Hypergeometric Type Research Articles

Schlesinger Equations for Upper Triangular Matrices and Their Solutions

We consider explicit integral expressions of hypergeometric and hyperelliptic types for solutions of Schlesinger equations in classes of upper triangular matrices with eigenvalues producing arithmetic progressions with the same difference. These integral representations extend and generalize earlier known results.

Topological recursion for the extended Ooguri–Vafa partition function of colored HOMFLY-PT polynomials of torus knots

We prove that topological recursion applied to the spectral curve of colored HOMFLY-PT polynomials of torus knots reproduces the n-point functions of a particular partition function called the extended Ooguri-Vafa partition function. This generalizes and refines the results of Brini-Eynard-Marino and Borot-Eynard-Orantin. We also discuss how the statement of spectral curve topological recursion in this case fits into the program of Alexandrov-Chapuy-Eynard-Harnad of establishing the topological recursion for general weighted double Hurwitz numbers partition functions (a.k.a. KP tau-functions of hypergeometric type).

Некоторые формулы дробного интегрирования от одной функции Фокса с четырьмя параметрами

Solutions to many problems of mathematical physics, engineering and economics arc expressed through the so-called special functions. In the theory of special functions an important place is occupied by functions of the hypergeometric type. Many of them can be written in terms of the Meyer G-function. A generalization of the Meyer function is the Fox H-function. Some properties of this function can be obtained from its representation using the Mellin – Barnes integral. When deriving some formulas for this function for particular values of its parameters, due to the cumbersome writing of the Fox function, it is more convenient to use simplified notation. In this paper, we consider a special case of such a Fox function containing four parameters. For this function, Riemann-Liouville and Erdelyi-Kober fractional integration formulas arc obtained. An integral representation of the considered function h through the Mellin – Burns integral, we write out the conditions under which it converges absolutely, and the asymptotic expansions for this function for large and small values of the argument. The formulas proved in the paper are obtained using the indicated Mellin – Burns integral representation and the well-known integration formulas from power functions. For particular values of the parameters, the function under consideration yields some well-known elementary and special functions, and from the obtained formulas of fractional integration – the known integral values of these functions.

A Survey of Some Recent Developments on Higher Transcendental Functions of Analytic Number Theory and Applied Mathematics

Often referred to as special functions or mathematical functions, the origin of many members of the remarkably vast family of higher transcendental functions can be traced back to such widespread areas as (for example) mathematical physics, analytic number theory and applied mathematical sciences. Here, in this survey-cum-expository review article, we aim at presenting a brief introductory overview and survey of some of the recent developments in the theory of several extensively studied higher transcendental functions and their potential applications. For further reading and researching by those who are interested in pursuing this subject, we have chosen to provide references to various useful monographs and textbooks on the theory and applications of higher transcendental functions. Some operators of fractional calculus, which are associated with higher transcendental functions, together with their applications, have also been considered. Many of the higher transcendental functions, especially those of the hypergeometric type, which we have investigated in this survey-cum-expository review article, are known to display a kind of symmetry in the sense that they remain invariant when the order of the numerator parameters or when the order of the denominator parameters is arbitrarily changed.

Structural Formulas for Matrix-Valued Orthogonal Polynomials Related to 2times 2 Hypergeometric Operators

We give some structural formulas for the family of matrix-valued orthogonal polynomials of size 2times 2 introduced by C. Calderón et al. in an earlier work, which are common eigenfunctions of a differential operator of hypergeometric type. Specifically, we give a Rodrigues formula that allows us to write this family of polynomials explicitly in terms of the classical Jacobi polynomials, and write, for the sequence of orthonormal polynomials, the three-term recurrence relation and the Christoffel–Darboux identity. We obtain a Pearson equation, which enables us to prove that the sequence of derivatives of the orthogonal polynomials is also orthogonal, and to compute a Rodrigues formula for these polynomials as well as a matrix-valued differential operator having these polynomials as eigenfunctions. We also describe the second-order differential operators of the algebra associated with the weight matrix.

