation. We also make a conjecture about the true size of the sup-norm in the N-aspect that takes into account this central character phenomenon. Our results depend upon some explicit formulas and bounds for the Whittaker newvector over a non-archimedean local field, which may be of independent interest. [ABSTRACT FROM AUTHOR]

We obtain symmetric joint eigenfunctions for the commuting partial differential operators associated to the hyperbolic Calogero-Moser N-particle system. The eigenfunctions are constructed via a recursion scheme, which leads to representations by multidimensional integrals whose integrands are elementary functions. We also tie in these eigenfunctions with the Heckman-Opdam hypergeometric function for the root system AN-1. [ABSTRACT FROM AUTHOR]

Summary: Every year, a large number of novel algorithms are introduced to the scientific community for a myriad of applications, but using these across different research groups is often troublesome, due to suboptimal implementations and specific dependency requirements. This does not have to be the case, as public cloud computing services can easily house tractable implementations within self-contained dependency environments, making the methods easily accessible to a wider public. We have taken 14 popular methods, the majority related to expression data or promoter analysis, developed these up to a good implementation standard and housed the tools in isolated Docker containers which we integrated into the CyVerse Discovery Environment, making these easily usable for a wide community as part of the CyVerse UK project. [ABSTRACT FROM AUTHOR]

We consider the symmetric q-Painlev'e equations derived from the birational representation of affine Weyl groups by applying the projective reduction and construct the hypergeometric solutions. Moreover, we discuss continuous limits of the symmetric q-Painlev'e equations to Painlev'e equations together with their hypergeometric solutions. [ABSTRACT FROM AUTHOR]

We give asymptotic formulae for random matrix averages of derivatives of characteristic polynomials over the groups USp(2N), SO(2N) and O−(2N). These averages are used to predict the asymptotic formulae for moments of derivatives of L-functions which arise in number theory. Each formula gives the leading constant of the asymptotic in terms of determinants of hypergeometric functions. We find a differential recurrence relation between these determinants that allows the rapid computation of the (k+1)st constant in terms of the kth and (k−1)st. This recurrence is reminiscent of a Toda lattice equation arising in the theory of τ-functions associated with Painlevé differential equations. [ABSTRACT FROM PUBLISHER]

The asymptotic efficiency of the spatial sign covariance matrix relative to affine equivariant estimators of scatter is studied. In particular, the spatial sign covariance matrix is shown to be asymptotically inadmissible, i.e., the asymptotic covariance matrix of the consistency-corrected spatial sign covariance matrix is uniformly larger than that of its affine equivariant counterpart, namely Tyler’s scatter matrix. Although the spatial sign covariance matrix has often been recommended when one is interested in principal components analysis, its inefficiency is shown to be most severe in situations where principal components are of greatest interest. Simulation shows that the inefficiency of the spatial sign covariance matrix also holds for small sample sizes, and that the asymptotic relative efficiency is a good approximation to the finite-sample efficiency for relatively modest sample sizes. [ABSTRACT FROM PUBLISHER]

Functional enrichment analysis is conducted on high-throughput data to provide functional interpretation for a list of genes or proteins that share a common property, such as being differentially expressed (DE). The hypergeometric P-value has been widely used to investigate whether genes from pre-defined functional terms, for example, Gene Ontology (GO), are enriched in the DE genes. The hypergeometric P-value has three limitations: (1) computed independently for each term, thus neglecting biological dependence; (2) subject to a size constraint that leads to the tendency of selecting less-specific terms; (3) repeated use of information due to overlapping annotations by the true-path rule. We propose a Bayesian approach based on the non-central hypergeometric model. The GO dependence structure is incorporated through a prior on non-centrality parameters. The likelihood function does not include overlapping information. The inference about enrichment is based on posterior probabilities that do not have a size constraint. This method can detect moderate but consistent enrichment signals and identify sets of closely related and biologically meaningful functional terms rather than isolated terms. We also describe the basic ideas of assumption and implementation of different methods to provide some theoretical insights, which are demonstrated via a simulation study. A real application is presented. [ABSTRACT FROM AUTHOR]

