Search

We consider two recent solutions of the generalized Faddeev–Volkov model, which is an exactly solvable Ising-type lattice spin model. The first solution is obtained by using the noncompact quantum dilogarithm, and the second one is constructed in a recent study via the gauge/YBE correspondence. We show that the weight functions of these models obtained by different techniques are the same upto a constant. [ABSTRACT FROM AUTHOR]

In this paper, we investigate hypergeometric stringy corrections in the swampland program for rescaled gravity. Precisely, we study inflationary models from Gauss–Bonnet hypergeometric scalar couplings via the falsification scenario. We first derive generalized exponential potentials from such hypergeometric behaviors. Then, we examine certain selected scalar potentials by computing the relevant cosmological quantities using the slow-roll mechanism. Choosing specific points in the corresponding moduli space, we provide viable findings corroborated by Planck observational data and checked by the swampland criteria. [ABSTRACT FROM AUTHOR]

Let Dλd denote the family of monomial deformations of diagonal hypersurface over a finite field 픽q given by Dλd : X 1d + X 2d + ⋯ + X nd = λdX 1h1X2h2⋯Xnhn, where d,n ≥ 2, hi ≥ 1, ∑i=1nh i = d, and gcd(d,h1,h2,…,hn) = 1. The Dwork hypersurface is the case when d = n, that is, h1 = h2 = ⋯ = hn = 1. Formulas for the number of 픽q-points on the Dwork hypersurfaces in terms of McCarthy’s p-adic hypergeometric functions are known. In this paper, we provide a formula for the number of 픽q-points on Dλd in terms of McCarthy’s p-adic hypergeometric function which holds for d ≥ n. [ABSTRACT FROM AUTHOR]

We prove some weighted sum formulas for half multiple zeta values, half finite multiple zeta values, and half symmetric multiple zeta values. The key point of our proof is Dougall’s identity for the generalized hypergeometric function 5F4. Similar results for interpolated refined symmetric multiple zeta values and half refined symmetric multiple zeta values are also discussed. [ABSTRACT FROM AUTHOR]

We study the relations of multiple t -values of general level. The generating function of sums of multiple t -(star) values of level N with fixed weight, depth and height is represented by the generalized hypergeometric function 3 F 2 , which generalizes the results for multiple zeta(-star) values and multiple t -(star) values. As applications, we obtain formulas for the generating functions of sums of multiple t -(star) values of level N with height one and maximal height and a weighted sum formula for sums of multiple t -(star) values of level N with fixed weight and depth. Using the stuffle algebra, we also get the symmetric sum formulas and Hoffman's restricted sum formulas for multiple t -(star) values of level N. Some evaluations of multiple t -star values of level 2 with one–two–three indices are given. [ABSTRACT FROM AUTHOR]

In this paper, we use Padé approximations, constructed in [Kawashima and Poëls, Padé approximations for shifted functions and parametric geometry of numbers, J. Number Theory 243 (2023) 646–687] for binomial functions, to give a new upper bound for the number of the solutions of the S -unit equation in two variables. Combining explicit Padé approximants with a simple argument relying on Mahler measure as well as the local height, we refine the bound due to Evertse [On equations in S -units and the Thue-Mahler equation, Invent. Math. 75 (1984) 561–584]. [ABSTRACT FROM AUTHOR]

The k -convoluted operators related to the k -Whittaker function, confluent hypergeometric function of the first kind, have been developed using the k -symbol calculus in which this sort of calculus presents a generalization of the gamma function. K -symbol fractional calculus is employed to generalize and extend many differential and integral operators of fractional calculus. Based on this premise, a new geometric formula for normalized functions in the symmetric domain known as the open unit disk using the conformable fractional differential operator has been presented in this study. Thus, our technique entails investigating the most well-known geometric properties of this new operator, such as the subordination features and coefficient bounds so that the theory of differential subordination can be adjusted accordingly. By means of this technique, numerical results have been investigated for the proposed method. To this end, a few prominent corollaries of our primary findings as standout instances have been pointed out based on the positivity of the solutions, computational and numerical analyses. [ABSTRACT FROM AUTHOR]

