Double Hypergeometric Series Research Articles

Computation of integral transforms in terms of double hypergeometric series ψ2 with applications

The principal aim of the present paper is to evaluate a new integral transform involving the product of two Bessel Functions of the first kind expressed in terms of the Humbert function ψ2. We also derive some interesting special cases from the main result of this paper. The presented result will be applied to generate some new laser beams called as the Humbert beams of type-II and study there propagation behaviour. From graphical simulations this type of wave seems interesting and useful in optical trapping which is used in some applications for pushing and moving particles such as: biological cells, neutral atoms, fluorescent, DNA molecules and trapping productivity on nano-sized spheres.

Olsson.wl & ROC2.wl: Mathematica packages for transformations of multivariable hypergeometric functions & regions of convergence for their series representations in the two variables case

We present the Olsson.wlMathematica package which aims to find transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson [1] (1964) to derive the analytic continuations of the Appell F1 double hypergeometric series using the linear transformations of the Gauss F12 hypergeometric function. We provide a brief description of the method of Olsson and demonstrate the use of the commands of the Olsson.wl package using some examples that are presented in the text and in some ancillary Mathematica notebooks. In particular, we reproduce various results of the literature on multivariable hypergeometric functions and show practical applications of this package in the derivation of novel formulas. In the context of high energy physics, we also demonstrate how it can be used to disentangle some known results about the analytic continuation of some series representations of the one-loop pentagon in multi-Regge kinematics and D=6−2ϵ. We also provide a companion package, called ROC2.wl, which is dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson.wl. Program summaryProgram Title:Olsson.wlCPC Library link to program files:https://doi.org/10.17632/gc63xzwzz5.1Licensing provisions: GNU General Public License v3.0.Programming language: Wolfram Mathematica version 11.3 and beyond.Nature of problem: To find the transformation formulas of multivariable hypergeometric series appearing in Feynman integral calculus.Solution method: Mathematica implementation of the method of Olsson [1]. The method uses the transformation theory of lower variable hypergeometric functions to find the transformation formulas of higher variable hypergeometric functions.Companion package:ROC2.wl, which is a Mathematica package used inside Olsson.wl and can also be used as a standalone package. The package is used to find the region of convergence of double hypergeometric functions, using Horn's theorem.

Confluent Kampé de Fériet Series Arising in the Solutions of Cauchy Problem for the Degenerate Hyperbolic Equation of the Second Kind with the Spectral Parameter

We define the order of the double hypergeometric series, investigate the properties of the new confluent Kampé de Fériet series, and build systems of partial differential equations that satisfy the new Kampé de Fériet series. We solve the Cauchy problem for a degenerate hyperbolic equation of the second kind with a spectral parameter using the high-order Kampé de Fériet series. Thanks to the properties of the introduced Kampé de Fériet series, it is possible to obtain a solution to the problem in explicit forms.

On Some Branched Continued Fraction Expansions for Horn’s Hypergeometric Function H4(a,b;c,d;z1,z2) Ratios

The paper deals with the problem of representation of Horn’s hypergeometric functions by branched continued fractions. The formal branched continued fraction expansions for three different Horn’s hypergeometric function H4 ratios are constructed. The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. It is proven that the branched continued fraction, which is an expansion of one of the ratios, uniformly converges to a holomorphic function of two variables on every compact subset of some domain H,H⊂C2, and that this function is an analytic continuation of this ratio in the domain H. The application to the approximation of functions of two variables associated with Horn’s double hypergeometric series H4 is considered, and the expression of solutions of some systems of partial differential equations is indicated.

Finite summation formulas for the multivariable $A$-function

The object of this paper is to evaluate some finite double summations relations for the multivariable $A$-function using the summation of a double hypergeometric series. The formulas derived in this paper are most general in character, we also provide a few particular cases for derived summation formulas.

On the evaluation of the Appell F2 double hypergeometric function

The transformation theory of the Appell F2(a,b1,b2;c1,c2;x,y) double hypergeometric function is developed to obtain a set of series representations of F2 which provide an efficient way to evaluate F2 for real values of its arguments x and y and generic complex values of its parameters a,b1,b2,c1 and c2 (i.e. in the nonlogarithmic case). This study rests on a classical approach where the usual double series representation of F2 and other double hypergeometric series that appear in the intermediate steps of the calculations are written as infinite sums of one variable hypergeometric series, such as the Gauss F12 or the F23, various linear transformations of the latter being then applied to derive known and new formulas. Use of the three well-known Euler transformations of F2 on these results allows us to obtain a total of 44 series which form the basis of the Mathematica package AppellF2.wl, dedicated to the evaluation of F2. A brief description of the package and of the numerical analysis that we have performed to test it is also presented. Program summaryProgram Title: AppellF2.wlCPC Library link to program files:https://doi.org/10.17632/n9v6bwpsyd.1Licensing provisions: GPLv3Programming language: Wolfram Mathematica version 11.3 and beyondNature of problem: Numerical evaluation of the double hypergeometric Appell F2 function for real values of its variables and generic complex values of its Pochhammer parametersSolution method: Mathematica implementation of a set of transformation formulas of the Appell F2 function

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.

