Appell Function Research Articles

The mixed mock modularity of a new $U$-type function related to the Andrews-Gordon identities

In this paper we resolve a question by Bringmann, Lovejoy, and Rolen on a new vector-valued $U$-type function. We obtain an expression for a corresponding family of Hecke--Appell-type sums in terms of mixed mock modular forms; that is, we express the sum in terms of Appell functions and theta functions. This $U$-type function appears from considering the special polynomials related to generating functions for the partitions occurring in Gordon’s generalization of the Rogers--Ramanujan identities.

Certain integral transforms involving Appell and Bessel functions and their applications

Abstract This paper deals with the evaluation of certain integral transforms involving the product of certain Appell and Bessel functions with a weight e - γ ⁢ t 2 {e^{-\gamma t^{2}}} . The transformations of these integrals are evaluated in terms of the Appell, Kampé de Fériet and the triple hypergeometric functions. As an application, we studied propagation of generalized Humbert–Gaussian beams (GHGBs) and hypergeometric-Gaussian beams (HyGGBs) in turbulent atmosphere and through an ABCD paraxial optical system. The evaluation of these integral transforms has initiated a great interest in mathematical physics and its applications to laser physics and linear or non-linear optics.

Some results of new extended beta, hypergeometric, Appell and Lauricella matrix functions

Some results of new extended beta, hypergeometric, Appell and Lauricella matrix functions

Convolution of beta prime distribution

We establish some identities in law for the convolution of a beta prime distribution with itself, involving the square root of beta distributions. The proof of these identities relies on transformations on generalized hypergeometric series obtained via Appell series of the first kind and Thomae’s relationships for 3 F 2 ( 1 ) {}_3F_2(1) . Using a self-decomposability argument, the identities are applied to derive complete monotonicity properties for quotients of confluent hypergeometric functions having a doubling character. By means of probability, we also obtain a simple proof of Turán’s inequality for the parabolic cylinder function and the confluent hypergeometric function of the second kind. The case of Mill’s ratio is discussed in detail.

Exploring the strong-coupling region of SU(N) Seiberg-Witten theory

We consider the Seiberg-Witten solution of pure mathcal{N} = 2 gauge theory in four dimensions, with gauge group SU(N). A simple exact series expansion for the dependence of the 2(N − 1) Seiberg-Witten periods aI(u), aDI(u) on the N − 1 Coulomb-branch moduli un is obtained around the ℤ2N-symmetric point of the Coulomb branch, where all un vanish. This generalizes earlier results for N = 2 in terms of hypergeometric functions, and for N = 3 in terms of Appell functions. Using these and other analytical results, combined with numerical computations, we explore the global structure of the Kähler potential K = frac{1}{2}{sum}_I Im( overline{a} IaDI), which is single valued on the Coulomb branch. Evidence is presented that K is a convex function, with a unique minimum at the ℤ2N-symmetric point. Finally, we explore candidate walls of marginal stability in the vicinity of this point, and their relation to the surface of vanishing Kähler potential.

On the new bicomplex generalization of Hurwitz–Lerch zeta function with properties and applications

Abstract In the recent years, various authors introduced different generalizations of the Hurwitz–Lerch zeta function and discussed its various properties. The main aim of our study is to introduce a new bicomplex generalization of the Hurwitz–Lerch zeta function using the new generalized form of the beta function that involves the Appell series and Lauricella functions. The new bicomplex generalization of the Hurwitz–Lerch zeta function reduces to some already known functions like the Hurwitz–Lerch zeta function, Hurwitz zeta function, Riemann zeta function and polylogarithmic function. Its different properties such as recurrence relation, summation formula, differentiation formula, generating relations and integral representations are investigated. All results induced are general in nature and reducible to already known results. As an application of the new bicomplex generalization of the Hurwitz–Lerch zeta function, we have developed a new generalized form of fractional kinetic equation and obtained its solution using the natural transform.