Local system cohomology groups of the complements of hyperplane arrangements have played an important role in the theory of hypergeometric integrals, the topology of Milnor fibers and covering spaces. One of the important theorems is the vanishing theorem for generic ℂ -local systems which goes back to Aomoto's work. Later, Cohen, Dimca, and Orlik proved a stronger version of the vanishing theorem. In this paper, we prove a Cohen–Dimca–Orlik type theorem for ℤ -local systems. [ABSTRACT FROM AUTHOR]

Recent work of Griffin, Ono, Rolen, and Zagier shows that the Jensen polynomials for the Riemann xi function converge to the Hermite polynomials under a suitable normalization. We generalize this result, proving that the normalized Jensen polynomials for a large class of genus zero or one entire functions converge either to the Hermite polynomials, or to a class of polynomials which can be written as a confluent hypergeometric function. [ABSTRACT FROM AUTHOR]

In this paper, we introduce certain Kampé de Fériet-like matrix functions and investigate their regions of convergence and recurrence relations satisfied by them. The importance of studying these matrix functions stems from the fact that the derivatives of arbitrary integral order of confluent hypergeometric matrix function and Gauss hypergeometric matrix function with respect to their matrix parameters can be expressed in the form of these Kampé de Fériet-like matrix functions. Integral representations of these matrix functions in a special case are also given. [ABSTRACT FROM AUTHOR]

The asymptotic expansion of the generalized hypergeometric function p F q (z) as z → + ∞ involves a coefficient sequence { c k }. Explicit formulas are given for this sequence when 0 ≤ p < q. The result is based on an integral representation of the generalized hypergeometric function that allows application of Watson's lemma. [ABSTRACT FROM AUTHOR]

We compute the Laplace transforms of some integral functionals of the three-dimensional Bessel process in terms of modified Bessel functions, Gauss' hypergeometric functions, and confluent hypergeometric functions. Some new results are obtained, and several established results, such as Dufresne's perpetuity and a particular case of its translated version, are recovered. In particular, we derive the Laplace transform of the two-dimensional random variable (∫ 0 ∞ (exp (ρ t) − η) − 1 d t , ∫ 0 ∞ (exp (ρ t) − η) − 2 d t) , where ρ is a three-dimensional Bessel process and η ≤ 1. A crucial step of the derivation is to construct a martingale, as performed in the Bass solution of the Skorokhod embedding problem, with respect to an enlarged filtration. [ABSTRACT FROM AUTHOR]

This paper investigates the fractional Schrödinger equation (FSE) with the Caputo time fractional derivative for the generalized Morse potential, which has not yet been presented for this equation. This study depends on the analytical solution of the FSE by the method of integral transforms and the numerical solutions are presented by plotting the eigensolutions with the Python script. For this purpose, we apply a special ansatz solution together with the Fourier transform (for the space variable) and the Laplace transform (with respect to time) on the FSE and obtain the Gaussian hypergeometric differential equation. By applying the inverse Fourier transform on the solution of the hypergeometric function, the G-Meijer function in terms of the coordinate and the Laplace transformed variable are obtained. We then calculate the wave function of the time fractional Schrödinger using the inverse Laplace transform together considering the Schouten-Vanderpol theorem and some special circumstances of the problem. The obtained results show that for different values of the time fractional parameter, the probability of the particle presence is time-dependent, and in the limit case of α → 1 , the solutions obtained from the time FSE are consistent with the results of standard Schrödinger equation for the generalized Morse potential. The results also show that the amplitude of wave function of the particle presence decreases over time and the energy of the system decreases in small times for different values of the fractional parameter and for the large times, the amount of energy is almost constant. [ABSTRACT FROM AUTHOR]