ON THE MODULARITY OF SOLUTIONS TO CERTAIN DIFFERENTIAL EQUATIONS OF HYPERGEOMETRIC TYPE

AbstractWe answer some questions in a paper by Kaneko and Koike [‘On modular forms arising from a differential equation of hypergeometric type’, Ramanujan J.7(1–3) (2003), 145–164] about the modularity of the solutions of a certain differential equation. In particular, we provide a number-theoretic explanation of why the modularity of the solutions occurs in some cases and does not occur in others. This also proves their conjecture on the completeness of the list of modular solutions after adding some missing cases.

On General Properties of Degenerate Systems of Second Order Partial Differential Equations of Hypergeometric Type

For the first time, the general properties of degenerate related hypergeometric systems such as Horn, Whittaker, Bessel and Laguerre are investigated together. The joint research allowed to reveal their various common properties and to establish a number of new degenerate related systems. They are all private cases of the common system offered by the authors for consideration. For the full study, it is important to classify its regular and irregular special curves and to identify the types of corresponding solutions. In this paper, they are implemented using simple rules. Special attention is paid to the construction of normal and regular solutions, because the solutions of all related degenerate systems such as Horn, Whittaker, Bessel and Laguerre near the irregular singularity on infinity relate to this species. Peculiarities of building normal-regular solutions by the Frobenius-Latysheva method are shown. All constructed normal-regular solutions are expressed through the function of Humbert  variables, which is the solution of degenerate hypergeometric system of Horn type. As an example, the cases  where, along with the application of the Frobenius-Latysheva method, the possibility of outputting new degenerate related systems is demonstrated.

On a Symmetric Generalization of Bivariate Sturm–Liouville Problems

A new class of partial differential equations having symmetric orthogonal solutions is presented. The general equation is presented and orthogonality is obtained using the Sturm–Liouville approach. Conditions on the polynomial coefficients to have admissible partial differential equations are given. The general case is analyzed in detail, providing orthogonality weight function, three-term recurrence relations for the monic orthogonal polynomial solutions, as well as explicit form of these monic orthogonal polynomial solutions, which are solutions of an admissible and potentially self-adjoint linear second-order partial differential equation of hypergeometric type.

The Asymptotic Expansion of a Function Introduced by L.L. Karasheva

The asymptotic expansion for x→±∞ of the entire function Fn,σ(x;μ)=∑k=0∞sin(nγk)sinγkxkk!Γ(μ−σk),γk=(k+1)π2n for μ>0, 0<σ<1 and n=1,2,… is considered. In the special case σ=α/(2n), with 0<α<1, this function was recently introduced by L.L. Karasheva (J. Math. Sciences, 250 (2020) 753–759) as a solution of a fractional-order partial differential equation. By expressing Fn,σ(x;μ) as a finite sum of Wright functions, we employ the standard asymptotics of integral functions of hypergeometric type to determine its asymptotic expansion. This was found to depend critically on the parameter σ (and to a lesser extent on the integer n). Numerical results are presented to illustrate the accuracy of the different expansions obtained.

CDF of non-central [formula omitted] distribution revisited. Incomplete hypergeometric type functions approach

The cumulative distribution function of the non-central chi-square distribution χν′2(λ),ν∈R+ possesses an integral representation in terms of a generalized Marcum Q-function. Regarding some already known-results, here we derive a simpler form of the cumulative distribution function for ν=2n∈N degrees of freedom. Also, we express these representations in terms of an incomplete Fox–Wright function pΨq(γ) and the generalized incomplete hypergeometric functions concerning the important special cases such as 1Γ1,2Γ1 and 2γ1. New identities are established between 1Γ1 and 2Γ1 as well.

