Generalized Hypergeometric Functions Research Articles

Two q-congruences involving double basic hypergeometric sums

Two q-congruences involving double basic hypergeometric sums

Supercongruences for rigid hypergeometric Calabi–Yau threefolds

We establish the supercongruences for the fourteen rigid hypergeometric Calabi–Yau threefolds over Q conjectured by Rodriguez-Villegas in 2003. Our first method is based on Dwork's theory of p-adic unit roots and it allows us to establish the supercongruences between the truncated hypergeometric series and the corresponding unit roots for ordinary primes. The other method makes use of the theory of hypergeometric motives, in particular, adapts the techniques from the recent work of Beukers, Cohen and Mellit on finite hypergeometric sums over Q. Essential ingredients in executing the both approaches are the modularity of the underlying Calabi–Yau threefolds and a p-adic perturbation method applied to hypergeometric functions.

Corollaries and multiple extensions of Gessel and Stanton hypergeometric summation formulas

We find some new simple hypergeometric formulas in the footsteps of the important article by Gessel and Stanton. These are multiple reduction formulas, multiple summation formulas, as well as multiple transformation formulas for special Kampé de Fériet functions and Appell functions. The hypergeometric summation formulas have special function arguments in Q and parameter values in N or C. The proofs use Pfaff-Kummer transformation, Euler transformation, or an improved form of Slater reversion.

(q-)Supercongruences hit again

Using an intrinsic q-hypergeometric strategy, we generalise Dwork-type congruences H(p^{s+1})/H(p^s) ≡ H(p^s)/H(p^{s−1}) (mod p 3) for s = 1, 2,. .. and p a prime, when H(N) are truncated hypergeometric sums corresponding to the periods of rigid Calabi-Yau threefolds.

Some q-congruences on double basic hypergeometric sums

ABSTRACT We give three q-congruences on double basic hypergeometric sums. One of them is a q-analogue of the following supercongruence: for any prime p>3, Our proof uses q-analogues of two Ramanujan-type supercongruences of Van Hamme and a q-analogue of a ‘divergent’ Ramanujan-type supercongruence.

Inclusion Relation between Various Subclasses of Harmonic Univalent Functions Associated with Wright’s Generalized Hypergeometric Functions

The purpose of the present paper is to obtain some inclusion relation between various subclasses of harmonic univalent functions by applying certain convolution operators associated with Wright’s generalized hypergeometric functions.

Bayesian cross-product quality control via transfer learning

ABSTRACT Quality control is essential for modern business success. The traditional statistical process control (SPC), however, lacks efficacy in current high-variety low-volume industrial practices since the historical reference data in Phase I are usually too scarce to infer the in-control process parameters accurately. To solve this ‘small data’ challenge, a novel Bayesian process monitoring scheme via transfer learning is proposed to facilitate a cross-product data sharing. In particular, a joint prior distribution is taken to explicitly capture the relatedness between the process data of two similar products, through which the process information can be transferred from one product (source domain) to improve the Bayesian inference for the other product (target domain). The posteriors can be derived analytically in closed forms by using generalised hypergeometric functions, thereby leading to a computationally efficient control chart for the online real-time monitoring in Phase II. A user-specified parameter is also provided to enable a better theoretical understanding of the transferability matter and a free practical control of the transferred information across domains. Extensive numerical simulations and real example studies of an assembly process validate the superiority of our proposed scheme in terms of both the false alarm rate and detection capability.

Some new q-congruences on double sums

Swisher confirmed an interesting congruence: for any odd prime p, $$\begin{aligned} \sum _{k=0}^{(p-1)/2}(-1)^k(6k+1)\frac{(\frac{1}{2})_k^{3}}{k!^{3}8^k} \sum _{j=1}^{k}\Bigg (\frac{1}{(2j-1)^{2}}-\frac{1}{16j^{2}}\Bigg ) \equiv 0 \pmod {p}, \end{aligned}$$ which was conjectured by Long. Recently, its q-analogue was proved by Gu and Guo. Inspired by their work, we obtain a new similar q-congruence modulo $$\Phi _n(q)$$ and two q-supercongruences modulo $$\Phi _n(q)^{2}$$ on double basic hypergeometric sums, where $$\Phi _n(q)$$ is the n-th cyclotomic polynomial.

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.

Horn's hypergeometric functions with three variables

The work is devoted to obtaining explicit formulas for analytic continuation of quite general hypergeometric series depending on three variables and belonging to the Horn class. We have derived such continuation formulas with respect to one variable for an arbitrary series belonging to the class under consideration. In addition, on the base of several examples we have demonstrated the use of the obtained formulas for construction of analytic continuation with respect to all three variables. The obtained continuation formulas give representations for a triple hypergeometric series outside the domain of its convergence in the form of a combination of other hypergeometric series that also belong to the Horn class. These series satisfy to the same system of partial differential equations as the initial series. The results of the paper are useful for qualitative analysis and calculation of triple hypergeometric functions.

