Appell Function Research Articles

Special Lie—Mei Symmetry and Conserved Quantities of Appell Equations Expressed by Appell Function

Special Lie—Mei symmetry and conserved quantities for Appell equations expressed by Appell functions in a holonomic mechanical system are investigated. On the basis of the Appell equation in a holonomic system, the definition and the criterion of special Lie—Mei symmetry of Appell equations expressed by Appell functions are given. The expressions of the determining equation of special Lie—Mei symmetry of Appell equations expressed by Appell functions, Hojman conserved quantity and Mei conserved quantity deduced from special Lie—Mei symmetry in a holonomic mechanical system are gained. An example is given to illustrate the application of the results.

A type of new conserved quantity deduced from Mei symmetry for Appell equations in a holonomic system with unilateral constraints

A type of new conserved quantity deduced from Mei symmetry of Appell equations for a holonomic system with unilateral constraints is investigated. The expressions of new structural equation and new conserved quantity deduced from Mei symmetry of Appell equations for a holonomic system with unilateral constraints expressed by Appell functions are obtained. An example is given to illustrate the application of the results.

Hypergeometric transformations relating Gauss, Appell and Srivastava–Daoust functions

The object of the present paper is to obtain hypergeometric transformation formulas involving Appell's double hypergeometric function of the fourth kind F 4 and Srivastava–Daoust hypergeometric function of three variables, using Laplace-type double integral technique. Watson's summation theorem for Clausenian function 3 F 2 having unit argument is deduced as a special case. A hypergeometric transformation formula for Gauss function 2 F 1 [H. Exton, A Note on a hypergeometric transformation, Bull. Calcutta Math. Soc. 74(1979), pp. 337–340] is also corrected here.

EXACT ANALYTICAL EXPRESSIONS AND NUMERICAL VALUES OF SLATER'S AND MARVIN'S RADIAL INTEGRALS OF THE TYPE $R_k^{(\mu, \nu)} (11, 21; 12, 22)$, $F_k^{(\mu, \nu)} (1, 2)$ AND $G_k^{(\mu, \nu)} (1; 2)$ WITH THE KERNEL $r_<^{k+\mu}/r_>^{k+\nu}$ (AS k ≥ 0, μ ≥ 0 IS AN EVEN AND ν - μ = 1, 3, 5, …) FOR ARBITRARY nl-STATES OF H-LIKE ATOMS AND

This article deals with the connection between a.m. the hydrogenic radial integrals in the theory of atomic spectra and new properties of Appell's function F2(x, y) to the vicinity of the singular point (1, 1) and to the domain |x|+|y| < 1. In Tables 1–7 exact numerical values of these integrals are first given, i.e., actually for all Mendeleev's table. (Author has all values these of integrals for arbitrary nl-states of H-like atoms, k ≥ 0 and Z ≤ 137).

NEW PROPERTIES OF THE P. E. APPELL HYPERGEOMETRIC SERIES F2(α;β, β′;γ, γ′;x, y) TO THE VICINITY OF THE SINGULAR POINT (1, 1) AND NEAR THE BOUNDARY OF ITS DOMAIN OF CONVERGENCE D2:|x|+|y|<1

Exact analytical representations for Appell's series F2(x, y) to the vicinity of the singular point (1, 1) and near the boundary of its domain of convergence are given. It is shown that Appell's functions F2(1, 1) and F3(1, 1) have the property of mirror-like symmetry with respect to the center j0 ↦ -(1/2) under the change j ↦ -j-1, z∈ℤ, and they correlate between each other.

Anharmonic Electron−Phonon Coupling in Condensed Media: 1. Formalism

Three different schemes for calculating anharmonic line shape functions are reported and discussed for the first time in this article using eigenstate representation. First, the linear dipole-moment time correlation function (DMTCF), homogeneous (single-site) absorption line shape function, and the respective Franck-Condon factors (FCF) are derived and explored as a molecule makes a transition from a harmonic to an anharmonic (Morse potential) electronic state. Second, the linear DMTCF, homogeneous absorption line shape function, and FCFs are also derived as a molecule makes a transition from one anharmonic to another linearly displaced anharmonic state; FCFs in this case are reported in an exact closed-form expressed in terms of Appell's hypergeometric function. Third, same as the latter set of results are reported but with both linearly displaced and distorted shape of the upper Morse potential. FCFs of the zero-phonon line in all three cases are reported. The first case is rather mathematically complex as a result of taking the overlap integral of the Morse oscillator eigenfunctions, whose spatial decay is a simple exponential, with those of harmonic oscillator, whose decay is a Gaussian. This form of a functional disparity gives rise to some challenges. Model calculations are presented and discussed.