In our previous work, a unified description as polynomial Hamiltonian systems was established for a broad class of the Schlesinger systems including the sixth Painlevé equation and Garnier systems. The main purpose of this paper is to present particular solutions of this Hamiltonian system in terms of a certain generalization of Gauß's hypergeometric function. Key ingredients of the argument are the linear Pfaffian system derived from an integral representation of the hypergeometric function (with the aid of twisted de Rham theory) and Lax formalism of the Hamiltonian system. [ABSTRACT FROM PUBLISHER]

We introduce a generalization of elliptic 6j-symbols, which can be interpreted as matrix elements for intertwiners between corepresentations of Felder’s elliptic quantum group. For special parameter values, they can be expressed in terms of multivariable elliptic hypergeometric series related to the root system An. As a consequence, we obtain new biorthogonality relations for such series. [ABSTRACT FROM PUBLISHER]

Methods for the numerical evaluation of the Weber parabolic cylinder functions W(a, ± x), which are independent solutions of the inverted harmonic oscillator y″ + (x2/4 – a)y = 0, are described. The functions appear in the solution of many physical problems and notably in quantum mechanics. It is shown that the combined use of Maclaurin series, Chebyshev series, uniform asymptotic expansions for large a and/or x and the integration of the differential equation by local Taylor series are enough for computing the functions accurately in a wide rage of parameters. Differently from previous methods, the computational scheme is stable in the sense that high accuracy is retained: only two or three digits may be lost in double precision computations. [ABSTRACT FROM PUBLISHER]

We describe the structure of all codimension-2 lattice configurations A which admit a stable rational A-hypergeometric function, that is a rational function F all the partial derivatives of which are nonzero, and which is a solution of the A-hypergeometric system of partial differential equations defined by Gel′ fand, Kapranov, and Zelevinsky. We show, moreover, that all stable rational A-hypergeometric functions may be described by toric residues and apply our results to study the rationality of bivariate series the coefficients of which are quotients of factorials of linear forms. [ABSTRACT FROM PUBLISHER]

We obtain “large gap” asymptotics for a Fredholm determinant with a confluent hypergeometric kernel. We also obtain asymptotics for determinants with two types of Bessel kernels which appeared in random matrix theory. [ABSTRACT FROM PUBLISHER]

Connection matrices associated with the generalized hypergeometric function n+1Fn are determined by using the intersection number of twisted cycles. [ABSTRACT FROM PUBLISHER]

We consider the q-Painlevé III equation arising from the birational representation of the affine Weyl group of type (A2 + A1)(1). We study the reduction of the q-Painlevé III equation to the q-Painlevé II equation from the viewpoint of affine Weyl group symmetry. In particular, the mechanism of apparent inconsistency between the hypergeometric solutions to both equations is clarified by using factorization of difference operators and the τ functions. [ABSTRACT FROM PUBLISHER]

The quantum group analog of the normalizer of SU(1, 1) in is an important and nontrivial example of a noncompact quantum group. The general theory of locally compact quantum groups in the operator algebra setting implies the existence of the dual quantum group. The first main goal of this article is to give an explicit description of the dual quantum group for this example involving the quantized enveloping algebra . It turns out that does not suffice to generate the dual quantum group. The dual quantum group is graded with respect to commutation and anticommutation with a suitable analog of the Casimir operator characterized by an affiliation relation to a von Neumann algebra. This is used to obtain an explicit set of generators. Having the dual quantum group the left regular corepresentation of the quantum group analog of the normalizer of SU(1, 1) in is decomposed into irreducible corepresentations. Upon restricting the irreducible corepresentations to -representation one finds combinations of the positive and negative discrete series representations with the strange series representations as well as combinations of the principal unitary series representations. The detailed analysis of this example involves the analysis of special functions of basic hypergeometric type and, in particular, some results on these special functions are obtained, which are stated separately. This article is split into two parts: the first part gives almost all of the statements and the results, and the statements of this part are independent of the second part. The second part contains the proofs of all the statements. [ABSTRACT FROM PUBLISHER]