In this paper, we solve the Duffin–Kemmer–Petiau (DKP) equation in the presence of hyperbolic tangent potential for spin-one particles. By partitioning the spin-one spinor, we show that the DKP equation is equivalent to the Klein–Gordon equation formalism. The scattering solutions are derived in terms of hypergeometric functions. The reflection R and transmission T coefficients are calculated in terms of the Gamma functions. The results show the presence of the superradiance phenomenon when R for a specific region in the potential becomes greater than one. [ABSTRACT FROM AUTHOR]

In this paper, we construct a (p , k) -hypergeometric function by using the Hadamard product, which we call the generalized (p , k) -hypergeometric function. Several properties, namely, convergence properties, derivative formulas, integral representations and differential equations are indicated of this function. The latter function is a generalization of the usual hypergeometric function, the k-analogue of hypergeometric function and other hypergeometric functions are recently presented. As an application, we obtain the solution of the generalized fractional kinetic equations involving of the generalized (p , k) -hypergeometric function. [ABSTRACT FROM AUTHOR]

In the paper, on account of the Kummer confluent hypergeometric series of first type, we suggest the Turań-type inequalities for the hypergeometric supersine and supercosine functions by using the Cauchy–Bunyakovsky–Schwarz inequality. [ABSTRACT FROM AUTHOR]

Based on the q -Mellin operator properties, we verify rigorously new class of q -index transforms. Moreover, we establish a convolution operator and inversion formulas. As applications, we give new transformations with the hypergeometric functions kernel. [ABSTRACT FROM AUTHOR]

An efficient and accurate method for calculating the sound radiated by a baffled circular rigid piston is using spherical harmonics, and the solution is a series containing the integral of spherical Bessel functions. The integral is usually calculated with the generalized hypergeometric functions in existing literatures, which shows poor convergence at middle and high frequencies due to the overflow and the loss of significant figures. A rigorous and closed form solution of the integral is derived in this paper based on the recurrence method, which is accurate in the whole frequency range and thousands of times faster than the existing methods. It is shown that the proposed method can be extended for the calculation of the sound radiated by a baffled piston and an unbaffled resilient disk with axisymmetric velocity and pressure profiles, respectively, and some baffled rotating sources where the velocity profile is asymmetric. [ABSTRACT FROM AUTHOR]

In this paper, we use an identity connecting a modified q -Bessel function and a 1 ϕ 1 function to give m -versions of entries in the Lost Notebook of Ramanujan. We also establish an identity which gives an m -version of a partition identity. We prove new relations and identities involving theta functions, the Ramanujan function, the Stieltjes–Wigert, q -Lommel and q -Bessel polynomials. We introduce and study q -analogues of the spherical Bessel functions. [ABSTRACT FROM AUTHOR]

In this paper, we present an exact solution of the Klein–Gordon equation in the framework of the fractional-dimensional space, in which the momentum and position operators satisfying the R -deformed Heisenberg algebras. Accordingly, three essential problems have been solved such as: the free Klein–Gordon equation, the Klein–Gordon equation with mixed scalar and vector linear potentials and with mixed scalar and vector inversely linear potentials of Coulomb-type. For all these considered cases, the expressions of the eigenfunctions are determined and expressed in terms of the special functions: the Bessel functions of the first kind for the free case, the biconfluent Heun functions for the second case and the confluent hypergeometric functions for the end case, and the corresponding eigenvalues are exactly obtained. [ABSTRACT FROM AUTHOR]

Recently the first author proved a congruence proposed in 2006 by Adamchuk: ∑ k = 1 ⌊ 2 p / 3 ⌋ 2 k k ≡ 0 (mod p 2) for any prime p = 1 (mod 3). In this paper, we provide more examples (with proofs) of congruences of the same kind ∑ k = 1 ⌊ a p / r ⌋ 2 k k x k (mod p 2) where p is a prime such that p ≡ 1 (mod r) , a / r is a fraction in (1 / 2 , 1) and x is a p -adic integer. The key ingredients are the p -adic Gamma function Γ p and a special class of computer-discovered hypergeometric identities. [ABSTRACT FROM AUTHOR]

