An integral representation of the Gauss hypergeometric functions and its applications

In this paper, we present two new integral representations for the generalized hypergeometric function obtained by employing Edwards's double integral. These results represent a generalization of those previously obtained by Chammam et al. [A study of two new integral representations for Gauss's hypergeometric function with applications. Afr Mat. 2025;36(29). doi: 10.1007/S13370-025-01239-3; A note on two new integral representations of the Gauss hypergeometric function with an application. Bull Belgian Math Soc. 2024;31(3):336–340. doi: 10.36045/j.bbms.231230.]. We also derive several results by applying the Dixon summation theorem and the Watson summation theorem. Furthermore, we introduce two additional generalizations of these integral forms. As a significant application of our results, we derive two novel integral representations for a series related to the Fuss–Catalan–Qi numbers. In addition, we establish two integral representations for a series involving the Catalan–Qi numbers, the Catalan numbers, the Super Catalan numbers and the Wallis ratio. The aim of this paper is to generalize the results obtained by Chammam et al. [A study of two new integral representations for Gauss's hypergeometric function with applications. Afr Mat. 2025;36(29). doi: 10.1007/S13370-025-01239-3; A note on two new integral representations of the Gauss hypergeometric function with an application. Bull Belgian Math Soc. 2024;31(3):336–340. doi: 10.36045/j.bbms.231230.], extending them beyond the Gauss hypergeometric function to include the generalized hypergeometric function. This provides a broader framework for applying integral representations in various mathematical contexts, particularly in applied mathematics, functional analysis, and the theory of special functions.

Contour integral representations of Riemann's Zeta function and Dirichlet's Eta (alternating Zeta) function are presented and investigated. These representations flow naturally from methods developed in the 1800s, but somehow they do not appear in the standard reference summaries, textbooks, or literature. Using these representations as a basis, alternate derivations of known series and integral representations for the Zeta and Eta function are obtained on a unified basis that differs from the textbook approach, and results are developed that appear to be new.

The problem of calculating the Mittag-Leffler function is considered in the paper. To solve this problem integral representations for the function are transformed in such a way that they could not contain complex variables and parameters. Integral representations written in this form allow one to use standard methods of numerical integration to calculate integrals contained in them. To verify the correctness of the integral representations obtained the function was calculated both with the use of obtained formulas and with the use of known representations of the Mittag-Leffler function. The calculation results demonstrate their exact matching. This fact is indicative of the correctness of new integral representations of the function that were obtained.

Integral representations play a prominent role in the analysis of entire functions. The representations of generalized Mittag-Leffler type functions and their asymptotics have been (and still are) investigated by plenty of authors in various conditions and cases.The present paper explores the integral representations of a special function extending to two variables the two-parametric Mittag-Leffler type function. Integral representations of this functions within different variation ranges of its arguments for certain values of the parameters are thus obtained. Asymptotic expansion formulas and asymptotic properties of this function are also established for large values of the variables. This yields corresponding theorems providing integral representations as well as expansion formulas.

Integral representations play a prominent role in the analysis of entire functions. The representations of generalized Mittag-Leffler type functions and their asymptotics have been (and still are) investigated by plenty of authors in various conditions and cases. The present paper explores the integral representations of a special function extending to two variables the two-parametric Mittag-Leffler type function. Integral representations of this functions within different variation ranges of its arguments for certain values of the parameters are thus obtained. Asymptotic expansion formulas and asymptotic properties of this function are also established for large values of the variables. This yields corresponding theorems providing integral representations as well as expansion formulas.

In this paper, by means of the Faà di Bruno formula, with the help of explicit formulas for partial Bell polynomials at specific arguments of two specific sequences generated by derivatives at the origin of the inverse sine and inverse cosine functions, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes power series expansions for real powers of the inverse cosine (sine) functions and the inverse hyperbolic cosine (sine) functions. By comparing different series expansions for the square of the inverse cosine function and for the positive integer power of the inverse sine function, the author not only finds infinite series representations of the circular constant π and its real powers, but also derives several combinatorial identities involving central binomial coefficients and the Stirling numbers of the first kind.

On Representations of Distance Functions in the Plane

The paper presents an integral representation of the two-parameter Mittag-Leffler function $E_{\rho,\mu}(z)$ and singular points of this representation have been studied. It has been found that there are two singular points for this integral representation: $\zeta=1$ and $\zeta=0$. The point $\zeta=1$ is a pole of the first order and the point $\zeta=0$, depending on the values of parameters $\rho,\mu$ is either a pole or a branch point, or a regular point. The subsequent study showed that at some values of parameters $\rho,\mu$ with the help of the residue theory one can calculate the integral included in the studied integral representation and express the function $E_{\rho,\mu}(z)$ through elementary functions.

Complete triple Hypergeometric functions of the second order which were denoted by . Each of these triple Hypergeometric functions has been investigated extensively in many different ways including for example in the problem of finding their integral representations of one kind or other. Here in this paper we aim at presenting further integral representations for each triple Hypergeometric functions .

In this paper, using the Lipschitz–Hankel identities we obtain some new integral representations for the toroidal functions in terms of the elementary, Bessel, parabolic cylinder and hypergeometric functions. Manipulating the integrands of Lipschitz–Hankel identities with several integral representations lead us to present the results. In this sense, we also derive some integral representations for the products of toroidal functions.

Various integral representations of hypergeometric functions have been introduced and investigated due to their important applications in divers fields. In this article, we define some new Euler-type integral representations for the Horn's functions of two variables G1, G2, G3 and H1.

