In this article we propose an extension of Appell hypergeometric function $F_2$ (or equivalently $F_3$). It is derived from a particular solution of a higher order Painlevé system in two variables. On the other hand, an extension of Appell's $F_1$ was introduced by Tsuda. We also show that those two extensions are equivalent at the level of systems of linear partial differential equations.
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.
In a previous work ([Eb]), the author proposed a method employing contiguity relations to derive hypergeometric series in closed form. In [Eb], this method was used to derive Gauss's hypergeometric series $_2F_1$ possessing closed forms. Here, we consider the application of this method to Appell's hypergeometetric series $F_1$ and derive several $F_1$ possessing closed forms. Moreover, analyzing these $F_1$, we obtain values of $_2F_1$ with no free parameters. Some of these results provide new examples of algebraic values of $_2F_1$. Key Words and Phrases: Gauss's hypergeometric series, algebr…
This survey article (which will appear as a chapter in the book ``Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions'', Springer-Verlag) provides a small collection of basic material on multiple hypergeometric series of Appell-type and of more general series of related type.
Inspired by certain interesting recent extensions of the gamma, beta and hypergeometric matrix functions, we introduce here new extension of the gamma and beta matrix function. We also introduce new extensions of the Gauss hypergeometric matrix function, confluent hypergeometric matrix function, Appell matrix function and Lauricella matrix function of three variables in terms of the new extended beta matrix function. Then we investigate certain properties of these extended matrix functions such as the integral representations, differential formulae and recurrence relations.
We extend Schwarz' list of irreducible algebraic Gauss functions to the four classes of Appell-Lauricella functions in several variables and the 14 complete Horn functions in two variables. This gives an example of a family of functions such that for any number of variables there are infinitely many algebraic functions, namely the Lauricella $F_C$ functions.
Our present investigation is mainly based on the $k$-hypergeometric functions which are constructed by making use of the Pochhammer $k$-symbol \cite{Diaz} which are one of the vital generalization of hypergeometric functions. We introduce $k$-analogues of $F_{2}\ $and $F_{3}$ Appell functions denoted by the symbols $F_{2,k}\ $and $F_{3,k}\ $respectively, just like Mubeen et al. did for $F_{1}$ in 2015 \cite{Mubeen6}. Meanwhile, we prove some main properties namely integral representations, transformation formulas and some reduction formulas which help us to have relations between not only $k$-…
In this paper, we present a new real-valued Appell-type polynomial family $A_n^{(k)}(m,x), $ every member of which is expressed by mean of a generalized hypergeometric function. The generating exponential function of this type of polynomials is firstly discovered and the proof that they are of Appell-type ones is given. We present the differential operator formal power series representation as well as an explicit formula over the standard basis, and establish a new identity for the generalized hypergeometric function. Besides, we derive the addition, the multiplication and some other formulas …
We consider the system $\mathcal{F}_4(a,b,c)$ of differential equations annihilating Appell's hypergeometric series $F_4(a,b,c;x)$. We find the integral representations for four linearly independent solutions expressed by the hypergeometric series $F_4$. By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation of $\mathcal{F}_4(a,b,c)$ and the twisted period relations for the fundamental systems of solutions of $\mathcal{F}_4$.
We investigate the recurrence coefficients of discrete orthogonal polynomials on the non-negative integers with hypergeometric weights and show that they satisfy a system of non-linear difference equations and a non-linear second order differential equation in one of the parameters of the weights. The non-linear difference equations form a pair of discrete Painlevé equations and the differential equation is the $σ$-form of the sixth Painlevé equation. We briefly investigate the asymptotic behavior of the recurrence coefficients as $n\to \infty$ using the discrete Painlevé equations.
We construct several expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta functions and the Appell generalized hypergeometric functions of two variables of the fist kind. The coefficients of different expansions obey four-, five-, or six-term recurrence relations that are reduced to ones involving less number of terms only in a few exceptional cases. The conditions for deriving finite-sum solutions via termination of the series are discussed.