In this paper, we study the Duffin–Kemmer–Petiau (DKP) equation in the presence of a smooth barrier in dimensions space–time (1+1) dimensions. The eigenfunctions are determined in terms of the confluent hypergeometric function M (α ; β ; y). The transmission and reflection coefficients are calculated, special cases as a rectangular barrier and step potential are analyzed. A numerical study is presented for the transmission and reflection coefficients graphs for some values of the parameters (V 0 , E) are plotted. [ABSTRACT FROM AUTHOR]

This paper deals with the linear buckling problem for inhomogeneous Euler–Bernoulli column having both mechanical and geometrical properties variable along its length. Four classes of longitudinally functionally graded material columns with variable cross-sections are considered. The solutions of the relevant differential equations are obtained in terms of both hypergeometric functions and elementary functions. The normalized buckling loads are computed for five typical boundary conditions and they are validated by a comparison with approximate numerical results available in literature. The proposed formulation may provide a further benchmark for the accuracy assessment of numerical and approximated solutions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of sense preserving harmonic mappings f = h + g ¯ in the open unit disk and prove that functions in this class are close-to-convex. We give some basic properties such as coefficient bounds, growth estimates, convolution and determine the radius of convexity for the sections of functions belonging to this family. In addition, we construct certain harmonic univalent polynomials belonging to this family. [ABSTRACT FROM AUTHOR]

Inspired by the recent work by Abd-Elmageed et al., who established recursion formulas satisfied by the first Appell matrix function, namely F 1 , Sahai et al. presented various recursion formulas for the Gauss hypergeometric matrix function and all four Appell matrix functions. In this paper, we obtain recursion formulas and infinte summation formulas for Srivastava's triple hypergeometric matrix functions H 𝒜 , H ℬ and H 𝒞 . [ABSTRACT FROM AUTHOR]

We describe an extension of Fine's method of deriving basic hypergeometric series transformations and derive new transformations from the method. Combinatorial proofs of two of the examples are provided. [ABSTRACT FROM AUTHOR]

A typical quandary in geometric functions theory is to study a functional composed of amalgamations of the coefficients of the pristine function. Conventionally, there is a parameter over which the extremal value of the functional is needed. The present paper deals with consequential functional of this type. By making use of Hohlov operator, a new subclass ℛ a , b c of analytic functions defined ins the open unit disk is introduced. For both real and complex parameter, the sharp bounds for the Fekete–Szegö problems are found. An attempt has also been taken to found the sharp upper bound to the second and third Hankel determinant for functions belonging to this class. All the extremal functions are express in term of Gauss hypergeometric function and convolution. Finally, the sufficient condition for functions to be in ℛ a , b c is derived. Relevant connections of the new results with well-known ones are pointed out. [ABSTRACT FROM AUTHOR]

We present new expressions for the densities and distributions of the largest eigenvalue and the trace of a Beta–Wishart matrix. The series expansions for these expressions involve fewer terms than previously known results. For the trace, we also present a new algorithm that is linear in the size of the matrix and the degree of truncation, which is optimal. [ABSTRACT FROM AUTHOR]

The method of Frobenius is a standard technique to construct series solutions of an ordinary linear differential equation around a regular singular point. In the classical case, when the roots of the indicial polynomial are separated by an integer, logarithmic solutions can be constructed by means of perturbation of a root. The method for a regular A -hypergeometric system is a theme of the book by Saito, Sturmfels and Takayama. Whereas they perturbed a parameter vector to obtain logarithmic A -hypergeometric series solutions, we adopt a different perturbation in this paper. [ABSTRACT FROM AUTHOR]

We solve the one-dimensional time-independent Klein{Gordon equation in the presence of a smooth potential well. The bound state solutions are given in terms of the Whittaker Mκ,μ(x) function, and the antiparticle bound state is discussed in terms of potential parameters. [ABSTRACT FROM AUTHOR]

