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We study the spectral bounds of self-adjoint operators on the Hilbert space of square-integrable functions, arising from the representation theory of the Heisenberg group. Interestingly, starting either with the von Neumann lattice or the hexagonal lattice of density 2, the spectral bounds obey well-known arithmetic–geometric mean iterations. This follows from connections to Jacobi theta functions and Ramanujan's corresponding theories. As a consequence, we rediscover that these operators resemble the identity operator as the density of the lattice grows. We also prove that the conjectural value of Landau's constant is obtained as half the cubic arithmetic–geometric mean of 2 3 and 1, which we believe to be a new result. [ABSTRACT FROM AUTHOR]

In this paper, first we present two new integral representations of the Gauss hypergeometric function in the form of double integrals obtained recently by us Chammam et al. (Bull Belgian Math Soc 31(3), 336–340, 2024). Two specific cases of our representations allow for an alternative proofs of the well-known Gauss summation theorem and Watson summation theorem. Additionally, as an application, we introduce a new class of double integrals and provide interesting integral representations for Catalan–Qi numbers and Fuss–Catalan–Qi numbers. [ABSTRACT FROM AUTHOR]

In this paper, we have developed novel proofs of two known formulas for the Meijer integral transform and obtained non-trivial extensions of these formulas (in one of them, the Bessel function is replaced by the Coulomb wave function). Also we have derived a new formula for a series involving product of Whittaker functions and modified Bessel functions of the second kind. Our proofs are based on a group theoretical approach and involves the use of an intertwining operator for two representations of one matrix group. [ABSTRACT FROM AUTHOR]

In this article, we aim to present a new extension of the beta-logarithmic function with the help of an extended beta function and logarithmic mean. Then we introduce its statistical properties and investigate some basic important formulas of extended beta-logarithmic functions, such as integral representation, summation formula and Mellin transform. Further, we consider a new type of hypergeometric and confluent hypergeometric functions using this extended beta-logarithmic function and discuss their various result. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a finite field analog for a transformation satisfied by 2 F 1 -classical hypergeometric series. We use properties of character sums over finite fields to deduce these results. Using these, we further evaluate certain special values for hypergeometric series over finite fields. [ABSTRACT FROM AUTHOR]

For r ∈ (0 , 1) , let E (r) be the complete elliptic integral of the second kind. In this paper, the authors obtain a accuracy approximation for E (r) , which improve the related inequalities satisfied by E (r) . [ABSTRACT FROM AUTHOR]

Properties of the Lommel polynomials $ R_{n,\nu }(z) $ R n , ν (z) which appear in the theory of the Bessel functions, are studied. New functional relations for hypergeometric and Horn functions, including reduction formulas, are derived. [ABSTRACT FROM AUTHOR]

In this paper, we express a generalization of the Ramanujan integral I ⁢ (α) with the analytical solutions, using the Laplace transform technique and some algebraic relation or the Pochhammer symbol. Moreover, we evaluate some consequences of a generalized definite integral ϕ * ⁢ (υ , β , a) . The well-known special cases appeared, whose solutions are possible by Cauchy's residue theorem, and many known applications of the integral I ⁢ (a , β , υ) are discussed by the Leibniz rule for differentiation under the sign of integration. Further, one closed-form evaluation of the infinite series of the F 0 1 ⁢ (⋅) function is deduced. In addition, we establish some integral expressions in terms of the Euler numbers, which are not available in the tables of the book of Gradshteyn and Ryzhik. [ABSTRACT FROM AUTHOR]

