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While working on financing efficiency of securities-based crowdfunding, David C. Brown and Shaun William Davies found that in order to establish their major results analytically they needed to find a proof of their conjecture (based on numerical attempts) that a certain quotient of quotients of 2 F 1 hypergeometric functions is bounded above by one. In this paper, the author extends his proof of the conjecture, which is contained in Addendum 1 of the published 2020 Brown and David paper, to generalizations of hypergeometric and basic hypergeometric functions. [ABSTRACT FROM AUTHOR]

In this paper, we present several explicit formulas for Appell polynomials in terms of the weighted Stirling numbers of the second kind. We also provide a unified approach to obtain a three-term recurrence formula for the computation of Appell polynomials. Several examples are given to illustrate our results. As an application, we define and study a new class of polynomials that we call SPI polynomials. They are related to sums of powers of integers. [ABSTRACT FROM AUTHOR]

The classical AGM produces wonderful infinite sequences of arithmetic and geometric means with common limit. For finite fields F q , with q ≡ 3 (mod 4) , we introduce a finite field analogue AGM F q that spawns directed finite graphs instead of infinite sequences. The compilation of these graphs reminds one of a jellyfish swarm, as the 3D renderings of the connected components resemble jellyfish (i.e., tentacles connected to a bell head). These swarms turn out to be more than the stuff of child's play; they are taxonomical devices in number theory. Each jellyfish is an isogeny graph of elliptic curves with isomorphic groups of F q -points, which can be used to prove that each swarm has at least (1 / 2 − ε) q jellyfish. This interpretation also gives a description of the class numbers of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish. [ABSTRACT FROM AUTHOR]

In this paper, we consider the characteristic Goursat problem with regular singular data for a class of linear partial differential equations of second order in the complex domain. We give an explicit representation of the solutions. Such representations permit to see the structure of the solutions and to show that they are singular on the union of the characteristic surfaces issued from data. [ABSTRACT FROM AUTHOR]

Over the last two decades, special matrix functions have become a major area of study for mathematicians and physicists. The famous four Appell hypergeometric matrix functions have received considerable attention by many authors from different points of view. The present paper is devoted to provide further investigations on the two variables second Appell hypergeometric matrix functions. Precisely, we derive certain mathematical properties such as limit formulae of Humbert hypergeometric matrix functions, recursion formulae using contiguous matrix relations, some differentiation and summation formulae. We also find general differential operators which induce some differential equations. This enriches the theory of special matrix functions. The obtained results are believed to be newly presented. [ABSTRACT FROM AUTHOR]

On page 206 in his lost notebook, Ramanujan recorded the following enigmatic identity for his theta function φ (q) : φ (e - 7 π 7 ) = 7 - 3 / 4 φ (e - π 7 ) { 1 + () 2 / 7 + () 2 / 7 + () 2 / 7 }. We give the three missing terms. In addition, we calculate the class invariant G 343 and further special values of φ (e - n π) , for n = 7 , 21 , 35 , and 49. [ABSTRACT FROM AUTHOR]

In the article, we establish some monotonicity and convexity (concavity) properties for certain combinations of polynomials and the Ramanujan R-function by use of the monotone form of L’Hôpital’s rule and present serval new asymptotically sharp bounds for the Ramanujan R-function that improve some previously known results.

A geometric quantization using a partial differential equation version of topological recursion is established for the compactified cotangent bundle of a smooth projective curve of an arbitrary genus. In this quantization, the Hitchin spectral curve of a rank 2 meromorphic Higgs bundle on the base curve corresponds to a quantum curve, which is a Rees D-module on the base. The topological recursion then gives an all-order asymptotic expansion of its solution, thus determining a state vector corresponding to the spectral curve as a meromorphic Lagrangian. We propose a generalization of the topological recursion for a singular spectral curve. We show that the PDE version of the topological recursion automatically selects the normal ordering of the canonical coordinates, and determines the unique quantization of the spectral curve. The quantum curve thus constructed has the semi-classical limit that agrees with the original spectral curve. Typical examples of our construction includes classical differential equations, such as Airy, Hermite, and Gauß hypergeometric equations. The topological recursion gives an asymptotic expansion of solutions to these equations at their singular points, relating Higgs bundles and various quantum invariants.