In [M. Asakura, N. Otsubo and T. Terasoma, An algebro-geometric study of special values of hypergeometric functions 3 F 2 , to appear in Nagoya Math. J.; https://doi.org/10.1017/nmj.2018.36 ], we proved that the value of 3 F 2 a , b , q a + b , q ; 1 of the generalized hypergeometric function is a ℚ ¯ -linear combination of log of algebraic numbers if rational numbers a , b , q satisfy a certain condition. In this paper, we present a method to obtain an explicit description of it. [ABSTRACT FROM AUTHOR]

In this paper, we give some characterizations for the boundedness of weighted fractional composition operator D φ , u β from α -Bloch spaces into weighted type spaces by deriving the bounds of its norm. Also, estimates for essential norm are obtained which gives necessary and sufficient conditions for the compactness of the operator D φ , u β . [ABSTRACT FROM AUTHOR]

This paper deals with the accretion onto some spherically symmetric singularity-free black holes with barotropic equation of state. We consider the black hole with Gauss hypergeometric function, regular phantom and charged noncommutative black hole. We formulate the general formalism for the spherically symmetric accretion and then found the dynamical quantities, fluid four-velocity, energy density and rate of accretion for the specific black holes. The critical accretion is discussed by finding critical radius, critical velocity and critical sound velocity. It is found that the black hole mass decreases with the accretion of phantom-like fluid. [ABSTRACT FROM AUTHOR]

In Phys. Rev. Lett. 115, 143001 (2015), H. Mera et al. developed a new simple but precise hypergeometric resummation technique. In this work, we suggest to obtain half of the parameters of the hypergeometric function from the strong-coupling expansion of the physical quantity. Since these parameters are taking now their exact values they can improve the precision of the technique for the whole range of the coupling values. The second order approximant 2 F 1 of the algorithm is applied to resum the perturbation series of the ground state energy of the 𝒫 𝒯 -symmetric (i ϕ 3) 0 + 1 field theory. It gives accurate results compared to exact calculations from the literature specially for very large coupling values. The 𝒫 𝒯 -symmetry breaking of the Yang–Lee model has been investigated where third, fourth and fifth orders were able to get very accurate results when compared to other resummation methods involving 1 5 0 orders. The critical exponent ν of the O (4) -symmetric model in three dimensions has been precisely obtained using only first order of perturbation series as input. The algorithm can be extended easily to accommodate any order of perturbation series in using the generalized hypergeometric function k + 1 F k as it shares the same analytic properties with 2 F 1 . [ABSTRACT FROM AUTHOR]

We study a local system associated with a system ℱ of hypergeometric differential equations in two variables of rank 9 with seven parameters a 1 , a 2 , a 3 and b 1 , b 2 , b 3 , b 4 . We modify the fundamental system of solutions to ℱ given in [A system of hypergeometric differential equations in two variables of rank 9, Internat. J. Math. 28 (2017), 1750015, 34 pp] so that it is valid even in cases where b 1 , b 2 , b 3 , b 4 satisfy some integral conditions. By using this fundamental system, we show the irreducibility of the monodromy representation of ℱ under some conditions on the parameters. We characterize the fundamental group of the base space of this local system as the group generated by three loops with four relations among them. [ABSTRACT FROM AUTHOR]

We study the problem of pair creation of scalar particles by an electric field in curved Snyder space. We find exact solutions for the Klein–Gordon equation with a constant electric field in terms of hypergeometric functions. Then we calculate the pair creation probability and the number of created pairs of particles through Bogoliubov transformation technique. [ABSTRACT FROM AUTHOR]