New families of double hypergeometric series for constants involving ${1}/{\pi ^2}$

We apply a Fourier–Legendre-based technique recently introduced by Campbell et al. [J. Math. Anal. Appl. 479 (2019), 90–121], to prove new rational double hypergeometric series formulas for expressions involving $ {1}/{\pi ^2}$, especi

Conformal 3-Point Functions and the Lorentzian OPE in Momentum Space

In conformal field theory in Minkowski momentum space, the 3-point correlation functions of local operators are completely fixed by symmetry. Using Ward identities together with the existence of a Lorentzian operator product expansion (OPE), we show that the Wightman function of three scalar operators is a double hypergeometric series of the Appell F_4 type. We extend this simple closed-form expression to the case of two scalar operators and one traceless symmetric tensor with arbitrary spin. Time-ordered and partially-time-ordered products are constructed in a similar fashion and their relation with the Wightman function is discussed.

On the Laguerre fractional integro-differentiation

A fractional power interpretation of the Laguerre derivative is discussed. The corresponding fractional integrals are introduced. Mapping and semigroup properties, integral representations and Mellin transform analysis are presented. A relationship with the Riemann–Liouville fractional integrals is demonstrated. Finally, a second kind integral equation of the Volterra type, involving the Laguerre fractional integral is solved in terms of the double hypergeometric type series as the resolvent kernel.

Three-loop QED contributions to the g−2 of charged leptons with two internal fermion loops and a class of Kampé de Fériet series

The three-loop QED mass-dependent contributions to the $g-2$ of each of the charged leptons with two internal closed fermion loops, sometimes called $A^{(6)}_3\left(\frac{m_1}{m_2}, \frac{m_1}{m_3}\right)$ in the $g-2$ literature, is revisited using the Mellin-Barnes (MB) representation technique. Results for the muon and $\tau$ lepton anomalous magnetic moments $A^{(6)}_{3,\mu}$ and $A^{(6)}_{3,\tau}$, which were known as series expansions in the lepton mass ratios up to the first few terms only, are extended to their exact expressions. The contribution to the anomalous magnetic moment of the electron $A^{(6)}_{3,e}$ is also explicitly given in closed form. In addition to this, we show that the different series representations derived from the MB representation collectively converge for all possible values of the masses. Such unexpected behavior is related to the fact that these series bring into play double hypergeometric series that belong to a class of Kamp\'e de F\'eriet series which we prove to have the same simple convergence and analytic continuation properties as the Appell $F_1$ double hypergeometric series.

Analytic continuation of the Horn hypergeometric series with an arbitrary number of variables

The problem of analytic continuation is considered for the general hypergeometric Horn series with an arbitrary number of variables. An approach is proposed that allows one to find formulas for the continuation of such series into the exterior of the set of their convergence in the form of linear combinations of other hypergeometric series. These new hypergeometric series belong also to the Horn class, and are solutions of the same system of partial differential equations. Using this general result, we construct formulas for the analytic continuation for a double hypergeometric series of Horn's type, which plays an important role in the theory of the Appell function .

Two-loop kite master integral for a correlator of two composite vertices

We consider the most general two-loop massless correlator I(n1, n2, n3, n4, n5; x, y; D) of two composite vertices with the Bjorken fractions x and y for arbitrary indices {ni} and space-time dimension D; this correlator is represented by a “kite” diagram. The correlator I({ni}; x, y; D) is the generating function for any scalar Feynman integrals related to this kind of diagrams. We calculate I({ni}; x, y; D) and its Mellin moments in a direct way by evaluating hypergeometric integrals in the α representation. The result for I({ni}; x, y; D) is given in terms of a double hypergeometric series — the Kampé de Férriet function. In some particular but still quite general cases it reduces to a sum of generalized hypergeometric functions 3F2. The Mellin moments can be expressed through generalized Lauricella functions, which reduce to the Kampé de Férriet functions in several physically interesting situations. A number of Feynman integrals involved and relations for them are obtained.