We describe the set of bounded harmonic functions for the Heckman–Opdam Laplacian when the multiplicity function is larger than 1/2. We prove that this set is a vector space of dimension the cardinality of the Weyl group. We give some consequences in terms of the associated hypergeometric functions. [ABSTRACT FROM PUBLISHER]

Motivation: Genome-wide association (GWA) studies may identify multiple variants that are associated with a disease or trait. To narrow down candidates for further validation, quantitatively assessing how identified genes relate to a phenotype of interest is important. Results: We describe an approach to characterize genes or biological concepts (phenotypes, pathways, diseases, etc.) by ontology fingerprint—the set of Gene Ontology (GO) terms that are overrepresented among the PubMed abstracts discussing the gene or biological concept together with the enrichment p-value of these terms generated from a hypergeometric enrichment test. We then quantify the relevance of genes to the trait from a GWA study by calculating similarity scores between their ontology fingerprints using enrichment p-values. We validate this approach by correctly identifying corresponding genes for biological pathways with a 90% average area under the ROC curve (AUC). We applied this approach to rank genes identified through a GWA study that are associated with the lipid concentrations in plasma as well as to prioritize genes within linkage disequilibrium (LD) block. We found that the genes with highest scores were: ABCA1, lipoprotein lipase (LPL) and cholesterol ester transfer protein, plasma for high-density lipoprotein; low-density lipoprotein receptor, APOE and APOB for low-density lipoprotein; and LPL, APOA1 and APOB for triglyceride. In addition, we identified genes relevant to lipid metabolism from the literature even in cases where such knowledge was not reflected in current annotation of these genes. These results demonstrate that ontology fingerprints can be used effectively to prioritize genes from GWA studies for experimental validation. Contact: zhengw@musc.edu Supplementary information: Supplementary data are available at Bioinformatics online. [ABSTRACT FROM AUTHOR]

We examine the ability of two recent methods – the smoothed implied volatility smile method (SML) and the density functionals based on confluent hypergeometric functions (DFCH) – for estimating implied risk-neutral densities (RNDs) from European-style options. Two complementary Monte Carlo experiments are conducted and the performance of the two RND estimators is evaluated by the root mean integrated squared error (RMISE) criterion. Results from both experiments show that the DFCH method outperforms the SML method for the overall quality of the estimated RNDs concerning both accuracy and stability. An application of the two methods to the OTC currency options market is also presented. [ABSTRACT FROM AUTHOR]

The paper describes a distribution generated by the Gaussian hypergeometric function that may be seen as a generalization of the beta–binomial distribution. It is expressed as a generalized beta mixture of a binomial distribution. This new mixing distribution allows the existence of a mode and an antimode, which is very useful for fitting some data sets. Two examples illustrate the greater versatility of the new distribution compared with the beta–binomial distribution. [ABSTRACT FROM AUTHOR]

We describe the exceptional set of a function studied by Picard. This is the set of algebraic points at which the function takes algebraic values. In particular, we deduce necessary and sufficient conditions for the finiteness of the exceptional set. Some of our results depend on recent conjectures of Pink. [ABSTRACT FROM AUTHOR]

Hypergeometric solutions to the q-Painlevé equations are constructed by direct linearization of discrete Riccati equations. The decoupling factors are explicitly determined so that the linear systems give rise to q-hypergeometric equations. [ABSTRACT FROM AUTHOR]

By taking an appropriate limit, we obtain the Bessel functions related to root systems as limits of Heckman-Opdam hypergeometric functions. A more general class of Bessel functions is also investigated, which we will call the Θ-Bessel functions. Explicit formulas for the Θ-Bessel functions are obtained when the multiplicity functions are even and positive integer-valued. This class encloses the Bessel functions on the tangent space at the origin of noncompact causal symmetric spaces, where an integral representation for these special functions is shown. [ABSTRACT FROM AUTHOR]