Constellations and $\tau$-functions for rationally weighted Hurwitz numbers

Weighted constellations give graphical representations of weighted branched coverings of the Riemann sphere. They were introduced to provide a combinatorial interpretation of the 2D Toda \tau -functions of hypergeometric type serving as generating functions for weighted Hurwitz numbers in the case of polynomial weight generating functions. The product over all vertex and edge weights of a given weighted constellation, summed over all configurations, reproduces the \tau -function. In the present work, this is generalized to constellations in which the weighting parameters are determined by a rational weight generating function. The associated \tau -function may be expressed as a sum over the weights of doubly labelled weighted constellations, with two types of weighting parameters associated to each equivalence class of branched coverings. The double labelling of branch points, referred to as “colour” and “flavour” indices, is required by the fact that, in the Taylor expansion of the weight generating function, a particular colour from amongst the denominator parameters may appear multiply, and the flavour labels indicate this multiplicity.

Constructing canonical Feynman integrals with intersection theory

Canonical Feynman integrals are of great interest in the study of scattering amplitudes at the multi-loop level. We propose to construct dlog-form integrals of the hypergeometric type, treat them as a representation of Feynman integrals, and project them into master integrals using intersection theory. This provides a constructive way to build canonical master integrals whose differential equations can be solved easily. We use our method to investigate both the maximally cut integrals and the uncut ones at one and two loops, and demonstrate its applicability in problems with multiple scales.

Weighted Average Consensus-Based Optimization of Advection-Diffusion Systems

As a fundamental algorithm for collaborative processing over multi-agent systems, distributed consensus algorithm has been studied for optimizing its convergence rate. Due to the close analogy between the diffusion problem and the consensus algorithm, the previous trend in the literature is to transform the diffusion system from the spatially continuous domain into the spatially discrete one. In this transformation, the optimality is not necessarily preserved. In this paper, the reverse of this approach has been adopted, and it has been shown that the optimality can be preserved. This paper studies optimization of the Continuous-Time Consensus (CTC) problem on a weighted digraph with given average weight. Based on the detailed balance property, the CTC algorithm is converted into the weighted-average CTC algorithm. For the given distribution and average weight, a possible solution procedure has been provided. For finding the optimal weights corresponding to the weighted-average CTC algorithm with optimal convergence rate on a general graph. This solution procedure has been implemented based on the min-max theorem. For path topology, it is shown that the linearity of the drift term is the necessary and sufficient condition for the optimality of the consensus algorithm (and the corresponding diffusion system). Thus, the Pearson's class of discrete (continuous) distributions are optimal, where the closed-form formulas for the convergence rate, spectrum and other characteristics of the corresponding optimal consensus algorithm (diffusion system), i.e., the Hypergeometric types have been provided.

Seven-point conformal blocks in the extended snowflake channel and beyond

Seven-point functions have two inequivalent topologies or channels. The comb channel has been computed previously and here we compute scalar conformal blocks in the extended snowflake channel in $d$ dimensions. Our computation relies on the known action of the differential operator that sets up the operator product expansion in embedding space. The scalar conformal blocks in the extended snowflake channel are obtained as a power series expansion in the conformal cross-ratios whose coefficients are a triple sum of the hypergeometric type. This triple sum factorizes into a single sum and a double sum. The single sum can be seen as originating from the comb channel and is given in terms of a ${}_3F_2$-hypergeometric function, while the double sum originates from the snowflake channel which corresponds to a Kamp\'e de F\'eriet function. We verify that our results satisfy the symmetry properties of the extended snowflake topology. Moreover, we check that the behavior of the extended snowflake conformal blocks under several limits is consistent with known results. Finally, we conjecture rules leading to a partial construction of scalar $M$-point conformal blocks in arbitrary topologies.

Six-point conformal blocks in the snowflake channel

We compute d-dimensional scalar six-point conformal blocks in the two possible topologies allowed by the operator product expansion. Our computation is a simple application of the embedding space operator product expansion formalism developed recently. Scalar six-point conformal blocks in the comb channel have been determined not long ago, and we present here the first explicit computation of the scalar six-point conformal blocks in the remaining inequivalent topology. For obvious reason, we dub the other topology the snowflake channel. The scalar conformal blocks, with scalar external and exchange operators, are presented as a power series expansion in the conformal cross-ratios, where the coefficients of the power series are given as a double sum of the hypergeometric type. In the comb channel, the double sum is expressible as a product of two 3F2-hypergeometric functions. In the snowflake channel, the double sum is expressible as a Kampé de Fériet function where both sums are intertwined and cannot be factorized. We check our results by verifying their consistency under symmetries and by taking several limits reducing to known results, mostly to scalar five-point conformal blocks in arbitrary spacetime dimensions.