Recently, the applications of special functions of matrix arguments have received more attention in many fields, such as theoretical physics, number theory, probability theory, engineering and theory of group representations. Gaining enlightenment from these works, in this paper, we introduce the degenerate gamma matrix function, the degenerate zeta matrix function, the degenerate digamma matrix function, the degenerate polygamma matrix function and the degenerate Gauss hypergeometric matrix function. Basic properties of these functions are discussed. In addition, we derive some interesting formulas related to these functions and special cases. [ABSTRACT FROM AUTHOR]

In this paper, by using the residue method of complex analysis, we obtain an explicit partial fraction decomposition for the general rational function x M (x + 1) n λ (M is any nonnegative integer, λ and n are any positive integers). As applications, we deduce the corresponding algebraic identities and combinatorial identities which are the corresponding extensions of Chu' results. We also give some explicit formulas of Apostol-type polynomials and harmonic Stirling numbers of the second kind. [ABSTRACT FROM AUTHOR]

We aim to introduce extended generalized Mittag-Leffler function (EGMLF) via the extended Beta function and obtain certain integral and differential representation of them. Further, we present some formulas of the Riemann - – Liouville fractional integration and differentiation operators. Also, we derive various integral transforms, including Euler transform, Laplace transform, Whittakar transform and K-transform. The operator and transform images are expressed in terms of the Wright generalized hypergoemetric hypergeometric type function. Interesting special cases of the main results are also considered. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the extensions of the Bessel matrix function and the modified Bessel matrix function. We first introduce the extended Bessel matrix function and the extended modified Bessel matrix function of the first kind by using the extended Beta matrix function. Then we establish the integral representations, differentiation formula, and hypergeometric representation of such functions. Finally, as an application, we study a kind of second-order matrix differential equations. We prove that the extended modified Bessel matrix function is a particular solution to this kind of differential equations. [ABSTRACT FROM AUTHOR]

We derive an expansion for the fundamental solution of Laplace's equation in flat-ring cyclide coordinates in three-dimensional Euclidean space. This expansion is a double series of products of functions that are harmonic in the interior and exterior of coordinate surfaces which are peanut shaped and orthogonal to surfaces which are flat-rings. These internal and external peanut harmonic functions are expressed in terms of Lamé—Wangerin functions. Using the expansion for the fundamental solution, we derive an addition theorem for the azimuthal Fourier component in terms of the odd-half-integer degree Legendre function of the second kind as an infinite series in Lamé—Wangerin functions. We also derive integral identities over the Legendre function of the second kind for a product of three Lamé—Wangerin functions. In a limiting case we obtain the expansion of the fundamental solution in spherical coordinates. [ABSTRACT FROM AUTHOR]

We develop an approach to establish 1 / π -series from bimodular forms. Utilizing this approach, we obtain new families of 2-variable 1 / π -series associated to Zagier's sporadic Apéry-like sequences. We first establish a general form of 1 / π -series involving constants related to the evaluation of modular forms at CM-points. Then we discuss situations when the series have rational terms. In addition, we discuss strategies for evaluating the constants involved and present a number of explicit 1 / π -series. [ABSTRACT FROM AUTHOR]

Many product formulas are known classically for generalized hypergeometric functions over the complex numbers. In this paper, we establish some analogous formulas for generalized hypergeometric functions over finite fields. [ABSTRACT FROM AUTHOR]

In this article, we obtain finite field analogues of classical summation identities connecting F 3 -Appell series and 4 F 3 -classical hypergeometric series. As an application, we establish a new summation formula satisfied by the 4 F 3 -Gaussian hypergeometric series. We further express certain special values of 4 F 3 -Gaussian hypergeometric series in terms of traces of the Frobenius endomorphisms of certain families of elliptic curves. We also explicitly find some special values of 4 F 3 -Gaussian hypergeometric series. [ABSTRACT FROM AUTHOR]

We discuss the spectral decomposition of the hypergeometric differential operators on the line R e z = 1 / 2 , such operators arise in the problem of decomposition of tensor products of unitary representations of the universal covering of the group S L (2 , R). Our main purpose is a search of natural bases in generalized eigenspaces and variants of the inversion formula. [ABSTRACT FROM AUTHOR]

Recently, Hong, Mertens, Ono, and Zhang [8] proved a conjecture of Căldăraru, He, and Huang [4] that expresses the Taylor series of the modular j-function around the elliptic points i and ρ = e π i / 3 as rational functions arising from the signature 2 and 3 cases of Ramanujan's theory of elliptic functions to alternative bases. We extend these results and give inversion formulas for the j-function around i and ρ arising from Gauss' hypergeometric functions and Ramanujan's theory in signatures 4 and 6. [ABSTRACT FROM AUTHOR]

© 2018 World Scientific Publishing Co. Pte. Ltd. This chapter aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the topological recursion, that has its origin in random matrix theory and has been effectively applied to many enumerative geometry problems; and the other is the quantization of Hitchin spectral curves associated with Higgs bundles. Our emphasis is on explaining the motivation and examples. Concrete examples of the direct relation between Hitchin spectral curves and enumeration problems are given. A general geometric framework of quantum curves is also discussed.