The Sklyanin algebra admits realizations by difference operators acting on theta functions. Sklyanin found an invariant metric for the action and conjectured an explicit formula for the corresponding reproducing kernel. We prove this conjecture and also give natural biorthogonal and orthogonal bases for the representation space. Moreover, we discuss connections with elliptic hypergeometric series and integrals and with elliptic 6j-symbols. [ABSTRACT FROM AUTHOR]

Hypergeometric solutions to seven q-Painlevé equations in Sakai's classification are constructed. Geometry of plane curves is used to reduce the q-Painlevé equations to the three-term recurrence relations for q-hypergeometric functions. [ABSTRACT FROM AUTHOR]

We introduce elliptic Macdonald polynomials associated with the 3-dimensional irreducible representation of sl2. Suitable limits of elliptic Macdonald polynomials are the standard Macdonald polynomials and conformal blocks. Identities for elliptic Macdonald polynomials, in particular, their modular properties, are studied. [ABSTRACT FROM AUTHOR]

We show that the N × N Toeplitz determinants with the symbols zμexp(−(1/2)t(z+1/z)) and (1+z)μ(1+1/z)ν exp(tz)—known τ-functions for the PIII′ and PV systems—are characterised by nonlinear recurrences for the reflection coefficients of the corresponding orthogonal polynomial system on the unit circle. It is shown that these recurrences are entirely equivalent to the discrete Painlevé equations associated with the degenerations of the rational surfaces D6(1)→E7(1) (discrete Painlevé II) and D5(1)→E6(1) (discrete Painlevé IV), respectively, through the algebraic methodology based upon the affine Weyl group symmetry of the Painlevé system, originally due to Okamoto. In addition, it is shown that the difference equations derived by methods based upon the Toeplitz lattice and Virasoro constraints, when reduced in order by exact summation, are equivalent to our recurrences. Expressions in terms of generalised hypergeometric functions 0F1(1),1F1(1) are given for the reflection coefficients, respectively. [ABSTRACT FROM AUTHOR]

With the aim of treating the statistics of palaeomagnetic directions and intensities jointly and consistently, we represent the mean and the variance of palaeomagnetic vectors, at a particular site and of a particular polarity, by a probability density function in a Cartesian three-space of orthogonal magnetic-field components consisting of a single (unimodal) non-zero mean, spherically-symmetrical (isotropic) Gaussian function. For palaeomagnetic data of mixed polarities, we consider a bimodal distribution consisting of a pair of such symmetrical Gaussian functions, with equal, but opposite, means and equal variances. For both the Gaussian and bi-Gaussian distributions, and in the spherical three-space of intensity, inclination, and declination, we obtain analytical expressions for the marginal density functions, the cumulative distributions, and the expected values and variances for each spherical coordinate (including the angle with respect to the axis of symmetry of the distributions). The mathematical expressions for the intensity and off-axis angle are closed-form and especially manageable, with the intensity distribution being Rayleigh-Rician. In the limit of small relative vectorial dispersion, the Gaussian (bi-Gaussian) directional distribution approaches a Fisher (Bingham) distribution and the intensity distribution approaches a normal distribution. In the opposite limit of large relative vectorial dispersion, the directional distributions approach a spherically-uniform distribution and the intensity distribution approaches a Maxwell distribution. We quantify biases in estimating the properties of the vector field resulting from the use of simple arithmetic averages, such as estimates of the intensity or the inclination of the mean vector, or the variances of these quantities. With the statistical framework developed here and using the maximumlikelihood method, which gives unbiased estimates in the limit of large data numbers, we demonstrate how to... [ABSTRACT FROM AUTHOR]

Two unitary integral transforms with a very well poised 7F6-function as a kernel are given. For both integral transforms the inverse is the same as the original transform after an involution on the parameters. The 7F6-function involved can be considered as a nonpolynomial extension of the Wilson polynomial, and is therefore called a Wilson function. The two integral transforms are called a Wilson function transform of types I and II. Furthermore, a few explicit transformations of hypergeometric functions are calculated, and it is shown that the Wilson function transform of type I maps a basis of orthogonal polynomials onto a similar basis of polynomials. [ABSTRACT FROM AUTHOR]