Overdetermined boundary value problems and the minimal operators generated by them are extremely important in the description of regular boundary value problems for differential equations, and are also widely used in the study of local properties of solutions. In addition, for inverse problems of mathematical physics arising from applications, when determining unknown data, it is necessary to study problems with overdetermined boundary conditions, which is reflected in the study of problems, including for hyperbolic equations and systems, arising in physics, geophysics, seismic tomography, geoelectrics, electrodynamics, medicine, ecology, economics and many other practical areas. Thus, the study of overdetermined boundary value problems is of both theoretical and applied interest. In this paper, a criterion for the regular solvability of the overdetermined Cauchy problem for the Gellerstedt equation and the minimal differential operator generated by it in a hyperbolic domain is established, as which both the case of a characteristic triangle and the case of a more general domain with fairly general assumptions about the boundary of the domain are considered. Due to overdetermined boundary conditions, the problem under consideration will be ill-posed in the general case, therefore, for its regular solvability, additional conditions must be imposed on the initial data. In other words, we have considered the inverse problem: to determine what requirements the initial data of the problem, in particular the right part of the Gellerstedt equation, should meet, in question, so that the overdetermined Cauchy problem is regularly solvable. The proof is based on the Gellerstedt potential, the properties of solutions of the Goursat problem in the characteristic triangle, and the properties of special functions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with determining the explicit solution of a Sea-Breeze flow model by selecting a special viscosity function. Firstly, we examine the exact solution when the viscosity function is related to a nonnegative constant coefficient. Further, by employing suitable transformations and forcing terms, we transform the original second order differential equation corresponding to the Sea-Breeze flow model into the Bessel equation and derive the corresponding exact solution. Finally, we determine the exact solution when the viscosity function is related to a nonnegative quadratic function. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the incomplete beta matrix function $ \mathbb {B}^{z}(M,K),\;z\in \mathbb {C} $ B z (M , K) , z ∈ C , for parameter matrices M and K in $ \mathbb {C}^{{m}\times {m}} $ C m × m using the neutrix limit and evaluation methods on their the partial derivatives. Also, we define a new extension of the Pochhammer matrix symbol in terms of the incomplete beta matrix functions. Furthermore, we then apply this extended matrix symbol for the incomplete first Appell hypergeometric matrix functions and obtain some of their properties. [ABSTRACT FROM AUTHOR]

The purpose of this article is to find summation, transformation and reduction formulas for Lauricella functions of n variables and for their special cases Appell functions ( n = 2 ) and 2 F 1 functions. The method of proof is always to start with the Eulerian integral for the Lauricella function or its special cases, and then make an integral substitution, which maps (0, 1) to itself, and finally rewrite the new integral in terms of another Lauricella function or its special cases. In this way, quite interesting formulas are obtained, and the number of variables in the new Lauricella function can change. Some of the formulas generalize formulas for Appell functions, and some formulas are of general type, which are then specialized to concrete formulas. The important object reduced roots for a polynomial f (τ) is introduced in order to factorize f (τ) . When the arguments of the double functions are > 1 , the results are only formal. Often these formulas are specialized to new summation formulas for Gaussian hypergeometric functions. Several of the new formulas have been checked by Mathematica. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Atkin-like polynomials that appeared in the study of normalized extremal quasimodular forms of depth 1 on S L 2 (Z) by Kaneko and Koike as orthogonal polynomials and clarify their properties. Using them, we show that the normalized extremal quasimodular forms have a certain expression by the linear functional corresponding to the Atkin inner product and prove an unexpected connection between generalized Faber polynomials, which are closely related to certain bases of the vector space of weakly holomorphic modular forms, and normalized extremal quasimodular forms. In particular, we reveal that the orthogonal polynomial expansion coefficients of the generalized Faber polynomials by the Atkin-like polynomials appear in the Fourier coefficients of normalized extremal quasimodular forms multiplied by certain (weakly) holomorphic modular forms. [ABSTRACT FROM AUTHOR]

For a , b , c , d , e ∈ Q × , denote by C a , b , c , d , e the nonsingular projective curve over Q given by the affine equation C a , b , c , d , e : a x 2 + b y 2 = c + d x y + e x 2 y 2. In this paper, we find a relation between the number of points on C a , b , c , d , e over F q and the finite field Appell series F 4 ∗ . As a consequence, in view of certain transformations for finite field Appell series F 4 ∗ , we prove the existence of isogenies between C a , b , c , d , ab c and the twisted Edward family of elliptic curves E α , 16 a b α d 2 . [ABSTRACT FROM AUTHOR]

Ramanujan in his notebook recorded two modular equations involving multipliers with moduli of degrees (1,7) and (1,23). In this paper, we find some new Ramanujan-type modular equations involving multipliers with moduli of degrees (3,5) and (1,15), and give concise proofs by employing Ramanujan's multiplier functional equation. [ABSTRACT FROM AUTHOR]