We suggest an approximation for the zero-balanced Appell hypergeometric function $F_1$ near the singular point $(1,1)$. Our approximation can be viewed as a generalization of Ramanujan's approximation for zero-balanced ${_2F_1}$ and is expressed in terms of ${_3F_2}$. We find an error bound and prove some basic properties of the suggested approximation which reproduce the similar properties of the Appell function. Our approximation reduces to the approximation of Carlson-Gustafson when the Appell function reduces to the first incomplete elliptic integral.
Hypergeometric solutions to the q-Painlevé equations are constructed by direct linearization of disrcrete Riccati equations. The decoupling factors are explicitly determined so that the linear systems give rise to q-hypergeometric equations.
A class of classical solutions to the $q$-Painlevé equation of type $(A_1+A_1')^{(1)}$ (a $q$-difference analog of the Painlevé II equation) is constructed in a determinantal form with basic hypergeometric function elements. The continuous limit of this $q$-Painlevé equation to the Painlevé II equation and its hypergeometric solutions are discussed. The continuous limit of these hypergeometric solutions to the Airy function is obtained through a uniform asymptotic expansion of their integral representation.
We consider the q-Painlevé equation of type $A_4^{(1)}$ (a version of q-Painlevé V equation) and construct a family of solutions expressible in terms of certain basic hypergeometric series. We also present the determinant formula for the solutions.
We consider a $q$-Painlevé III equation and a $q$-Painlevé II equation arising from a birational representation of the affine Weyl group of type $(A_2+A_1)^{(1)}$. We study their hypergeometric solutions on the level of $τ$ functions.
In this paper, we obtain a $(p,v)$-extension of the Appell hypergeometric function $ F_{1}(\cdot)$, together with its integral representation, by using the extended Beta function $B_{p,v}(x,y)$ introduced in arXiv:1502.06200. Also, we give some of its main properties, namely the Mellin transform, a differential formula, recursion formulas and a bounded inequality. In addition, some new integral representations of the extended Appell function$ F_{1,p,v}(\cdot)$ involving Meijer's $G$-function are obtained.
We present the $τ$-functions for the hypergeometric solutions to the $q$-Painlevé system of type $E_7^{(1)}$ in a determinant formula whose entries are given by the basic hypergeometric function ${}_8W_7$. By using the $W(D_5)$ symmetry of the function ${}_8W_7$, we construct a set of twelve solutions and describe the action of $\widetilde{W}(D_6^{(1)})$ on the set.
We construct a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painlevé equation (or higher-order analogues), and admitting a large family of monodromy-preserving deformations. The solutions are certain semiclassical biorthogonal functions (and their Cauchy transforms), biorthogonal with respect to higher-order analogues of Spiridonov's elliptic beta integral.
We consider an application of Grothendieck's dessins d'enfants to the theory of the sixth Painlevé and Gauss hypergeometric functions: two classical special functions of the isomonodromy type. It is shown that, higher order transformations and the Schwarz table for the Gauss hypergeometric function are closely related with some particular Belyi functions. Moreover, we introduce a notion of deformation of the dessins d'enfants and show that one dimensional deformations are a useful tool for construction of algebraic the sixth Painlevé functions.
The transformation theory of the Appell $F_2(a,b_1,b_2;c_1,c_2;x,y)$ double hypergeometric function is used to obtain a set of series representations of $F_2$ which provide an efficient way to evaluate $F_2$ for real values of its arguments $x$ and $y$ and generic complex values of its parameters $a,b_1, b_2, c_1$ and $c_2$ (i.e. in the nonlogarithmic case). This study rests on a classical approach where the usual double series representation of $F_2$ and other double hypergeometric series that appear in the intermediate steps of the calculations are written as infinite sums of one variable hy…
In this article, we propose a class of six-dimensional Painleve systems given as the monodromy preserving deformations of the Fuchsian systems. They are expressed as polynomial Hamiltonian systems of sixth order. We also discuss their particular solutions in terms of the hypergeometric functions defined by fourth order rigid systems.
In a recent work, we proposed the coupled Painlevé VI system with $A^{(1)}_{2n+1}$-symmetry, which is a higher order generalization of the sixth Painlevé equation ($P_{\rm VI}$). In this article, we present its particular solution expressed in terms of the hypergeometric function ${}_{n+1}F_n$. We also discuss a degeneration structure of the Painlevé system derived from the confluence of ${}_{n+1}F_n$.