We consider the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates and derive a functional analytic integral representation of the associated propagator using the spectral theorem for unbounded self-adjoint operators, Stone's formula, and quantities arising in the analysis of Chandrasekhar's separation of variables. This integral representation describes the dynamics of Dirac particles outside and across the event horizon, up to the Cauchy horizon. In the derivation, we first write the Dirac equation in Hamiltonian form and show the essential self-adjointness of the Hamiltonian. For the latter purpose, as the Dirac Hamiltonian fails to be elliptic at the event and the Cauchy horizon, we cannot use standard elliptic methods of proof. Instead, we employ a new, general method for mixed initial-boundary value problems that combines results from the theory of symmetric hyperbolic systems with near-boundary elliptic methods. In this regard and since the time evolution may not be unitary because of Dirac particles impinging on the ring singularity, we also impose a suitable Dirichlet-type boundary condition on a time-like inner hypersurface placed inside the Cauchy horizon, which has no effect on the dynamics outside the Cauchy horizon. We then compute the resolvent of the Dirac Hamiltonian via the projector onto a finite-dimensional, invariant spectral eigenspace of the angular operator and the radial Green's matrix stemming from Chandrasekhar's separation of variables. Applying Stone's formula to the spectral measure of the Hamiltonian in the spectral decomposition of the Dirac propagator, that is, by expressing the spectral measure in terms of this resolvent, we obtain an explicit integral representation of the propagator.

Integral representation formulas play an essential role in the modern function theory. They serve to solve boundary value problems for differential equations. As an example of such integral representations is the Cauchy formula for analytic functions but it, thus, is just a special case of the Cauchy–Pompieu formula. Higher Cauchy–Pompieu formulas in the complex, hypercomplex and Clifford analysis have been presented by iteration in [Begehr, H., 2000, Integral representations for differentiable functions. Ricci, P. E. (Ed.), Current Problems in Analysis and Mathematical Physics. Papers of the 2nd international symposium dedicated to the memory of Prof. Gaetano Fichera (1922–1996). (Roma: Dipartimento di Matematica, Universita di Roma), 111–130, Begehr, H., 2002, Integral representations in complex, hypercomplex and Clifford analysis. Integral Transforms and Special Functions, 13(3), 223–241]. In this paper, certain special cases of higher order Cauchy–Pompeiu integral representations are given.

Looking for a quantum field theory model of Archimedean algebraic geometry a class of infinite-dimensional integral representations of classical special functions was introduced. Precisely the special functions such as Whittaker functions and Gamma-function were identified with correlation functions in topological field theories on a two-dimensional disk. Mirror symmetry of the underlying topological field theory leads to a dual finite-dimensional integral representations reproducing classical integral representations for the corresponding special functions. The mirror symmetry interchanging infinite- and finite-dimensional integral representations provides an incarnation of the local Archimedean Langlands duality on the level of classical special functions. In this note we provide some directions to higher-dimensional generalizations of our previous results. In the first part we consider topological field theory representations of multiple local L-factors introduced by Kurokawa and expressed through multiple Barnes's Gamma-functions. In the second part we are dealing with generalizations based on consideration of topological Yang-Mills theories on non-compact four-dimensional manifolds. Presumably, in analogy with the mirror duality in two-dimensions, S-dual description should be instrumental for deriving integral representations for a particular class of quantum field theory correlation functions and thus providing a new interesting class of special functions supplied with canonical integral representations.

We propose to define the Horn's double hypergeometric functions H3 and H4 of matrix arguments and deduce some integral representations for these two functions. Utilizing the first author's definitions (Upadhyaya, Lalit Mohan and Dhami, H.S., Matrix generalizations of multiple hypergeometric functions; #1818, Nov.2001, IMA Preprint Series, University of Minnesota, Minneapolis, U.S.A. (Retrieved from the University of Minnesota Digital Conservancy, http://hdl.handle.net/11299/3706); Upadhyaya, Lalit Mohan, Matrix Generalizations of Multiple Hypergeometric Functions by Using Mathai's Matrix Transform Techniques (Ph.D. Thesis, Kumaun University, Nainital, Uttarakhand, India), #1943, Nov. 2003, IMA Preprint Series, University of Minnesota, Minneapolis, U.S.A. (https://www.ima.umn.edu/sites/default/files/1943.pdfhttp://www.ima.umn.edu/preprints/abstracts/1943ab.pdfhttp://www.ima.umn.edu/preprints/nov2003/1943.pdf http://hdl.handle.net/11299/3955https://zbmath.org/?q=an:1254.33008http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.192.2172\r Choi, Junesang, Hasanov, Anvar and Turaev, Mamasali, Integral representations for Srivastava's hyper-geometric function HA, Honam Mathematical J., Vol. 34, No. 1, (2012), 113–124: http://dx.doi.org/10.5831/HMJ.2012.34.1.113; Choi, Junesang, Hasanov, Anvar and Turaev, Mamasali, Decomposition formulas and integral representations for some Exton hypergeometric functions, Journal of the Chungcheong Mathematical Society., Vol. 24, No. 4 (December 2011), (2011), 745–758) for these two of the Horn's double and the Sri-vastava's triple hypergeometric functions. For proving our results for these functions of matrix arguments we invoke the Mathai's matrix transform technique for real symmetric positive definite matrices as arguments. We conclude by stating the corresponding parallel results for these Horn's double and the Srivastava's triple hypergeometric functions, when their argument matrices are complex Hermitian positive definite, with the remark that these parallel results can be easily proved by following our given lines of proofs and by employing the corresponding known results available in the literature.