Multiple orthogonal polynomials associated with confluent hypergeometric functions

We introduce and analyse a new family of multiple orthogonal polynomials of hypergeometric type with respect to two measures supported on the positive real line which can be described in terms of confluent hypergeometric functions of the second kind. These two measures form a Nikishin system. Our focus is on the multiple orthogonal polynomials for indices on the step line. The sequences of the derivatives of both type I and type II polynomials with respect to these indices are again multiple orthogonal and they correspond to the original sequences with shifted parameters. For the type I polynomials, we provide a Rodrigues-type formula. We characterise the type II polynomials on the step line, also known as d-orthogonal polynomials (where d is the number of measures involved so that here d=2), via their explicit expression as a terminating generalised hypergeometric series, as solutions to a third-order differential equation and via their recurrence relation. The latter involves recurrence coefficients which are unbounded and asymptotically periodic. Based on this information we deduce the asymptotic behaviour of the largest zeros of the type II polynomials. We also discuss limiting relations between these polynomials and the multiple orthogonal polynomials with respect to the modified Bessel weights. Particular choices on the parameters for the 2-orthogonal polynomials under discussion correspond to the cubic components of the already known threefold symmetric Hahn-classical multiple orthogonal polynomials on star-like sets.

Adjoint difference equation for the Nikiforov–Uvarov–Suslov difference equation of hypergeometric type on non-uniform lattices

In this article, we obtain the adjoint difference equation for the Nikiforov–Uvarov–Suslov difference equation of hypergeometric type on non-uniform lattices, and prove it to be a difference equation of hypergeometric type on non-uniform lattices as well. The particular solutions of the adjoint difference equation are then obtained. As an application of these particular solutions, we use them to obtain the particular solutions for the original difference equation of hypergeometric type on non-uniform lattices. In addition, we give another kind of fundamental theorems for the Nikiforov–Uvarov–Suslov difference equation of hypergeometric type, which are essentially new results and their expressions are different from the Suslov Theorem. Finally, we give an example to illustrate the application of the new fundamental theorems.

On the Laguerre fractional integro-differentiation

A fractional power interpretation of the Laguerre derivative is discussed. The corresponding fractional integrals are introduced. Mapping and semigroup properties, integral representations and Mellin transform analysis are presented. A relationship with the Riemann–Liouville fractional integrals is demonstrated. Finally, a second kind integral equation of the Volterra type, involving the Laguerre fractional integral is solved in terms of the double hypergeometric type series as the resolvent kernel.

Inclusion properties of hypergeometric type functions and related integral transforms

In this work, conditions on the parameters $a, b$ and $c$ are given so that the normalized Gaussian hypergeometric function $zF(a,b;c;z)$, where \begin{align*} F(a,b;c;z)=\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_n(1)_n}z^n, \quad |z|<1, \end{align*} is in certain class of analytic functions. Using Taylor coefficients of functions in certain classes, inclusion properties of the Hohlov integral transform involving $zF(a,b;c;z)$ are obtained. Similar inclusion results of the Komatu integral operator related to the generalized polylogarithm are also obtained. Various results for the particular values of these parameters are deduced and compared with the existing literature.

Harmonic manifolds of hypergeometric type and spherical Fourier transform

The spherical Fourier transform on a harmonic Hadamard manifold (Xn,g) of positive volume entropy is studied. If (Xn,g) is of hypergeometric type, namely spherical functions of X are represented by the Gauss hypergeometric functions, the inversion formula, the convolution rule together with the Plancherel theorem are shown by the representation of the spherical functions in terms of the Gauss hypergeometric functions. A geometric characterization of hypergeometric type is derived in terms of volume density of geodesic spheres. Geometric properties of (Xn,g) are also discussed.