We study the linear Pfaffian systems satisfied by a certain class of hypergeometric functions, which includes Gauß's ${}_2 F_{1}$, Thomae's ${}_L F_{L-1}$ and Appell-Lauricella's $F_D$. In particular, we present a fundamental system of solutions with a characteristic local behavior by means of Euler-type integral representations. We also discuss how they are related to the theory of isomonodromic deformations or Painlevé equations.
In this paper, the incomplete Pochhammer ratios are defined in terms of the incomplete beta function $B_{y}(x,z)$. With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric and Appell's functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, and recurrence relation. Furthermore, an incomplete Riemann-Liouville fractional derivative operators are introduced. This definition helps us to obtain linear and bilinear generating relations for the new incomplete Gauss hyp…
HYPERDIRE is a project devoted to the creation of a set of Mathematica based programs for the differential reduction of hypergeometric functions. The current version includes two parts: one, pfq, is relevant for manipulations of hypergeometric functions_{p+1}F_p, and the second one, AppellF1F4, for manipulations with Appell hypergeometric functions F_1,F_2,F_3,F_4 of two variables.
We study Wronskians of Appell polynomials indexed by integer partitions. These families of polynomials appear in rational solutions of certain Painlevé equations and in the study of exceptional orthogonal polynomials. We determine their derivatives, their average and variance with respect to Plancherel measure, and introduce several recurrence relations. In addition, we prove an integrality conjecture for Wronskian Hermite polynomials previously made by the first and last authors. Our proofs all exploit strong connections with the theory of symmetric functions.
In this paper, we aim to present new extensions of incomplete gamma, beta, Gauss hypergeometric, confluent hypergeometric function and Appell-Lauricella hypergeometric functions, by using the extended Bessel function due to Boudjelkha [4]. Some recurrence relations, transformation formulas, Mellin transform and integral representations are obtained for these generalizations. Further, an extension of the Riemann-Liouville fractional derivative operator is established.
In this paper, we present a new method for finding identities for hypergeoemtric series, such as the (Gauss) hypergeometric series, the generalized hypergeometric series and the Appell-Lauricella hypergeometric series. Furthermore, using this method, we get identities for the hypergeometric series $F(a,b;c;x)$; we show that values of $F(a,b;c;x)$ at some points $x$ can be expressed in terms of gamma functions, together with certain elementary functions. We tabulate the values of $F(a,b;c;x)$ that can be obtained with this method. We find that this set includes almost all previously known value…
Recently, there emerges different versions of beta function and hypergeometric functions containing extra parameters. Gaining enlightenment from these ideas, we will first introduce a new extension of generalized hypergeometric function and then put forward some fundamental results in the paper. Next, we will derive some properties of certain functions like extended Gauss hypergeometric functions, extended Appell's hypergeometric functions $\mathbf{F}_{1}^{(κ_{l})}$, $\mathbf{F}_{2}^{(κ_{l})}$, and extended Lauricella's hypergeometric function $\mathbf{F}_{D,(κ_{l})}^{(r)}$,$\mathbf{F}_{A,(κ_{…
In theory and practice of elastic straight rods, the statically indeterminate reactions acted by perfect constraints are commonly believed not to depend on the flexural stiffness $EJ$. We solve exactly two elastica problems in order to obtain hypergeometrically (helped by Lagrange, Lauricella, Appell), the true displacements upon which the forces method is founded. As a consequence, the above reactions are found to depend on stiffness: the presumptive independence credited as general, is far from being always true, but, quite the contrary, is valid only within a first-order approximation.
We introduce two variants of $q$-hypergeometric equation. We obtain several explicit solutions of variants of $q$-hypergeometric equation. We show that a variant of $q$-hypergeometric equation can be obtained by a restriction of $q$-Appell equation of two variables.
