Narrow Results

We present the spectral and scattering theory of the Casimir operator acting on radial functions in $L^2(\mathrm{\text{SL}}(2,ℝ))$. After a suitable decomposition, these investigations consist in studying a family of differential operators acting on the half-line. For these operators, explicit expressions can be found for the resolvent, for the spectral density, and for the Møller wave operators, in terms of the Gauss hypergeometric function. An index theorem is also introduced and discussed. The resulting equality, generically called Levinson’s theorem, links various asymptotic behaviors of the hypergeometric function. This work is a first attempt to connect group theory, special functions, scattering theory, $C^{\ast }$-algebras, and Levinson’s theorem.

The Kappa function is introduced as the function κ satisfying J(κ(τ)) = λ(τ), where J and λ are the elliptic modular functions. A Fourier expansion of κ is studied.

The hypergeometric equations with polyhedral monodromy groups derive 3-integral-parameter families of polynomials.

The real Lie group G=SU(2,2) has a standard maximal parabolic subgroup P_J corresponding to the long simple roots in its restricted root system. We have an explicit formula of the holonomic system of the radial part of the matrix coefficients with corner K-type of the generalized principal series representations of G obtained by parabolic induction with respect to P_J from the discrete series representations of M_J. It is equivalent to the modified F₂ system of Appell.

In this paper, we determine all irreducible spherical functions Φ of any K-type associated to the dual Hermitian symmetric pairs (G, K) = (SU(3), U(2)) and (SU(2,1), U(2)). This is accomplished by associating to Φ a vector valued function H = H(u) of a real variable u, analytic at u = 0, which is a simultaneous eigenfunction of two second order differential operators with matrix coefficients. One of them comes from the Casimir operator of G and we prove that it is conjugated to a hypergeometric operator, allowing us to express the function H in terms of a matrix valued hypergeometric function. For the compact pair (SU(3), U(2)), this project was started in [4].

The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in [4]. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao's equation in order to get a generalized matrix valued hypergeometric equation. We take advantage of the tools and strategies developed in [4] to identify the corresponding matrix hypergeometric functions _nF_m. We prove that, if n = m + 1, these functions are analytic for |z| < 1 and we give a necessary condition for the convergence on the unit circle |z| = 1.

We compute generalized Bernstein–Reznikov integrals associated with standard complex symplectic forms by studying Knapp–Stein intertwining operators between spherical degenerate principal series of $\mathrm{\text{Sp}}(n,ℂ)$.

In this paper, the extended double shuffle relations for interpolated multiple zeta values (MZVs) are established. As an application, Hoffman’s relations for interpolated MZVs are proved. Furthermore, a generating function for sums of interpolated MZVs of fixed weight, depth and height is represented by hypergeometric functions, and we discuss some special cases.

It has been shown earlier [D. Bazeia and A. K. Das, Phys. Lett. B 715, 256 (2012)] that the solubility of the Legendre and the associated Legendre equations can be understood as a consequence of an underlying supersymmetry and shape invariance. We have extended this result to the hypergeometric equation. Since the hypergeometric equation as well as the hypergeometric function reduce to various orthogonal polynomials, this study shows that the solubility of all such systems can also be understood as a consequence of an underlying supersymmetry and shape invariance. Our analysis leads naturally to closed form expressions (Rodrigues' formula) for the orthogonal polynomials.

By using the plane-wave expansion for the electromagnetic-field vector potential, relativistic bound–bound, bound–unbound and unbound–unbound transition matrix elements for hydrogenic atoms are expressed universally in terms of hypergeometric functions. By applying the obtained formulas, these transition matrix elements can be evaluated analytically and numerically with arbitrarily high precision. The newfound representation for the matrix elements is very convenient for direct numerical evaluation of the Lamb shift because of its universality, conciseness and reliance on functions already built in the standard computational packages. All of this is highly favorable for programming of computationally efficient algorithms.

In this paper, some new fractal versions of Fejér–Hermite–Hadamard (FHH) type variants for generalized Raina $(\mathrm{\Psi },\hslash )$-convex mappings are established benefiting from Raina’s function and fractal set ${ℝ}^{\phi },0<\phi <1$. By means of three integral identities coupled with Raina’s function and local differentiation, we established some bounds for the difference between the left and central parts and also the difference between the center and right parts in FHH inequality. Besides that, some illustrative examples and noted special cases are apprehended. Additionally, we developed various generalizations for random variables, cumulative distribution functions, and special function theory as applications of local fractional integrals. The consequences established can provide contribution to inequality theory, fractional calculus and probability theory from the viewpoint of application to establish the other associated classes of functions. With the aid of these methodologies, it is promising to comprise further bounds of other type of variants which involve local fractional techniques.