Method of Continual Addition Theorems and Integral Relations between the Coulomb Functions and the Appell Function F1

Method of Continual Addition Theorems and Integral Relations between the Coulomb Functions and the Appell Function F1

A GENERALISATION OF A SUPERCONGRUENCE ON THE TRUNCATED APPELL SERIES

AbstractRecently, Lin and Liu [‘Congruences for the truncated Appell series $F_3$ and $F_4$ ’, Integral Transforms Spec. Funct.31(1) (2020), 10–17] confirmed a supercongruence on the truncated Appell series $F_3$ . Motivated by their work, we give a generalisation of this supercongruence by establishing a q-supercongruence modulo the fourth power of a cyclotomic polynomial.

An Analytic Representation of the Second Symmetric Standard Elliptic Integral in Terms of Elementary Functions

We derive new convergent expansions of the symmetric standard elliptic integral R_D(x,y,z), for x, y,zin {mathbb {C}}{setminus }(-infty ,0], in terms of elementary functions. The expansions hold uniformly for large and small values of one of the three variables x, y or z (with the other two fixed). We proceed by considering a more general parametric integral from which R_D(x,y,z) is a particular case. It turns out that this parametric integral is an integral representation of the Appell function F_1(a;b,c;a+1;x,y). Therefore, as a byproduct, we deduce convergent expansions of F_1(a;b,c;a+1;x,y). We also compute error bounds at any order of the approximation. Some numerical examples show the accuracy of the expansions and their uniform features.

Effect of Cattaneo-Christov approximation for viscoelastic fluid with carbon nanotubes on flow and heat transfer.

The current work studies the motion of viscoelastic liquid saturated with carbon nanotubes over a stretching surface in a Darcy porous medium analytically below an influence of Cattaneo-Christov heat flux. The carbon nanotubes (CNTs) act as nanoparticles which are then appended into the base fluid. Water and kerosene are used as a base fluid with two types of CNTs, namely, Single-wall carbon nanotubes and Multiwall carbon nanotubes. Carbon nanotubes possess a wide range of industrial and biomedical applications including energy production, nuclear reactor cooling, and galaxy cooling applications because they can expand the thermal and mechanical properties of base things. As a result, the carbon nanotubes used in the mentioned fields are being investigated for their potential in heat transfer applications. Governing equations formulated using the Partial differential equations have converted to Ordinary differential equations exhausting the appropriate comparison transformation process. An influence of some relevant constraints on velocity and temperature is evaluated in details. The Cattaneo-Christov heat transfer model is utilized to investigate the heat transfer individualities with varying thermal conductivity consuming the attributes of the Appell hypergeometric function. The impacts of the emerging parameters on the profiles are depicted through graphical representations and analytically constructed tables. Considering its usefulness in modulating temperature distribution in different industrial application, including solar collector design, electronic cooling, building ventilation, etc. According to our findings, the temperature profile exhibits an enhancement with the thermal radiation parameter and the viscous-elastic fluids. In addition, when compared to the classical Fourier's law of heat conduction, the temperature profile and thermal boundary layer thickness for the Cattaneo-Christov heat flux model are lower.

Functional reduction of one-loop Feynman integrals with arbitrary masses

A method of functional reduction for the dimensionally regularized one-loop Feynman integrals with massive propagators is described in detail.The method is based on a repeated application of the functional relations proposed by the author. Explicit formulae are given for reducing one-loop scalar integrals to a simpler ones, the arguments of which are the ratios of polynomials in the masses and kinematic invariants. We show that a general scalar n-point integral, depending on n(n + 1)/2 generic masses and kinematic variables, can be expressed as a linear combination of integrals depending only on n variables. The latter integrals are given explicitly in terms of hypergeometric functions of (n − 1) dimensionless variables. Analytic expressions for the 2-, 3- and 4-point integrals, that depend on the minimal number of variables, were also obtained by solving the dimensional recurrence relations. The resulting expressions for these integrals are given in terms of Gauss’ hypergeometric function 2F1, the Appell function F1 and the hypergeometric Lauricella — Saran function FS. A modification of the functional reduction procedure for some special values of kinematic variables is considered.