We derive formulas for the construction of all inequivalent Jacobian elliptic fibrations on the Kummer surface of two non-isogeneous elliptic curves from extremal rational elliptic surfaces by rational base transformations and quadratic twists. We then show that each such decomposition yields a description of the Picard-Fuchs system satisfied by the periods of the holomorphic two-form as either a tensor product of two Gauss' hypergeometric differential equations, an Appell hypergeometric system, or a GKZ differential system. As the answer must be independent of the fibration used, identities r…
A correlation function of the classical orthogonal polynomials is defined and determined. The correlation function obeys a second order difference equation in two variables. The correlation function for the Gegenbauer, Chebyshev and Legendre polynomials can be written as a 4F3 hypergeometric function. For the Jacobi polynomials the result is an F2 Appell function. For the Generalized Laguerre polynomials the result is a confluent hypergeometric function and for the Hermite polynomials there rests only a single term.
New explicit expressions are derived for the one-loop two-point Feynman integral with arbitrary external momentum and masses $m_1^2$ and $m_2^2$ in D dimensions. The results are given in terms of Appell functions, manifestly symmetric with respect to the masses $m_i^2$. Equating our expressions with previously known results in terms of Gauss hypergeometric functions yields reduction relations for the involved Appell functions that are apparently new mathematical results.
Over the last decade it has become clear that discrete Painlevé equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painlevé equation and determining its type according to Sakai's classification scheme, understanding whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such an example, becomes one of the central ones. Fortunately, Sakai's geometric theory provides an almost algorithmic procedure for answer…
We propose a systematic study of transformations of $A$-hypergeometric functions. Our approach is to apply changes of variables corresponding to automorphisms of toric rings, to Euler-type integral representations of $A$-hypergeometric functions. We show that all linear $A$-hypergeometric transformations arise from symmetries of the corresponding polytope. As an application of the techniques developed here, we show that the Appell function $F_4$ does not admit a certain kind of Euler-type integral representation.
Gauss hypergeometric functions with a dihedral monodromy group can be expressed as elementary functions, since their hypergeometric equations can be transformed to Fuchsian equations with cyclic monodromy groups by a quadratic change of the argument variable. The paper presents general elementary expressions of these dihedral hypergeometric functions, involving finite bivariate sums expressible as terminating Appell's F2 or F3 series. Additionally, trigonometric expressions for the dihedral functions are presented, and degenerate cases (logarithmic, or with the monodromy group Z/2Z) are consid…
In this paper, using similar symbolical method of Burchnall and Chaundy formulas of expansion for the generalized hypergeometric function were constructed. By means of the found formulas of expansion the formulas of an analytic continuation for hypergeometric function of Clausen is defined. The obtained formulas of an analytic continuation express known hypergeometric Appell function $ F_2 ({a;b_1, b_2 ;c_1, c_2 ;x,y}) $ which theory is well studied.
We find two-sided inequalities for the generalized hypergeometric function of the form ${_{q+1}}F_{q}(-x)$ with positive parameters restricted by certain additional conditions. Both lower and upper bounds agree with the value of ${_{q+1}}F_{q}(-x)$ at the endpoints of positive semi-axis and are asymptotically precise at one of the endpoints. The inequalities are derived from a theorem asserting the monotony of the quotient of two generalized hypergeometric functions with shifted parameters. The proofs hinge on a generalized Stieltjes representation of the generalized hypergeometric function. T…
It is tempting to evaluate F2(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 2F1(1). But this kind of naive evaluation can lead to errors as the 2F1(1) coefficients might eventually diverge; then the actual power series at x=0 might involve branching terms. This paper demonstrates these complications on concrete examples.
Special matrix functions have recently been investigated for regions of convergence, integral representations and the systems of matrix differential equation that these functions satisfy. In this paper, we find the recursion formulas for the Gauss hypergeometric matrix function. We also give the recursion formulas for the two variable Appell matrix functions
Two integral operator involving the Appell's functions, or Horn's function in the kernel are considered. Composition of such functions with generalized Bessel functions of the first kind are expressed in term of generalized Wright function and generalized hypergeometric series. Many special cases, including cosine and sine function are also discussed.
In this paper we are interested in extending Bailey's identity to other classical hypergeometric functions. Bailey's identity states that under a suitable choice of parameters, Appell's $F_4$ decomposes into a product of two ${}_2F_1$'s. We will show how Bailey-type factorizations can be found for Horn's hypergeometric functions $H_1, H_4$ and $H_5$.