This work explores the possibility of developing the analog of some classic results from elliptic PDEs for a class of fractional ODEs involving the composition of both left- and right-sided Riemann-Liouville (R-L) fractional derivatives, D a + α D b - β , 1 < α + β < 2 . Compared to one-sided non-local R-L derivatives, these composite operators are completely non-local, which means that the evaluation of D a + α D b - β u (x) at a point x will have to retrieve the information not only to the left of x all the way to the left boundary but also to the right up to the right boundary, simultaneously. Therefore, only limited tools can be applied to such a situation, which is the most challenging part of the work. To overcome this, we do the analysis from a non-traditional perspective and eventually establish elliptic-type results, including Hopf's Lemma and maximum principles. As α → 1 - or α , β → 1 - , those operators reduce to the one-sided fractional diffusion operator and the classic diffusion operator, respectively. For these reasons, we still refer to them as "elliptic diffusion operators", however, without any physical interpretation. [ABSTRACT FROM AUTHOR]

We exhibit an infinite family of discrete subgroups of Sp 4 (R) which have a number of remarkable properties. Our results are established by showing that each group plays ping-pong on an appropriate set of cones. The groups arise as the monodromy of hypergeometric differential equations with parameters N - 3 2 N , N - 1 2 N , N + 1 2 N , N + 3 2 N at infinity and maximal unipotent monodromy at zero, for any integer N ≥ 4 . Additionally, we relate the cones used for ping-pong in R 4 with crooked surfaces, which we then use to exhibit domains of discontinuity for the monodromy groups in the Lagrangian Grassmannian. These domains of discontinuity lead to uniformizations of variations of Hodge structure with Hodge numbers (1, 1, 1, 1). [ABSTRACT FROM AUTHOR]

In this manuscript, we present a new definition of $ (k,\tau) $ (k , τ) -Gauss hypergeometric matrix function and study its analytical properties, like derivative formulas and integral representations. Furthermore, as an application we establish $ {\rm k} $ k -fractional calculus operators for the novel matrix function. We also give some new and known results as special cases of our proposed generalization of the Wright hypergeometric matrix function. [ABSTRACT FROM AUTHOR]

The Mellin transforms of some products of two shifted Airy functions are evaluated in a closed form. The relation between the Mellin transforms of a single Airy function and the square of the function is obtained. The shifted Airy function moments of integer and half-integer order are expressed in terms of Gegenbauer polynomials. [ABSTRACT FROM AUTHOR]

The Fröbenius method of power series has been applied to several linear random differential equations. The interest relies on the derivation of a closed-form mean-square solution and on the possibility of approximating statistical measures at exponential convergence rate. In this paper, we deal with the hypergeometric differential equation with random coefficients and initial conditions. On the interval (0, 1), random power series centered at the regular singular point 0 are employed, which are given in terms of the hypergeometric function. We find the stochastic basis of mean-square solutions and solve random initial-value problems. The approximation of the expectation and the variance is studied and illustrated computationally. [ABSTRACT FROM AUTHOR]

In this work, we propose some unified integral formulas for the Mittag-Leffler confluent hypergeometric function (MLCHF), and our findings are assessed in terms of generalized special functions. Additionally, certain unique cases of confluent hypergeometric function have been corollarily presented. [ABSTRACT FROM AUTHOR]

Many authors investigated the characteristics of the Bell, Euler, Bernoulli, and Genocchi polynomials because of their numerous uses in statistics, number theory, and other branches of science. A generating function for mixed-type Apostol–Euler polynomials of order η related with Bell polynomials is presented in this study. We also construct essential identities of Apostol–Euler polynomials of order η connected with Bell polynomials, such as the correlation formula, the implicit summation formula, the derivative formula, some association with Stirling numbers, and specific cases. Also we establish certain symmetric identities and their known consequences. In addition, we investigate several interesting relationships related to umbral calculus. [ABSTRACT FROM AUTHOR]

We investigate subclasses of generalized Bernstein functions related to complete Bernstein and Thorin–Bernstein functions. Representations in terms of incomplete beta and gamma as well as hypergeometric functions are presented. Several special cases and examples are discussed. [ABSTRACT FROM AUTHOR]