Positivity of the Veneziano amplitude in D = 4

The Veneziano amplitude was put forward as a solution to the axioms of the S-matrix bootstrap. However, unitarity, reflected in the positivity of the coefficients in the Gegenbauer expansion of the amplitude is not obvious. In this note we compute the generating function of these coefficients in terms of the Appell hypergeometric function. We use this to read off an exact form of this coefficient on the leading Regge trajectory in D = 4. We find that it decays with the spin but always remains positive. Since for large spin these coefficients are expected to be smaller than those on the subleading trajectories, our result indicates the positivity of the full Veneziano amplitude in D = 4.

Yangian Ward identities for fishnet four-point integrals

We derive and study Yangian Ward identities for the infinite class of four-point ladder integrals and their Basso-Dixon generalisations. These symmetry equations follow from interpreting the respective Feynman integrals as correlation functions in the biscalar fishnet theory. Alternatively, the presented identities can be understood as anomaly equations for a momentum space conformal symmetry. The Ward identities take the form of inhomogeneous extensions of the partial differential equations defining the Appell hypergeometric functions. We employ a manifestly conformal tensor reduction in order to express these inhomogeneities in compact form, which are given by linear combinations of Basso-Dixon integrals with shifted dimensions and propagator powers. The Ward identities naturally generalise to a one-parameter family of D-dimensional integrals representing correlators in the generalised fishnet theory of Kazakov and Olivucci. When specified to two spacetime dimensions, the Yangian Ward identities decouple. Using separation of variables, we explicitly bootstrap the solution for the conformal 2D box integral. The result is a linear combination of Yangian invariant products of Legendre functions, which reduce to elliptic K integrals for an isotropic choice of propagator powers. We comment on differences in the transcendentality patterns in two and four dimensions and their relations to discontinuities.

New Expression for CDF of chi _nu '^2(lambda ) Distribution and Marcum Q_1 Function

Closed form expression is established for the cumulative distribution function of the non-central chi _nu '^2(lambda ) probability distribution with integer degrees of freedom nu . This expression includes the Humbert Phi _1 (confluent Appell) hypergeometric function of two variables. A new representation of the Marcum Q_1 function of the first order is obtained, interconnection formula accompanied with a reduction formula for specific, equal unit-parameter case of the Humbert function Phi _1 and the function Phi _3 are established.

Performance analysis of non‐orthogonal multiple access assisted cooperative maritime communication system over two‐wave with diffuse power fading

AbstractNon‐orthogonal multiple access (NOMA) and millimeter‐wave (mmWave) communication are two promising technologies to fulfill the high throughput and reliable communication requirements. Maritime communication systems use a generalized fading at near‐sea‐surface channels (for seashore to ship/boat/fishing fleet communication) to provide reliable communication due to atmospheric turbulence. Experimental research on generalized two‐wave diffuse power (TWDP) fading reveal the footprints of mmWave. This article proposes novel exact and asymptotic closed‐form expressions of average symbol error probability (ASEP) of NOMA users for cooperative maritime communication systems over TWDP fading with the different modulations. The end‐to‐end closed‐form expressions are derived in terms of Appell's and Lauricella's hypergeometric functions. Also, the ASEP is shown using the power allocation factor for different modulation schemes. The proposed system provides better transmission reliability using generalized TWDP fading. Further, the outage probability of the system described above is analyzed using the distance, path loss exponent, transmission rate, and Rician factor as performance metrics for both the NOMA users. Effects of fading outage parameters on the system performance are studied. The end‐to‐end exact numerical results have been verified with the Monte‐Carlo simulations that validate with asymptotic results at a high SNR regime.