Solution of Abel-type integral equation involving the Appell hypergeometric function

The present paper gives a formal solution of a certain Abel-type integral equation involving the Appell hypergeometric function in the kernel. The integral equation and its solution give rise to new forms of generalized fractional calculus operators (viz. the generalized fractional integrals and generalized fractional derivatives). These and their various consequences are also mentioned. The concluding remarks briefly point out possibilities of further work concerning the operators studied in this paper.

Fractional calculus of the $ bar H $-function

The subject of this paper is to derive a fractional calculus formula for $ bar H $-function due to Inayat-Hussain whose based upon generalized fractional integration and differentiation operators of arbitrary complex order involving Appell function $F_3$ due to Saigo & Meada. The results are obtained in a compact form containing the Reimann-Liouville, Eredlyi-Kober and Saigo operators of fractional calculus.

Analytic result for the one-loop scalar pentagon integral with massless propagators

The method of dimensional recurrences proposed by Tarasov (1996, 2000) [1,2] is applied to the evaluation of the pentagon-type scalar integral with on-shell external legs and massless internal lines. For the first time, an analytic result valid for arbitrary space–time dimension d and five arbitrary kinematic variables is presented. An explicit expression in terms of the Appell hypergeometric function F3 and the Gauss hypergeometric function F12, both admitting one-fold integral representations, is given. In the case when one kinematic variable vanishes, the integral reduces to a combination of Gauss hypergeometric functions F12. For the case when one scalar invariant is large compared to the others, the asymptotic values of the integral in terms of Gauss hypergeometric functions F12 are presented in d=2−2ε, 4−2ε, and 6−2ε dimensions. For multi-Regge kinematics, the asymptotic value of the integral in d=4−2ε dimensions is given in terms of the Appell function F3 and the Gauss hypergeometric function F12.

A transformation formula for Appell's hypergeometric function $F_1$ and common limits of triple sequences by mean iterations

In this paper, we give a transformation formula for Appell's hypergeometric function $F_1$. As applications of this formula, we show that some common limits of triple sequences given by mean iterations of 3-terms can be expressed by $F_1$.

Rank-Crank type PDE's for higher level Appell functions

In this paper we consider level l Appell functions, and find a partial differential equation for all odd l. For l=3 this recovers the Rank-Crank PDE, found by Atkin and Garvan, and for l=5 we get a similar PDE found by Garvan.

A new type of conserved quantity induced by Mei symmetry of Appell equation

A New type of conserved quantity induced by Mei symmetry of Appell equations for a holonomic system is studied. The definition and the criterion of Mei symmetry of Appell equations for a holonomic system are given. A new type of conserved quantity induced by Mei symmetry of Appell equations expressed by Appell functions is obtained. And an example is given to illustrate the application of the results.

A variant of Jacobi type formula for Picard curves

The classical Jacobi formula for the elliptic integrals (Gesammelte Werke I, p. 235) shows a relation between Jacobi theta constants and periods of ellptic curves E ( λ ) : w 2 = z ( z - 1 ) ( z - λ ) . In other words, this formula says that the modular form ϑ 00 4 ( τ ) with respect to the principal congruence subgroup Γ ( 2 ) of PSL ( 2 , Z ) has an expression by the Gauss hypergeometric function F ( 1 / 2 , 1 / 2 , 1 ; 1 - λ ) via the inverse of the period map for the family of elliptic curves E ( λ ) (see Theorem 1.1). In this article we show a variant of this formula for the family of Picard curves C ( λ 1 , λ 2 ) : w 3 = z ( z - 1 ) ( z - λ 1 ) ( z - λ 2 ) , those are of genus three with two complex parameters. Our result is a two dimensional analogy of this context. The inverse of the period map for C ( λ 1 , λ 2 ) is established in [S] and our modular form ϑ 0 3 ( u , v ) (for the definition, see (2.7)) is defined on a two dimensional complex ball D = { 2 Re v + | u | 2 < 0 } , that can be realized as a Shimura variety in the Siegel upper half space of degree 3 by a modular embedding. Our main theorem says that our theta constant is expressed in terms of the Appell hypergeometric function F 1 ( 1 / 3 , 1 / 3 , 1 / 3 , 1 ; 1 - λ 1 , 1 - λ 2 ) .