We find an explicit description of all GL(n, ℝ)-Whittaker functions as oscillatory integrals and thus construct equivariant mirrors of flag manifolds. As a consequence, we prove the Virasoro conjecture for flag manifolds. [ABSTRACT FROM AUTHOR]

A system of partially ordered polynomials will be referred to as partially orthogonal if the polynomials are orthogonal when comparable in the partial order. We present a formula describing the asymptotics for large degrees of a class of (partially) orthogonal multivariate polynomials associated with root systems. As an application, the asymptotics of the Macdonald polynomials is determined. [ABSTRACT FROM AUTHOR]

The exact solution for any eccentricity of the Peters-Mathews model of binary stars emitting gravitational waves leads to hypergeometric functions of Appell and Gauss type. Efficient (fast and accurate) numerical computations of such functions can be obtained by using the Padé approximation. Padé rational approximants, for equal numbers of coefficients, are more accurate than Taylor series representation. Unfortunately, the algorithm computing Padé coefficients numerically from Taylor series is known to be unstable. This numerical instability lowers the accuracy of the Padé approximant. In this paper, double precision accuracy Padé approximant coefficients are computed symbolically. Advantages of the symbolic computation with respect to the numerical treatment are shown, providing accurate representations of the Appell and Gauss hypergeometric functions. The symbolic approach can be extended to higher post-Newtonian models. [ABSTRACT FROM AUTHOR]

Motivated by applications in high‐dimensional settings, we suggest a test of the hypothesis H0 that two sampled distributions are identical. It is assumed that two independent datasets are drawn from the respective populations, which may be very general. In particular, the distributions may be multivariate or infinite‐dimensional, in the latter case representing, for example, the distributions of random functions from one Euclidean space to another. Our test uses a measure of distance between data. This measure should be symmetric but need not satisfy the triangle inequality, so it is not essential that it be a metric. The test is based on ranking the pooled dataset, with respect to the distance and relative to any fixed data value, and repeating this operation for each fixed datum. A permutation argument enables a critical point to be chosen such that the test has concisely known significance level, conditional on the set of all pairwise distances. [ABSTRACT FROM PUBLISHER]

For the first time a Bailey chain with all entries composed out of the Jacobi theta functions is constructed. This is an elliptic extension of the WP (well-poised) Bailey chain of Andrews and it generates an infinite sequence of identities for theta hypergeometric series. As a particular example, we obtained a new proof of the Frenkel-Turaev elliptic analogue of the Bailey transformation for a terminating 10Φ9 basic hypergeometric series. An elliptic generalization of the Andrews-Berkovich 10Φ9 → 12Φ11 transformation formula is derived by employing an elliptic extension of a Bressoud's Bailey pair. [ABSTRACT FROM AUTHOR]

Our aim is to find a general approach to the theory of classical solutions of the Garnier system in n-variables, Gn, based on the Riemann-Hilbert problem and on the geometry of the space of isomonodromy deformations. Our approach consists in determining the monodromy data of the corresponding Fuchsian system that guarantee to have a classical solution of the Garnier system Gn. This leads to the idea of the reductions of the Garnier systems. We prove that if a solution of the Garnier system Gn is such that the associated Fuchsian system has l monodromy matrices equal to ±1, then it can be reduced classically to a solution of a the Garnier system with n − 1 variables Gn − 1. When n monodromy matrices are equal to ±1, we have classical solutions of Gn. We give also another mechanism to produce classical solutions: we show that the solutions of the Garnier systems having reducible monodromy groups can be reduced to the classical solutions found by Okamoto and Kimura in terms of Lauricella hypergeometric functions. In the case of the Garnier system in 1-variables, i.e. for the PainlevéVI equation, we prove that all classical non-algebraic solutions have either reducible monodromy groups or at least onemonodromy matrix equal to ±1. [ABSTRACT FROM AUTHOR]