The Gudermannian function relates the circular angle to the hyperbolic one when their cosines are reciprocal. Whereas both such angles are halved areas of circular and hyperbolic sectors, it is natural to develop similar considerations within the study of a class of curves images of maps with constant areal speed. After a brief exposition of some use of the Gudermannian in applied sciences, we proceed to illustrate the class of curves, called Keplerian curves, which can be parametrised by a map m = (cos m , sin m) whose areal speed is 1. In the next Sections, after a detailed study of p-circular and hyperbolic Fermat curves F p and F p ∗ , we define the p-Gudermannian as the primitive of the derivative of the p-hyperbolic sine divided by the square of the p-hyperbolic cosine: all the analogues of the classical identities are proven. Having realised that such curves correspond to each other by means a homology, we extend our study to a wide class of Keplerian curves and their homologues; once again, defined the Gudermannian in an identical manner, all the analogues of classical identities subsist. Below, three examples are detailed. The last paragraph further extends this consideration, eliminating the hypothesis that the curves are parametrised by maps with areal speed 1. The Appendix illustrates integrating techniques for systems defining the Fermat curves and determining the inverse of their tangent function. [ABSTRACT FROM AUTHOR]

In this research article, we provide an extended Hurwitz–Lerch zeta function using the extended beta function. We have also investigated the new generalized Hurwitz–Lerch zeta function's exciting characteristics, such as integral representations, differential equations, the Mellin transform and special generating relations. In order to support the main result, we have additionally considered several notable specific cases. Furthermore, we have discussed the solution of the fractional kinetic equation using the new extended Hurwitz–Lerch zeta function. [ABSTRACT FROM AUTHOR]

We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental properties and prove summation formulas, transformation formulas and product formulas. An application to zeta functions of K3-surfaces is given. In the appendix, we give an elementary proof of the Davenport–Hasse multiplication formula for Gauss sums. [ABSTRACT FROM AUTHOR]

The special matrix functions have received significant attention in many fields, such as theoretical physics, number theory, probability theory, and the theory of group representations. In 2017, Dwivedi and Sahai introduced the generalized hypergeometric matrix function using matrix parameters and proved the convergence on | z | = 1 . Recently, hypergeometric matrix functions and their potential applications have played a major role in mathematical physics and engineering. Motivated by aforesaid works and in order to enrich this flourishing field, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and determine various transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have been provided for several particular cases. [ABSTRACT FROM AUTHOR]

One may consider the generalization of Jacobi polynomials and the Jacobi function of the second kind to a general function where the degree is allowed to be a complex number instead of a non-negative integer. These functions are referred to as Jacobi functions. In a similar fashion as associated Legendre functions, these break into two categories, functions which are analytically continued from the real line segment (− 1 , 1) and those analytically continued from the real ray (1 , ∞). Using properties of Gauss hypergeometric functions, we derive multi-derivative and multi-integral representations for the Jacobi functions of the first and second kind. [ABSTRACT FROM AUTHOR]

This paper deals with the monotonicity and concavity properties of certain functions involving the Gaussian hypergeometric function. With these results, we not only obtain sharp bounds for the ratio of hypergeometric functions which extend recently discovered inequalities for k-balanced hypergeometric functions, and but also give an affirmative answer to an open problem proposed by Qiu and Vuorinen. In addition, as by-products, some monotonicity theorems for complete p-elliptic integrals and inequalities for generalized Grötzsch ring function are established. [ABSTRACT FROM AUTHOR]

In this paper, a product formula for the one-dimensional (k , a) -generalized Fourier kernel is given for k ≥ 0 , a>0 and 2 k > 1 − a 2 , extending the special case of [Boubatra MA, Negzaoui S, Sifi M. A new product formula involving Bessel functions. Integral Transforms Spec Funct. 2022;33:247–263.] when a = 2 n , n ∈ N ∗ . [ABSTRACT FROM AUTHOR]

The multivariable autoregressive filter problem asks for a polynomial p (z) = p (z 1 , ... , z d) without roots in the closed d-disk based on prescribed Fourier coefficients of its spectral density function 1 / | p (z) | 2 . The conditions derived in this paper for the construction of a degree one symmetric polynomial reveal a major divide between the case of at most two variables vs. the the case of three or more variables. The latter involves multivariable elliptic functions, while the former (due to [Geronimo and Woerdeman (Ann Math 160(3):839-906, 2004)]) only involve polynomials. The three variable case is treated with more detail, and entails hypergeometric functions. Along the way, we identify a seemingly new relation between 2 F 1 1 3 , 2 3 1 ; z and 2 F 1 1 2 , 1 2 1 ; z ~ . [ABSTRACT FROM AUTHOR]