On singular univariate specializations of bivariate hypergeometric functions

It is tempting to evaluate F 2 ( x , 1 ) and similar univariate specializations of Appell's functions by evaluating the apparent power series at x = 0 straight away using the Gauss formula for 2F 1(1). But this kind of naive evaluation can lead to errors as the 2F 1(1) coefficients might eventually diverge; then the actual power series at x = 0 might involve branching terms. This paper demonstrates these complications by concrete examples.

Structural Equation and Mei Conserved Quantity of Mei Symmetry for Appell Equations in Holonomic Systems with Unilateral Constraints

Structural equation and Mei conserved quantity of Mei symmetry for Appell equations in holonomic systems with unilateral constraints are investigated. Appell equations and differential equations of motion for holonomic mechanic systems with unilateral constraints are established. The definition and the criterion of Mei symmetry for Appell equations in holonomic systems with unilateral constraints under the infinitesimal transformations of groups are also given. The expressions of the structural equation and Mei conserved quantity of Mei symmetry for Appell equations in holonomic systems with unilateral constraints expressed by Appell functions are obtained. An example is given to illustrate the application of the results.

Error performance of digital modulation schemes with MRC diversity reception over η-μ fading channels

This paper provides exact-form expressions for the average symbol error probability (ASEP) of various digital modulation schemes with maximal ratio combining (MRC) diversity over L independent, not necessarily identically distributed (n.i.d.) eta-mu fading channels. The derived expressions are given in terms of the Lauricella and Appell hypergeometric functions and include several others available in the literature such as those for Nakagami-m and Hoyt. General asymptotic ASEP expressions are also derived for all the considered modulation schemes which provide useful insights regarding the factors affecting the performance of the considered system.

A two-parameter generalization of the complete elliptic integral of second kind

A two-parameter generalization of the complete elliptic integral of second kind is expressed in terms of the Appell function F4. This function is further reduced to a quite simple bilinear form in the complete elliptic integrals K and E. Some physical applications are briefly mentioned.

MULTIPOLE MATRIX ELEMENTS ν FOR H-LIKE ATOMS, THEIR ASYMPTOTICS AND APPLICATIONS (AS β=1, n ≤ 4, n' ≤ 10)

This article deals with the connection between multipole matrix elements &lt;nl|rβ|n'l' &gt;ν for H-like atoms and new properties of Appell's function F2(x,y) to the vicinity of the singular point (1, 1), where ν is the so-called "auxiliary" parameter of Heun-Schrödinger's radial equation, |1 - ν| = o(1), [Formula: see text] is the "effective" nuclear charge. Exact numerical values for the dipole matrix elements, the average oscillator strengths, the transition probabilities and the line intensities, as n ≤ 4 and n' ≤ 10, in the form of regular rational fractions are given (in Tables 1–4), that make more precise the well-known Tables 13–16 by Hans A. Bethe.

Specialization of Appell's functions to univariate hypergeometric functions

Univariate specializations of Appell's hypergeometric functions F1, F2, F3, F4 satisfy ordinary Fuchsian equations of order at most 4. In special cases, these differential equations are of order 2, and could be simple (pullback) transformations of Euler's differential equation for the Gauss hypergeometric function. The paper classifies these cases, and presents corresponding relations between univariate specializations of Appell's functions and univariate hypergeometric functions. The computational aspect and interesting identities are discussed.

NEW CONCEPTS IN ELECTROMAGNETIC JERKY DYNAMICS AND THEIR APPLICATIONS IN TRANSIENT PROCESSES OF ELECTRIC CIRCUIT

In this paper, jerk function for a transient process of RL circuit is investigated. Some new concepts such as time rate of change of induced emf, time rate of change of displacement current, and Appell function have been introduced for the flrst time in electromagnetic jerky dynamics. The problems on Appell function of several simple models in electromagnetic jerky dynamics are discussed. In the last conclusions and remarks are also presented.