Approximation Properties of Phillips Operators

In the spirit of Hasanov, Srivastava, and Turaev (2006), we introduce new inverse operators together with a more general operator and find a summation formula for the last one. Based on these operators and the earlier known q-analogues of the Burchnall-Chaundy operators, we find 15 symbolic operator formulas. Then, 10 expansions for the q-analogues of Srivastava’s three triple hypergeometric functions in terms of ϕ34q-hypergeometric and q-Kampé de Fériet functions are derived. These expansions readily reduce to 10 new expansions for the three triple Srivastava hypergeometric functions in terms of F34 hypergeometric and Kampé de Fériet functions.

We present the fermionic representation for the q-deformed hypergeometric functions related to Schur polynomials. We show that these multivariate hypergeometric functions are tau-functions of the KP hierarchy, and at the same time they are the ratios of Toda lattice tau-functions, considered by Takasaki, evaluated at certain values of higher Toda lattice times. The variables of the hypergeometric functions are related to the higher times of those hierarchies via a Miwa change of variables. The discrete Toda lattice variable shifts parameters of hypergeometric functions. Hypergeometric functions of type pΦs can also be viewed as a group 2-cocycle for the ΨDO on the circle (the group times are higher times of TL hierarchy and the arguments of a hypergeometric function). We obtain the determinant representation and the integral representation of a special type of KP tau-functions, these results generalize some of the results of Milne concerning multivariate hypergeometric functions. We write down a system of partial differential equations for these tau-functions (string equations).

For primes p ≡ 1 (mod 12), we present an explicit relation between the traces of Frobenius on a family of elliptic curves with j-invariant 1728 / t and values of a particular 2 F 1 -hypergeometric function over F p . We also give a formula for traces of Hecke operators on spaces of cusp forms of weight k and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's T -function in terms of hypergeometric functions.

Abstract. In this paper, the expansion of a basic confluent hypergeometric function in terms of an incomplete q-gamma function is investigated and shown to be an asymptotic expansion for a basic confluent hypergeometric function and a q-analogue of the Hadamard expansion. The general expansion associated with a basic hypergeometric function is introduced. In special cases, this provides alternative derivations of the q-Hadamard expansion. Expansions and asymptotics for modified q-Bessel functions are derived as special cases from the general expansion.

ABSTRACTGeneralized Abel type integral equations with Gauss, Kummer's and Humbert's confluent hypergeometric functions in the kernel and generalized Abel type integral equations with localized fractional integrals are considered. The left-hand sides of these equations are inversed by using generalized fractional derivatives. Explicit solutions of the equations in the class of locally summable functions are obtained. They are represented in terms of hypergeometric functions. Asymptotic power exponential type expansions of the generalized and localized fractional integrals are obtained. The base solutions of the generalized Abel type integral equation are given in the form of asymptotic series.

The Gauss hypergeometric function ${}_2F_1(a,b,c;z)$ can be computed by using the power series in powers of $z, z/(z-1), 1-z, 1/z, 1/(1-z),(z-1)/z$. With these expansions ${}_2F_1(a,b,c;z)$ is not completely computable for all complex values of $z$. As pointed out in Gil, {\it et al.} [2007, \S2.3], the points $z=e^{\pm i\pi/3}$ are always excluded from the domains of convergence of these expansions. B\"uhring [1987] has given a power series expansion that allows computation at and near these points. But, when $b-a$ is an integer, the coefficients of that expansion become indeterminate and its computation requires a nontrivial limiting process. Moreover, the convergence becomes slower and slower in that case. In this paper we obtain new expansions of the Gauss hypergeometric function in terms of rational functions of $z$ for which the points $z=e^{\pm i\pi/3}$ are well inside their domains of convergence . In addition, these expansion are well defined when $b-a$ is an integer and no limits are needed in that case. Numerical computations show that these expansions converge faster than B\"uhring's expansion for $z$ in the neighborhood of the points $e^{\pm i\pi/3}$, especially when $b-a$ is close to an integer number.

Inspired by recent developments in stereology, rotational versions of the Crofton formula are derived. The first version involves rotation averages of Minkowski functionals. It is shown that for the special case where the Minkowski functional is surface area, the rotation average can be expressed in terms of hypergeometric functions. The second rotational version of the Crofton formula solves the ‘opposite’ problem of finding functions with rotation averages equal to the Minkowski functionals. For the case of surface area, hypergeometric functions appear again. The second type of rotational Crofton formula has applications in local stereology. As a by-product, a formula involving mixed volumes is found.