In this note we will generalize the results deduced in Figalli and Glaudo (Arch Ration Mech Anal 237(1):201–258, 2020) and Deng et al. (Sharp quantitative estimates of Struwe's Decomposition. Preprint http://arxiv.org/abs/2103.15360, 2021) to fractional Sobolev spaces. In particular we will show that for s ∈ (0 , 1) , n > 2 s and ν ∈ N there exists constants δ = δ (n , s , ν) > 0 and C = C (n , s , ν) > 0 such that for any function u ∈ H ˙ s (R n) satisfying, u - ∑ i = 1 ν U ~ i H ˙ s ≤ δ where U ~ 1 , U ~ 2 , ... U ~ ν is a δ -interacting family of Talenti bubbles, there exists a family of Talenti bubbles U 1 , U 2 , ... U ν such that u - ∑ i = 1 ν U i H ˙ s ≤ C Γ if 2 s < n < 6 s , Γ | log Γ | 1 2 if n = 6 s , Γ p 2 if n > 6 s for Γ = Δ u + u | u | p - 1 H - s and p = 2 ∗ - 1 = n + 2 s n - 2 s. [ABSTRACT FROM AUTHOR]

Many authors defined and extended the beta function in various forms because the beta function has wide uses in different fields of science and applied science. In this article, we define a new more generalized form of the extended beta matrix function via the Wiman matrix function and describe their significant properties and special cases. Furthermore, we define an extension of the Gauss hypergeometric and confluent hypergeometric matrix functions by adopting a novel type of beta matrix function. We also derive their Laplace transform, derivative formula and transformation formulae. [ABSTRACT FROM AUTHOR]

A model of the Bannai–Ito algebra in a superspace is introduced. It is obtained from the threefold tensor product of the basic realization of the Lie superalgebra osp (1 | 2) in terms of operators in one continuous and one Grassmanian variable. The basis vectors of the resulting Bannai–Ito algebra module involve Jacobi polynomials. [ABSTRACT FROM AUTHOR]

In this article, we obtain an explicit general solution of the Schrödinger equation with the squared secant potential ν (ν + 1) sec 2 x in terms of elementary functions for non-negative integer value of ν = n . Alternatively, we provide a general solution in terms of the Gauss hypergeometric function for any parameter value ν . Then, we derive some hypergeometric identities by comparing these two sets of solutions when ν is a non-negative integer. With the help of these hypergeometric formulas, we derive several existing explicit representations as well as new ones for some special functions and orthogonal polynomials including the Legendre, Ferrers, and the Bessel functions, the Jacobi and the Lommel polynomials containing specific parameters. [ABSTRACT FROM AUTHOR]

We propose a notion of stability of the Walsh–Hadamard spectrum of Boolean functions when inputs are independent and identically distributed Bernoulli random variables. We study the stability spectrum of bent Boolean functions and obtain a bound for it. We also derive the formula for the stability transform of Maiorana–McFarland type bent functions. We analyze the stability spectrum of symmetric Boolean functions and characterize it for symmetric bent Boolean functions and symmetric Boolean functions in an odd number of variables with maximum nonlinearity. We show that the stability spectrum is not, in general, invariant under extended affine transformations. Further, we display some non-bent symmetric Boolean functions whose stability spectra are flatter than that of symmetric bent Boolean functions. [ABSTRACT FROM AUTHOR]

The aim of this article is to define the bicomplex hypergeometric function. We obtain some useful results from the theory of special functions over a relatively large domain. We discuss the bicomplex holomorphicity of the bicomplex hypergeometric function and establish some significant relations based on it along with some bicomplex geometric representations of conditions of existence. [ABSTRACT FROM AUTHOR]

In this article, we use a particular case of convolution as an operator to discuss a number of problems concerning multiplier results between function spaces such as Hardy and B p -spaces. As a consequence, we extend certain well-known results on fractional derivatives and fractional integrals. Also, we find condition on the parameters b , c such that 풫 b , c in B p . [ABSTRACT FROM AUTHOR]

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.