In this paper we consider two subclasses of bi-univalent functions defined by the Horadam polynomials. Further, we obtain coefficient estimates for the defined classes.

Using the direct relation between the Gegenbauer polynomials C n λ (x) and the Ferrers function of the first kind P ν µ (x) , we compute interrelations between certain Jacobi polynomials, Meixner polynomials, and Ferrers functions of the first and second kind. We then compute Rodrigues-type, standard integral orthogonality and Sobolev orthogonality relations for Ferrers functions of the first and second kinds. In the remainder of the paper using the relation between Gegenbauer polynomials and the Ferrers function of the first kind we derive connection and linearization relations, some definite integral and series expansions, Christoffel-Darboux summation formulas, Poisson kernel and infinite series closure relations (Dirac delta distribution expansions). [ABSTRACT FROM AUTHOR]

Ramanujan recorded Eisenstein series identities of level 5 of weight 2. The objective of this article is to prove these identities by classical method and also we find some new Eisenstein series identities for level 7. Finally we make use of these results to obtain convolution sums of divisor functions. We will also be introducing certain restricted color partition functions which helps us to represent the convolution sums in a much elegant form than the existing ones. [ABSTRACT FROM AUTHOR]

In this paper we explore special values of Gaussian hypergeometric functions in terms of products of Euler Γ -functions and exponential functions of linear functions of the hypergeometric parameters. They include some classical evaluations, but the main inspiration is from the contiguity method recently applied by Akihito Ebisu. [ABSTRACT FROM AUTHOR]

The power series expansions of the hypergeometric functions p + 1 F p (a , b 1 , ... , b p ; c 1 , ... , c p ; z) converge either inside the unit disk | z | < 1 or outside this disk | z | > 1 . Nørlund's expansion in powers of z / (z - 1) converges in the half-plane R (z) < 1 / 2 . For arbitrary z 0 ∈ C , Bühring's expansion in inverse powers of z - z 0 converges outside the disk | z - z 0 | = max { | z 0 | , | z 0 - 1 | } . None of them converge on the whole indented closed unit disk | z | ≤ 1 , z ≠ 1 . In this paper, we derive new expansions in terms of rational functions of z that converge in different regions, bounded or unbounded, of the complex plane that contain the indented closed unit disk. We give either explicit formulas for the coefficients of the expansions or recurrence relations. The key point of the analysis is the use of multi-point Taylor expansions in appropriate integral representations of p + 1 F p (a , b 1 , ... , b p ; c 1 , ... , c p ; z) . We show the accuracy of the approximations by means of several numerical experiments. [ABSTRACT FROM AUTHOR]

Many indefinite integrals are derived for Bessel functions and associated Legendre functions from particular transformations of their differential equations which are closely linked to Wronskians. A large portion of the results for Bessel functions is known, but all the results for associated Legendre functions appear to be new. The method can be applied to many other special functions. All results have been checked by differentiation using Mathematica. [ABSTRACT FROM AUTHOR]

In this paper we investigate solutions to the characteristic Goursat problem with analytic data for a class of linear operators of second order without the Levi's generalized condition. We give an explicit representation of the solutions of the problem involving Kummer functions and show that they are generally singular on the surface K tangent to the characteristic surface S. [ABSTRACT FROM AUTHOR]

In this article we find finite field analogues of certain transformations satisfied by the classical hypergeometric series. Using properties of Gauss and Jacobi sums we evaluate certain character sums to establish these transformations. We then use these transformations to evaluate explicitly several special values of 2 F 1 hypergeometric functions over finite fields. Certain special values of 2 F 1 hypergeometric functions over finite fields containing trivial and quadratic characters obtained by Ono follow from our special values of 2 F 1 hypergeometric functions containing arbitrary characters as parameters. One of the finite field analogues of algebraic hypergeometric identities given by Fuselier, Long, Ramakrishna, Swisher, and Tu also follows from one of our transformation formulas. [ABSTRACT FROM AUTHOR]

In 1985, Howard introduced degenerate Cauchy polynomials together with degenerate Bernoulli polynomials. His degenerate Bernoulli polynomials have been studied by many authors, but his degenerate Cauchy polynomials have been forgotten. In this paper, we introduce some kinds of hypergeometric degenerate Cauchy numbers and polynomials from the different viewpoints. By studying the properties of the first one, we give their expressions and determine the coefficients. Concerning the second one, called H-degenerate Cauchy polynomials, we show several identities and study zeta functions interpolating these polynomials.