We investigate the stochastic process defined as the square of the (integrated) symmetric telegraph process. Specifically, we obtain its probability law and a closed form expression of the moment generating function. Some results on the first-passage time through a fixed positive level are also provided. Moreover, we analyze some functionals Φ (⋅ , ⋅) of two independent squared telegraph processes, both in the case Φ (u , v) = u + v and Φ (u , v) = u ⋅ v. Starting from this study, we provide some results on the probability density functions of the two-dimensional radial telegraph process and of the product of two independent symmetric telegraph processes. Some of the expressions obtained are given in terms of new results about derivatives of hypergeometric functions with respect to parameters. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalization of the q -Taylor expansion theorems. We expand a function in a neighborhood of two points instead of one in three different theorems. The first is a q -analog of the Lidstone theorem where the two points are 0 and 1 and we expand the function in q -analogs of Lidstone polynomials which are in fact q -Bernoulli polynomials as in the classical case. The definitions of these q -Bernoulli polynomials and numbers are introduced. We also introduce q -analogs of Euler polynomials and numbers. On the other two expansion theorems, we expand an analytic function around arbitrary points z 1 and z 2 either in terms of the polynomials { 1 , ∏ k = 0 n − 1 (z − z 1 q k) (z − z 2 q k) , n ∈ ℕ } or in terms of the polynomials { 1 , ∏ k = 0 n − 1 (z 1 − z q k) (z 2 − z q k) , n ∈ ℕ }. As an application, we introduce a new series expansion for the basic hypergeometric series 2 ϕ 1 . [ABSTRACT FROM AUTHOR]

This paper is concerned with weighted energy estimates for solutions to wave equation ∂ t 2 u − Δ u + a (x) ∂ t u = 0 with space-dependent damping term a (x) = | x | − α (α ∈ [ 0 , 1 ]) in an exterior domain Ω having a smooth boundary. The main result asserts that the weighted energy estimates with weight function like polynomials are given and these decay rates are almost sharp, even when the initial data do not have compact support in Ω. The crucial idea is to use special solution of ∂ t u = | x | α Δ u including Kummer's confluent hypergeometric functions. [ABSTRACT FROM AUTHOR]

We present a multi-spin solution to the Yang–Baxter equation (YBE). The solution corresponds to the integrable lattice spin model of statistical mechanics with positive Boltzmann weights and parametrized in terms of the basic hypergeometric functions. We obtain this solution from a nontrivial basic hypergeometric sum-integral identity which originates from the equality of supersymmetric indices for certain three-dimensional 𝒩 = 2 Seiberg dual theories. [ABSTRACT FROM AUTHOR]

In this paper, the sextic oscillator is discussed as a potential obtained from the bi-confluent Heun equation after a suitable variable transformation. Following earlier results, the solutions of this differential equation are expressed as a series expansion of Hermite functions with shifted and scaled arguments. The expansion coefficients are obtained from a three-term recurrence relation. It is shown that this construction leads to the known quasi-exactly solvable (QES) form of the sextic oscillator when some parameters are chosen in a specific way. By forcing the termination of the recurrence relation, the Hermite functions turn into Hermite polynomials with shifted arguments, and, at the same time, a polynomial expression is obtained for one of the parameters, the roots of which supply the energy eigenvalues. With the δ = 0 choice the quartic potential term is canceled, leading to the reduced sextic oscillator. It was found that the expressions for the energy eigenvalues and the corresponding wave functions of this potential agree with those obtained from the QES formalism. Possible generalizations of the method are also presented. [ABSTRACT FROM AUTHOR]

This paper continues the study of finite summation formulas of multivariable hypergeometric functions. Earlier, Wang and Chen [Finite summation formulas of double hypergeometric functions, Int. Transf. Spec. Funct.28(3) (2016) 239–253] established finite summation formulas of double hypergeometric functions by using some well-known summation theorems. We present here finite summation formulas for Srivastava's general triple hypergeometric function. Certain particular cases leading to finite summation formulas for Lauricella functions and Srivastava's triple hypergeometric functions are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we explore the existence of wormhole solutions exhibiting spherical symmetry in a modified gravity namely f (R , T) theory by involving some aspects of noncommutative geometry. For this purpose, we consider the anisotropic matter contents along with the well-known Gaussian and Lorentizian distributions of string theory. For the sake of simplicity in analytic discussions, we take a specific form of f (R , T) function given by f (R , T) = R + λ T. For both these noncommutative distributions, we get exact solutions in terms of exponential and hypergeometric functions. By taking some suitable choice of free parameters, we investigate different interesting aspects of these wormhole solutions graphically. We also explored the stability of these wormhole models using equilibrium condition. It can be concluded that the obtained solutions are stable and physically viable satisfying the wormhole existence criteria. Lastly, we discuss the constraints for positivity of the active gravitational mass for both these distributions. [ABSTRACT FROM AUTHOR]

