Generalized Hypergeometric Function 3F2 Ratios and Branched Continued Fraction Expansions

The paper is related to the classical problem of the rational approximation of analytic functions of one or several variables, particulary the issues that arise in the construction and studying of continued fraction expansions and their multidimensional generalizations—branched continued fraction expansions. We used combinations of three- and four-term recurrence relations of the generalized hypergeometric function 3F2 to construct the branched continued fraction expansions of the ratios of this function. We also used the concept of correspondence and the research method to extend convergence, already known for a small region, to a larger region. As a result, we have established some convergence criteria for the expansions mentioned above. It is proved that the branched continued fraction expansions converges to the functions that are an analytic continuation of the ratios mentioned above in some region. The constructed expansions can approximate the solutions of certain differential equations and analytic functions, which are represented by generalized hypergeometric function 3F2. To illustrate this, we have given a few numerical experiments at the end.

The Marichev-Saigo-Maeda Fractional Calculus Operators Pertaining to the V -Function

In the present paper, we establish some composition formulas for Marichev-Saigo-Maeda (MSM) fractional calculus operators with V -function as the kernel. In addition, on account of V -function, a variety of known results associated with special functions such as the Mittag-Leffler function, exponential function, Struve’s function, Lommel’s function, the Bessel function, Wright’s generalized Bessel function, and the generalized hypergeometric function have been discovered by defining suitable values for the parameters.

Integral transforms and probability distributions involving a generalized hypergeometric function

Various extensions of the beta function together with their associated extended hypergeometric and confluent hypergeometric functions have been introduced and investigated. In this paper, using the very recently contrived extended beta function, we aim to introduce an extension F v p , q ; λ ; σ , τ u {{}_{u}F_{v}^{p,q;\lambda;\sigma,\tau}} of the generalized hypergeometric function F v u {{}_{u}F_{v}} and investigate certain classes of transforms and several identities of a generalized probability distribution involving this extension. In fact, we present some interesting formulas of Jacobi, Gegenbauer, pathway, Laplace, and Legendre transforms of this extension multiplied by a polynomial. We also introduce a generalized probability distribution to investigate its several related properties. Further, we consider some special cases of our main results with an argument about the derived process of a known result.

Two variables generalized Laguerre polynomials

The objective of this paper is to introduce and study the generalized Laguerre polynomial for two variables. We prove that these polynomials are characterized by the generalized hypergeometric function. An explicit representation, generating functions and some recurrence relations are shown. Moreover, these polynomials appear as solutions of some differential equations.

Celestial double copy from the worldsheet

Using the ambitwistor string, we compute tree-level celestial amplitudes for biadjoint scalars, Yang-Mills and gravity to all multiplicities. They are presented in compact CHY-like formulas with operator-valued scattering equations and numerators acting on a generalized hypergeometric function. With these we extend the celestial double copy to tree-level amplitudes with arbitrary number of external states. We also show how color-kinematics duality is implemented in celestial amplitudes and its interpretation in terms of a generalized twisted cohomology theory.

On several new Laplace transforms of generalized hypergeometric functions 2F2(x)

By employing generalizations of Gauss's second, Bailey's and Kummer's summation theorems obtained earlier by Rakha and Rathie, we aim to establish unknown Laplace transform of six rather general formulas of generalized hypergeometric function 2F2[a,b;c,d;x]. The results obtained in this paper are simple, interesting, easily established and may be useful in theoretical physics, engineering and mathematics. Results obtained earlier by Kim et al. and Choi and Rathie follow special cases of our main findings.

The y function applied in the study of an anomalous diffusion

In this article, we propose a new family of the extended analogues to the Y function for the first time. The relationships among the Y function, Fox H function, Meijer G function, Wright generalized hypergeometric function, and Clausen hypergeometric function are discussed in detail. This result is used to represent the solutions for the anomalous diffusion problems.

Unified Approach to Fractional Calculus Images of Special Functions—A Survey

Evaluation of images of special functions under operators of fractional calculus has become a hot topic with hundreds of recently published papers. These are growing daily and we are able to comment here only on a few of them, including also some of the latest of 2019–2020, just for the purpose of illustrating our unified approach. Many authors are producing a flood of results for various operators of fractional order integration and differentiation and their generalizations of different special (and elementary) functions. This effect is natural because there are great varieties of special functions, respectively, of operators of (classical and generalized) fractional calculus, and thus, their combinations amount to a large number. As examples, we mentioned only two such operators from thousands of results found by a Google search. Most of the mentioned works use the same formal and standard procedures. Furthermore, in such results, often the originals and the images are special functions of different kinds, or the images are not recognized as known special functions, and thus are not easy to use. In this survey we present a unified approach to fulfill the mentioned task at once in a general setting and in a well visible form: for the operators of generalized fractional calculus (including also the classical operators of fractional calculus); and for all generalized hypergeometric functions such as pΨq and pFq, Fox H- and Meijer G-functions, thus incorporating wide classes of special functions. In this way, a great part of the results in the mentioned publications are well predicted and appear as very special cases of ours. The proposed general scheme is based on a few basic classical results (from the Bateman Project and works by Askey, Lavoie–Osler–Tremblay, etc.) combined with ideas and developments from more than 30 years of author’s research, and reflected in the cited recent works. The main idea is as follows: From one side, the operators considered by other authors are cases of generalized fractional calculus and so, are shown to be (m-times) compositions of weighted Riemann–Lioville, i.e., Erdélyi–Kober operators. On the other side, from each generalized hypergeometric function pΨq or pFq (p≤q or p=q+1) we can reach, from the final number of applications of such operators, one of the simplest cases where the classical results are known, for example: to 0Fq−p (hyper-Bessel functions, in particular trigonometric functions of order (q−p)), 0F0 (exponential function), or 1F0 (beta-distribution of form (1−z)αzβ). The final result, written explicitly, is that any GFC operator (of multiplicity m≥1) transforms a generalized hypergeometric function into the same kind of special function with indices p and q increased by m.