Discrete analogues of the classical Fourier-Jacobi transform are introduced and investigated. It involves series and integrals with respect to parameters of the Gauss hypergeometric function 2 F 1 (a + i n / 2 , a − i n / 2 ; c ; − x 2) , x > 0 , n ∈ N , a , c > 0 , i is the imaginary unit. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals are established. [ABSTRACT FROM AUTHOR]

We derive relations for a certain class of terminating 4 F 3 (4) hypergeometric series with three free parameters. The invariance group composed of these relations is shown to be isomorphic to the symmetric group S 3 . We further study relations for terminating 3 F 2 (4) series that fall under two families. By using a series reversal, we examine the corresponding terminating 4 F 3 (1 / 4) and 3 F 2 (1 / 4) series relations. We additionally derive formulas for the sums of the first n + 1 terms of several nonterminating 3 F 2 (4) and 3 F 2 (1 / 4) series. We also show how certain known summation formulas for terminating 2 F 1 (4) and 3 F 2 (4) series follow as limiting cases of some of our relations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the normalized Bessel function of index α > − 1 2 , we find an integral representation of the term x n j α + n (x) j α (y). This allows us to establish a product formula for the generalized Hankel function B λ κ , n on R . B λ κ , n is the kernel of the integral transform F κ , n arising from the Dunkl theory. Indeed we show that B λ κ , n (x) B λ κ , n (y) can be expressed as an integral in terms of B λ κ , n (z) with explicit kernel invoking Gegenbauer polynomials for all n ∈ N ∗ . The obtained result generalizes the product formulas proved by M. Rösler for Dunkl kernel when n = 1 and by S. Ben Said when n = 2. As application, we define and study a translation operator and a convolution structure associated to B λ κ , n . They share many important properties with their analogous in the classical Fourier theory. [ABSTRACT FROM AUTHOR]

In this paper, we are motivated to establish the generalization of a R q p ⁢ (α , β ; z) {{}_{p}R_{q}(\alpha,\beta;z)} function in terms of incomplete exponential functions and obtain some properties of an incomplete exponential R q p ⁢ (α , β ; z) {{}_{p}R_{q}(\alpha,\beta;z)} function. Further we give generating relations for incomplete exponential R q p ⁢ (α , β ; z) {{}_{p}R_{q}(\alpha,\beta;z)} function. Some applications related to ground water pumping (hydrology) modelling and probability theory are also discussed. [ABSTRACT FROM AUTHOR]

The aim of this manuscript is to evaluate the Boros integral with three parameters involving the incomplete I -functions and generalized multi-index Mittag-Leffler function. As a special case of our main result some interesting corollaries are given. [ABSTRACT FROM AUTHOR]

A derivation (of order 1) satisfies the reduction formula f (x k) = k x k - 1 f (x) for any integer k. In this article we find corresponding reduction formulas for derivations of higher order on commutative rings. More precisely, for every derivation f of order n and every positive integer k we find an explicit formula for f (x k) as a linear combination of x k - 1 f (x) , x k - 2 f (x 2) , ... , x k - n f (x n) . The proof hinges on the hypergeometric identity ∑ k ≥ 0 (- 1) k n k n + d + 1 - k n - j d + j - k j = n j for any positive integer n, nonnegative integer d, and integer j satisfying 0 ≤ j ≤ n . We prove this identity via the WZ-method. [ABSTRACT FROM AUTHOR]

Various extensions of the beta function together with their associated extended hypergeometric and confluent hypergeometric functions have been introduced and investigated. In this paper, using the very recently contrived extended beta function, we aim to introduce an extension F v p , q ; λ ; σ , τ u {{}_{u}F_{v}^{p,q;\lambda;\sigma,\tau}} of the generalized hypergeometric function F v u {{}_{u}F_{v}} and investigate certain classes of transforms and several identities of a generalized probability distribution involving this extension. In fact, we present some interesting formulas of Jacobi, Gegenbauer, pathway, Laplace, and Legendre transforms of this extension multiplied by a polynomial. We also introduce a generalized probability distribution to investigate its several related properties. Further, we consider some special cases of our main results with an argument about the derived process of a known result. [ABSTRACT FROM AUTHOR]

We utilize a combination of integral transforms, including the Laplace transform, with some classical results in analytic number theory concerning the Riemann ξ-function, to obtain a new integral equation. This integral equation is generalized to self-dual principal automorphic L-functions. We also provide a new proof of known functional-type identities from analytic number theory, and recast some criteria associated with the RH. An application of our integral equation to the Dirichlet problem in the half plane is stated, giving a new application of the Riemann ξ-function integral. [ABSTRACT FROM AUTHOR]

In this paper, using the Lipschitz–Hankel identities we obtain some new integral representations for the toroidal functions in terms of the elementary, Bessel, parabolic cylinder and hypergeometric functions. Manipulating the integrands of Lipschitz–Hankel identities with several integral representations lead us to present the results. In this sense, we also derive some integral representations for the products of toroidal functions. [ABSTRACT FROM AUTHOR]