Further investigations on the two variables second Appell hypergeometric matrix function

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.

A New q-Hypergeometric Symbolic Calculus in the Spirit of Horn, Borngässer, Debiard and Gaveau

The purpose of this article is to introduce a new complete multiple q-hypergeometric symbolic calculus, which leads to q-Euler integrals and a very similar canonical system of q-difference equations for multiple q-hypergeometric functions. q-analogues of recurrence formulas in Horns paper and Borngässers thesis lead to a more exact way to find these Frobenius solutions. To find the right formulas, the parameters in q-shifted factorials can be changed to negative integers, which give no extra q-factors. In proving these q-formulas, in the limit q→1 we obtain versions of the paper by Debiard and Gaveau for the solution of differential or q-difference equations. The paper is also a correction of some of the statements in the paper by Debiard and Gaveau, e.g., the Euler integrals and other solutions to differential equations for Appell functions, also without references to page numbers in the standard work of Appell and Kampé de Fériet. Sometimes the q-binomial theorem is used to simplify q-integral formulas. By the Horn method, we find another solution to the Appell Φ1 function partial differential equation, which was not mentioned in the thesis by Le Vavasseur 1893.

Extended incomplete Riemann-Liouville fractional integral operators and related special functions

<abstract><p>In this study, we introduce the extended incomplete versions of the Riemann-Liouville (R-L) fractional integral operators and investigate their analytical properties rigorously. More precisely, we investigate their transformation properties in $ L_{1} $ and $ L_{\infty} $ spaces, and we observe that the extended incomplete fractional calculus operators can be used in the analysis of a wider class of functions than the extended fractional calculus operator. Moreover, by considering the concept of analytical continuation, definitions for extended incomplete R-L fractional derivatives are given and therefore the full fractional calculus model has been completed for each complex order. Then the extended incomplete $ \tau $-Gauss, confluent and Appell's hypergeometric functions are introduced by means of the extended incomplete beta functions and some of their properties such as integral representations and their relations with the extended R-L fractional calculus has been given. As a particular advantage of the new fractional integral operators, some generating relations of linear and bilinear type for extended incomplete $ \tau $-hypergeometric functions have been derived.</p></abstract>

On convergence of function F4(1,2;2,2;z1,z2) expansion into a branched continued fraction

In the paper, the possibility of the Appell hypergeometric function F4(1,2;2,2;z1,z2) approximation by a branched continued fraction of a special form is analysed. The correspondence of the constructed branched continued fraction to the Appell hypergeometric function F4 is proved. The convergence of the obtained branched continued fraction in some polycircular domain of two-dimensional complex space is established, and numerical experiments are carried out. The results of the calculations confirmed the efficiency of approximating the Appell hypergeometric function F4(1,2;2,2;z1,z2) by a branched continued fraction of special form and illustrated the hypothesis of the existence of a wider domain of convergence of the obtained expansion.

The double hypergeometric series for the partition function of the 2D anisotropic Ising model

In 1944 Lars Onsager published the exact partition function of the ferromagnetic Ising model on the infinite square lattice in terms of a definite integral. Only in the literature of the last decade, however, has it been recast in terms of special functions. Until now all known formulas for the partition function in terms of special functions have been restricted to the important special case of the isotropic Ising model with symmetric couplings. Indeed, the anisotropic model is more challenging because there are two couplings and hence two reduced temperatures, one for each of the two axes of the square lattice. Hence, standard special functions of one variable are inadequate to the task. Here, we reformulate the partition function of the anisotropic Ising model in terms of the Kampé de Fériet function, which is a double hypergeometric function in two variables that is more general than the Appell hypergeometric functions. Finally, we present hypergeometric formulas for the generating function of multipolygons of given length on the infinite square lattice, for isotropic as well as anisotropic edge weights. For the isotropic case, the results allow easy calculation, to arbitrary order, of the celebrated series found by Cyril Domb.