We define a function in terms of quotients of the p-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the p-adic setting. We prove, for primes p >3, that the trace of Frobenius of any elliptic curve over Fp, whose j -invariant does not equal 0 or 1728, is just a special value of this function. This generalizes results of Fuselier and Lennon which evaluate the trace of Frobenius in terms of hypergeometric functions over Fp when p 1 .mod 12/. 1. Introduction and statement of results Let Fp denote the finite field with p, a prime, elements. Consider E=Q an elliptic curve with an integral model of discriminantA.E/. We denote Ep the reduction of E modulo p. We note that Ep is nonsingular, and hence an elliptic curve over Fp, if and only if p› A.E/, in which case we say p is a prime of good reduction. Regardless, we define

Letx 0 (t),x 0 ɛℝ4 be a homothetic solution of the planar three-body problem with total energyh, described in relative coordinates with respect to one body. It is shown that the variational equation of the problem atx 0 (t) can be solved explicitly in terms of hypergeometric functions. This is done by using the scaled true anomaly of the one-dimensional Kepler motion as the independent variable. The classical theorems about hypergeometric functions allow a simple calculation of all the values needed in applications. By means of this theory the past of a homothetic triple close encounter may be described in a linearized approximation.

Let $X_0^6(1)/W_6$ be the Atkin-Lehner quotient of the Shimura curve $X_0^6(1)$ associated to a maximal order in an indefinite quaternion algebra of discriminant $6$ over $\mathbb Q$. By realizing modular forms on $X_0^6(1)/W_6$ in two ways, one in terms of hypergeometric functions and the other in terms of Borcherds forms, and using Schofer's formula for values of Borcherds forms at CM-points, we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over $\overline\mathbb Q$ with complex multiplication.

The present paper is a further development, after the Great Proletarian Cultural Revolu-tion, of our statistical point of view that the element of homogeneous isotropic turbulenceis a small axially symmetrical vortex from the solution of the Navier-Stokes equations ofmotion. After long years of observation and analysis in practice, we recognize that thestructure of vortices, as elements of turbulence, does not obey the simple law of similarityor self-preservation. The structure undergoes a kind of stretching phenomena in the processof the decay of vortices. Therefore, we first bring out a vortex scale—vortex Reynoldsnumber relation, and then introduce a pseudo-similarity or psoudo-self-preservation conceptinto the theory. We solve the case of small vortex Reynolds number flows. The zerothorder approximation of the solution can be expressed in terms of confluent hypergeometricfunctions. The law of the decay of turbulent energy and the expression for the microscaleof turbulence from the initial to the final period of decay together with the double velocitycorrelation thus calculated all agree very well with the experiments of Batchelor and Town-send. We have also computed the energy spectrum transfer function corresponding to thetriple velocity correlation. But in this range of small Reynolds number flows there arestill no experimental measurements available for its verification. In the Appendix by using the method of multi-Fourier transform, we have derived theFourier transforms of the triple and n-tiple velocity correlations and the simpler form ofthe energy spectrum transfer function corresponding to the triple velocity correlation.

W. Wirtinger, in 1902, introduced an integral representation for the Gauss hypergeometric function in terms of theta functions. We give a generalization of this integral representation. More concretely, we study a definite integral of a power product of theta functions which has zeros at N-torsion points (N is a natural number greater than one) on a one-dimensional complex torus. We show that this integral gives a solution of a system of Fuchsian differential equations on the modular curve of level N and determine the spectral types of the system at its regular singularities.

Asymptotic non-null distribution of the likelihood ratio criterion for testing the linear hypothesis in multivariate analysis is obtained up to the order $N^{-2}$, where $N$ means the sample size, by using the characteristic function expressed in terms of hypergeometric function with matrix argument. This result holds without any assumption on the rank of non-centrality matrix. Asymptotic non-null distribution of the likelihood ratio criterion for independence between two sets of variates is also obtained up to the order $N^{-1}$.

One-dimensional effective mass Dirac equation is solved in asymmetric Hulthén potential. Scattering and bound state solutions are obtained in terms of hypergeometric functions. Transmission and reflection coefficients are calculated by matching conditions on the wave function. The necessary conditions for the existence of transmission resonances and supercriticality are derived.

We propose coherent (‘Schrödinger cat-like’) states adapted to the parity symmetry providing a remarkable variational description of the ground and first excited states of vibron models for finite (N)-size molecules. Vibron models undergo a quantum shape phase transition (from linear to bent) at a critical value ξc of a control parameter. These trial cat states reveal a sudden increase in vibration–rotation entanglement linear (L) and von Neumann (S) entropies from zero to (to be compared with L(N)max(ξ) = 1 − 1/(N + 1)) and , respectively, above the critical point, ξ > ξc, in agreement with exact numerical calculations. We also compute inverse participation ratios, for which these cat states capture a sudden delocalization of the ground-state wave packet across the critical point. Analytic expressions for entanglement entropies and inverse participation ratios of variational states, as functions of N and ξ, are given in terms of hypergeometric functions.