In this paper, we are concerned with the local existence and singularity structures of low regularity solution to the semilinear generalized Tricomi equation with typical discontinuous initial data (u(0, x), ∂_tu(0, x)) = (0, φ(x)), where m ∈ ℕ, x = (x₁,…,x_n), n ≥ 2, and f(t, x, u) is C^∞ smooth on its arguments. When the initial data φ(x) is homogeneous of degree zero or piecewise smooth along the hyperplane {t = x₁ = 0}, it is shown that the local solution u(t, x) ∈ L^∞([0, T] × ℝⁿ) exists and is C^∞ away from the forward cuspidal conic surface or the cuspidal wedge-shaped surfaces respectively. On the other hand, for n = 2 and piecewise smooth initial data φ(x) along the two straight lines {t = x₁ = 0} and {t = x₂ = 0}, we establish the local existence of a solution and further show that in general due to the degenerate character of the equation under study, where . This is an essential difference to the well-known result for solution to the two-dimensional semilinear wave equation with (v(0, x), ∂_tv(0, x)) = (0, φ(x)), where Σ₀ = {t = |x|}, and .

Under a certain condition, we find the explicit formulas for the trace functions of certain intertwining operators among gl(n)-modules, introduced by Etingof in connection with the solutions of the Calogero–Sutherland model. If n = 2, the master function of the trace function is exactly the classical Gauss hypergeometric function. When n > 2, the master functions of the trace functions are a new family of multiple hypergeometric functions, whose differential property and Euler type integral representation are dominated by certain polynomials of integral paths connecting pairs of positive integers. Moreover, we also find the trace functions for sp(2n) explicitly and prove that they give rise to solutions of the Olshanesky–Perelomov model of type C. The master functions of the trace functions for sp(2n) are similar new multiple path hypergeometric functions. Analogous multiple path hypergeometric functions for orthogonal Lie algebras are defined and studied.

In this paper, we obtain an asymptotic expansion for the Gauss hypergeometric function ₂F₁(a, b - λ; c + λ; -z), as |λ| → ∞. The expansion holds for fixed values of a, b, and c, and is uniformly valid for z in the domain |ph z| < π.

In this paper, we obtain an asymptotic expansion for the Gauss hypergeometric function ₂F₁(a + λ, b + 2λ; c; -z), as |λ| → ∞. The expansion holds for fixed values of a, b, and c, and is uniformly valid for z in the domain |ph z| < π.

In this paper, we discuss asymptotic expansions for the Gauss hypergeometric function ₂F₁(a + e₁λ, b + e₂λ; c + e₃λ; z), where e_j = 0, ±1, as |λ| → ∞. We complete the results of two previous publications, extend the sectors of validity, and give more details on the computation of the coefficients.

Bumby proved that the only positive integer solutions to the quartic Diophantine equation 3X⁴ - 2Y² = 1 are (X, Y) = (1, 1),(3, 11). In this paper, we use Thue's hypergeometric method to prove that, for each integer m ≥ 1, the only positive integers solutions to the Diophantine equation (m² + m + 1)X⁴ - (m² + m)Y² = 1 are (X,Y) = (1, 1),(2m + 1, 4m² + 4m + 3).

We study overconvergence phenomena for 𝔽-linear functions on a function field over a finite field 𝔽. In particular, an analog of the Dwork exponential is introduced.

We determine the exceptional sets of hypergeometric functions corresponding to the (2, 4, 6) triangle group by relating them to values of certain quaternionic modular forms at CM points. We prove a result on the number fields generated by exceptional values, and by using modular polynomials we explicitly compute some examples.

The zeta Mahler measure is the generating function of higher Mahler measures. In this article, explicit formulas of higher Mahler measures, and relations between higher Mahler measures and multiple zeta (star) values are showed by observing connections between zeta Mahler measures and the generating functions of multiple zeta (star) values. Additionally, connections between higher Mahler measures and Dirichlet L-values associated with primitive quadratic characters are discussed.