Analytic continuation of the Appell function F 1 and integration of the associated system of equations in the logarithmic case

The Appell function F 1 (i.e., a generalized hypergeometric function of two complex variables) and a corresponding system of partial differential equations are considered in the logarithmic case when the parameters of F 1 are related in a special way. Formulas for the analytic continuation of F 1 beyond the unit bicircle are constructed in which F 1 is determined by a double hypergeometric series. For the indicated system of equations, a collection of canonical solutions are presented that are two-dimensional analogues of Kummer solutions well known in the theory of the classical Gauss hypergeometric equation. In the logarithmic case, the canonical solutions are written as generalized hypergeometric series of new form. The continuation formulas are derived using representations of F 1 in the form of Barnes contour integrals. The resulting formulas make it possible to efficiently calculate the Appell function in the entire range of its variables. The results of this work find a number of applications, including the problem of parameters of the Schwarz–Christoffel integral.

Further Double Sums of Dunkl and Gasper

By combining the series rearrangement with a reformulation of Beta function, we prove a transformation theorem that reduces a double sum with seven free parameters to a terminating hypergeometric \({_4F_3}\)-series. Seven double sum identities are derived as consequences with one of them extending the recent double sum of Dunkl and Gasper (The sums of a double hypergeometric series and of the first m + 1 terms of \({_3F_2(a,b,c;(a+b+1)/2,2c;1)}\) when c = −m is a negative integer. arXiv:1412.4022v2 [math.CA], 2014).

Transformation and summation formulae for Kampé de Fériet series

Euler’s two transformations are employed to establish several general transformation formulae for double hypergeometric series. These transformations yield further a number of reduction formulae for Kampé de Fériet series of types F1:0;μ1:1;λ, F1:0;μ0:2;λ and F1:0;μ2:0;λ. By specializing certain parameters, a list of new summation formulae for these Kampé de Fériet series are obtained.

Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals

Focusing on examples associated with holonomic functions, we try to bring new ideas on how to look at phase transitions, for which the critical manifolds are not points but curves depending on a spectral variable, or even fill higher dimensional submanifolds. Lattice statistical mechanics often provides a natural (holonomic) framework to perform singularity analysis with several complex variables that would, in the most general mathematical framework, be too complex, or simply could not be defined. In a learn-by-example approach, considering several Picard–Fuchs systems of two-variables ‘above’ Calabi–Yau ODEs, associated with double hypergeometric series, we show that D-finite (holonomic) functions are actually a good framework for finding properly the singular manifolds. The singular manifolds are found to be genus-zero curves. We then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics, the n-fold integrals χ(n), corresponding to the n-particle decomposition of the magnetic susceptibility of the anisotropic square Ising model. In this anisotropic case, we revisit a set of so-called Nickelian singularities that turns out to be a two-parameter family of elliptic curves. We then find the first set of non-Nickelian singularities for χ(3) and χ(4), that also turns out to be rational or elliptic curves. We underline the fact that these singular curves depend on the anisotropy of the Ising model, or, equivalently, that they depend on the spectral parameter of the model. This has important consequences on the physical nature of the anisotropic χ(n)s which appear to be highly composite objects. We address, from a birational viewpoint, the emergence of families of elliptic curves, and that of Calabi–Yau manifolds on such problems. We also address the question of singularities of non-holonomic functions with a discussion on the accumulation of these singular curves for the non-holonomic anisotropic full susceptibility χ.This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.

A connection formula between double hypergeometric series Ψ2 and Φ3

The Humbert confluent hypergeometric functions Φ2, Φ3 and Ψ2 are considered. The connection Φ3 with the function Ψ2 is established. Representations for functions Φ3, Ψ2 as series with the Gauss function 2 F 1 are proposed. Some special values of Φ2 and Ψ2 are found in terms of the generalized hypergeometric function p F q .

The Hadamard product of hypergeometric series

Typically a hypergeometric function is a multi-valued analytic function with algebraic singularities. In this paper we give a complete description of the Newton polytope of the polynomial whose zero set naturally contains the singular locus of a nonconfluent double hypergeometric series. We show in particular that the Hadamard multiplication of such series corresponds to the Minkowski sum of the Newton polytopes of polynomials which define their singularities.

Transformation formulas for double hypergeometric series related to 9-j coefficients and their basic analogs

In a recent paper, Ališauskas deduced different triple sum expressions for the 9-j coefficient of su(2) and suq(2). For a singly stretched 9-j coefficient, these reduce to different double sum series. Using these distinct series, we deduce a set of new transformation formulas for double hypergeometric series of Kampé de Fériet type and their basic analogs. These transformation formulas are valid for rather general parameters of the series, although a common feature is that all the series appearing here are terminating. It is also shown that the transformation formulas deduced here generate a group of transformation formulas, thus yielding an invariance group or symmetry group of particular double series.