We propose a new bilinear Hirota equation for $τ$-functions associated with the $E_8$ root lattice, that provides a "lens" generalisation of the $τ$-functions for the elliptic discrete Painlevé equation. Our equations are characterized by a positive integer $r$ in addition to the usual elliptic parameters, and involve a mixture of continuous variables with additional discrete variables, the latter taking values on the $E_8$ root lattice. We construct explicit $W(E_7)$-invariant hypergeometric solutions of this bilinear Hirota equation, which are given in terms of elliptic hypergeometric sum/in…
The theoretical computing of special values assumed by the hypergeometric functions has a high interest not only on its own, but also in sight of the remarkable implications to both pure Mathematics and Mathematical Physics. Accordingly, in this paper we continue the path of research started in two our previous papers appeared on [30] and [31] providing some contribution to such functions computability inside and outside their disk of convergence. This is accomplished through two different approaches. The first is to provide some new results in the spirit of theorem 3.1 of 31] obtaining formul…
Szmytkowski derived a certain integral with Gegenbauer polynomials. A natural generalization is to derive lookalike integrals with Jacobi polynomials. Six methods are treated to derive the first integral. The first method should be enough to prove the first integral, but by the other methods there arises remarkable formula such as par example a zero-balanced F3 Appell function which can be converted into a 2F1 hypergeometric function. Another three integrals complete the paper.
The main aim of this present paper is to present a new extension of the fractional derivative operator by using the extension of Beta function recently defined by Shadab et al.[19]. Moreover, we establish some results related to the newly defined modified fractional derivative operator such as Mellin transform and relations to extended hypergeometric and Appell's function via generating functions.
We investigate semi-classical generalizations of the Charlier and Meixner polynomials, which are discrete orthogonal polynomials that satisfy three-term recurrence relations. It is shown that the coefficients in these recurrence relations can be expressed in terms of Wronskians of modified Bessel functions and confluent hypergeometric functions, respectively for the generalized Charlier and generalized Meixner polynomials. These Wronskians arise in the description of special function solutions of the third and fifth Painlevé equations.
In our previous work, a unified description as polynomial Hamiltonian systems was established for a broad class of the Schlesinger systems including the sixth Painleve equation and Garnier systems. The main purpose of this paper is to present particular solutions of this Hamiltonian system in terms of a certain generalization of Gauss' hypergeometric function. Key ingredients of the argument are the linear Pfaffian system derived from an integral representation of the hypergeometric function (with the aid of twisted de Rham theory) and Lax formalism of the Hamiltonian system.
We investigate the structure of $τ$-functions for the elliptic difference Painlevé equation of type $E_8$. Introducing the notion of ORG $τ$-functions for the $E_8$ lattice, we construct some particular solutions which are expressed in terms of elliptic hypergeometric integrals. Also, we discuss how this construction is related to the framework of lattice $τ$-functions associated with the configuration of generic nine points in the projective plane.
Hypergeometric functions provide a useful representation of Feynman diagrams occuring in precision phenomenology. In dimension regularization, the epsilon-expansion of these functions about d=4 is required. We discuss the current status of differential reduction algorithms. As an illustration, we consider the construction of the all-order epsilon-expansion of the Appell hypergeometric function about integer values of the parameters and present an explicit evaluations of the first few terms.
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 ${}_3F_2(1)$. 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 relationship between Feynman diagrams and hypergeometric functions is discussed. Special attention is devoted to existing techniques for the construction of the $ε$-expansion. As an example, we present a detailed discussion of the construction of the epsilon-expansion of the Appell function $F_3$ around rational values of parameters via an iterative solution of differential equations. As a by-product, we have found that the one-loop massless pentagon diagram in dimension $d=3-2ε$ is not expressible in terms of multiple polylogarithms. Another interesting example is the Puiseux-type solutio…
A recently formulated conjecture of Gamayun, Iorgov and Lisovyy gives an asymptotic expansion of the Jimbo--Miwa--Ueno isomonodromic $τ$-function for certain Painlevé transcendents. The coefficients in this expansion are given in terms of conformal blocks of a two-dimensional conformal field theory, which can be written as infinite sums over pairs of partitions. In this note a discrete matrix model is proposed on a lattice whose partition function can be used to obtain a multiple integral representation for the length restricted partial sums of the Painlevé conformal blocks. This leads to expr…