Kimura and Takano showed that taking limits of regular elements of 𝔤 𝔩 (n) corresponds to the process of confluence of Aomoto–Gel'fand systems. We introduce a hypergeometric system associated with a principal nilpotent p -tuple, and, by using the principal nilpotent p -tuple, we directly deform a hypergeometric system of Gauss type into that of Airy type. Moreover, we explicitly describe the deformation. [ABSTRACT FROM AUTHOR]

In this paper, incorporating the effect of the deformed commutation relation on de Sitter background, we studied the deformed Schwinger mechanism in (1 + 1) dimensions for scalars particle of spin-0 in a constant electric field. The Klein–Gordon equation is solved exactly and the wave function is given in term of hypergeometric functions. The canonical method based on a Bogoliubov transformation is applied. The pair creation probability and the density number of created particles are calculated. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to present five mock theta functions by using the Bailey pair. Identities are established between the double sums and classical mock theta functions. [ABSTRACT FROM AUTHOR]

This is a review of (q-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal polynomials, and consider their various generalizations. The review also includes the orthogonal polynomials into a generic framework of (q-)hypergeometric functions and their integral representations. In particular, this gives rise to relations with conformal blocks of the Virasoro algebra. To the memory of Ludwig Dmitrievich Faddeev [ABSTRACT FROM AUTHOR]

One of the spectacular results in mathematical physics is the expression of Racah matrices for symmetric representations of the quantum group through the Askey-Wilson polynomials, associated with the -hypergeometric functions . Recently it was shown that this is in fact the general property of symmetric representations, valid for arbitrary - at least for exclusive Racah matrices . The natural question then is what substitutes the conventional -hypergeometric polynomials when representations are more general? New advances in the theory of matrices , provided by the study of differential expansions of knot polynomials, suggest that these are multiple sums over Young sub-diagrams of the one which describes the original representation of . A less trivial fact is that the entries of the sum are not just the factorized combinations of quantum dimensions, as in the ordinary hypergeometric series, but involve non-factorized quantities, like the skew characters and their further generalizations - as well as associated additional summations with the Littlewood-Richardson weights. [ABSTRACT FROM AUTHOR]

In recent work, Hickerson and Mortenson introduced a dual notion between Appell-Lerch sums and partial theta functions. In this sense, Appell-Lerch sums and partial theta functions appear to be dual to each other. In this paper, by making the substitution in the tail of the associated bilateral series of mock theta functions and universal mock theta functions, we demonstrate how to obtain duals of the second type in terms of Appell-Lerch sums defined by Mortenson for such functions. Then by using the substitution in duals of the second type of each bilateral series, we present how to translate between identities expressing -hypergeometric series in terms of Appell-Lerch sums and identities expressing -hypergeometric series in terms of partial theta functions. Indeed, we obtain only four duals in terms of partial theta functions of duals of the second type in terms of Appell-Lerch sums of bilateral series associated to mock theta functions. As an application, we construct Ramanujan radial limits by using these bilateral series with mock modular behavior in terms of Appell-Lerch sums for some order mock theta functions. This method is well-suited for the other order mock theta functions. [ABSTRACT FROM AUTHOR]

In recent years, Sun proposed several sophisticated conjectures on congruences for finite sums with terms involving combinatorial sequences such as central trinomial coefficients, Domb numbers and Franel numbers. These sums are double summations of hypergeometric terms. Using the telescoping method and certain mathematical software packages, we transform such a double summation into a single sum. With this new approach, we confirm several open conjectures of Sun. [ABSTRACT FROM AUTHOR]