Polylogarithms from the Bound-State S-matrix

Higher-point functions of gauge invariant composite operators in N=4 super Yang-Mills theory can be computed via triangulation. The elementary tile in this process is the hexagon introduced for the evaluation of structure constants. A glueing procedure welding the tiles back together is needed to return to the original object. In this note we present work in progress on n-point functions of BPS operators. In this case, quantum corrections are entirely carried by the glueing procedure. The lowest non-elementary process is the glueing of three adjacent tiles by the exchange of two single magnons. This problem has been analysed before. With a view to resolving some conceptional questions and to generalising to higher processes we are trying to develop an algorithmic approach using the representation of hypergeometric sums as integrals over Euler kernels.

On One Integral Equation in the Theory of Transform Operators

Integral representations of solutions of one differential equation with singularities in the coefficients, containing the Bessel operator perturbed by some potential, are considered. The existence of integral representations of a certain type for such solutions is proved by the method of successive approximations using transform operators. Potentials with strong singularities at the origin are allowed. As compared with the known results, the Riemann function is expressed not via the general hypergeometric function, but, more specifically, via the Legendre function, which helps to avoid unknown constants in the estimates.

Reducing radicals in the spirit of Euclid

Let p be an odd natural number ≥3. Inspired by results from Euclid's Elements, we express the irrationaly=d+Rp, whose degree is 2p, as a polynomial function of irrationals of degrees ≤p. In certain cases y is expressed by simple radicals. This reduction of the degree exhibits remarkably regular patterns of the polynomials involved. The proof is based on hypergeometric summation, in particular, on Zeilberger's algorithm.

On some new Gaussian hypergeometric summation formulae with applications

The aim of this note is to provide some new Gaussian hypergeometric summation formulae. These are further used to obtain certain new expressions for the product of hypergeometric series. Already obtained results by Bailey, Choi and Rathie and Qureshi et al. follow special cases of our main findings.

Analytic continuation of the Horn hypergeometric series with an arbitrary number of variables

The problem of analytic continuation is considered for the general hypergeometric Horn series with an arbitrary number of variables. An approach is proposed that allows one to find formulas for the continuation of such series into the exterior of the set of their convergence in the form of linear combinations of other hypergeometric series. These new hypergeometric series belong also to the Horn class, and are solutions of the same system of partial differential equations. Using this general result, we construct formulas for the analytic continuation for a double hypergeometric series of Horn's type, which plays an important role in the theory of the Appell function .

Hypergeometric decomposition of symmetric K3 quartic pencils

We study the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard–Fuchs differential equations; we count points using Gauss sums and rewrite this in terms of finite-field hypergeometric sums; then we match up each differential equation to a factor of the zeta function, and we write this in terms of global L-functions. This computation gives a complete, explicit description of the motives for these pencils in terms of hypergeometric motives.

A Common q-Analogue of Two Supercongruences

We give a q-congruence whose specializations q=-1 and q=1 correspond to supercongruences (B.2) and (H.2) on Van Hamme’s list (in: p-Adic Functional Analysis (Nijmegen, 1996), Lecture Notes in Pure and Applied Mathematics, vol 192. Dekker, New York, pp 223–236, 1997): ∑k=0(p-1)/2(-1)k(4k+1)Ak≡p(-1)(p-1)/2(modp3)and∑k=0(p-1)/2Ak≡a(p)(modp2),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned}&\\sum _{k=0}^{(p-1)/2}(-1)^k(4k+1)A_k\\equiv p(-1)^{(p-1)/2}\\;({\\text {mod}}p^3) \\quad \\text {and}\\quad \\\\&\\sum _{k=0}^{(p-1)/2}A_k\\equiv a(p)\\;({\\text {mod}}p^2), \\end{aligned}$$\\end{document}where p>2 is prime, Ak=∏j=0k-1(1/2+j1+j)3=126k2kk3fork=0,1,2,…,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} A_k=\\prod _{j=0}^{k-1}\\biggl (\\frac{1/2+j}{1+j}\\biggr )^3=\\frac{1}{2^{6k}}{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) }^3 \\quad \\text {for}\\; k=0,1,2,\\ldots , \\end{aligned}$$\\end{document}and a(p) is the pth coefficient of the modular form qprod _{j=1}^infty (1-q^{4j})^6 (of weight 3). We complement our result with a general common q-congruence for related hypergeometric sums.

Some hypergeometric summations formulas and series identities with applications

In the paper we present two incomplete Gaussian hypergeometric formulas in summation form by specific known formulas.We also developed each of these formulas and how they use to derive double series identities in general forms.

Non-Koszulness of Operads and Positivity of Poincaré Series

We prove that the operad of mock partially associative $n$-ary algebras is not Koszul, as conjectured by the second and the third author in 2009, and utilise the Zeilberger's algorithm for hypergeometric summation to demonstrate that non-Koszulness of that operad cannot be established by hunting for negative coefficients in the inverse of its Poincar\'e series.

An Identity Involving Product of Generalized Hypergeometric Series 2F2

An Identity Involving Product of Generalized Hypergeometric Series 2F2