No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
The Special Functions and Their Approximations
... 11,12 A collection of polynomials like Legendre, Chebyshev, and other spherical polynomials on [−1,1] are solutions to the above problem, and these polynomials are generated from the Jacobi polynomials by assigning particular values to the parameters of Jacobi polynomials. In the case of [0,1], we use the shifted version of two-dimensional (2D) Legendre polynomials called two-diemnsional shifted Legendre polynomials (2DSLPs) 30 to find an approximate solution of (9) numerically. ...
... Other related results are available in Lukes and colleagues. 30,34,35 To begin, we define the Brownian motion, a typical illustration of a stochastic process. The probability space P (Ω, , ) is to be constructed on the collection of continuous real-valued functions on R + starting at 0. Next, we provide the idea of Hilbert space and Banach space where the concept of defining a norm has been established in the probability space P (Ω, , ). ...
... The Legendre polynomials P z ( ) n are the solutions of Legendre's differential equation. 30 The SLPs are derived from P z ( ) ...
The deflection of Euler–Bernoulli beams under stochastic dynamic loading, exhibiting purely viscous behavior, is characterized by partial differential equations of the fourth order. This paper proposes a computational method to determine the approximate solution to such equations. The functions are approximated using two‐dimensional shifted Legendre polynomials. An operational matrix of integration and an operational matrix of stochastic integration are derived. The operational matrices assist in breaking down the problem under consideration into a set of algebraic equations that may be solved using any known numerical technique that leads to the solution of the stochastic beam equation. The well‐posedness of the problem is studied. The proposed methodology is demonstrated to be practical for addressing the novel stochastic dynamic loading problem by confirming the outcome using a few numerical examples. Thus the effectiveness and applicability of the technique are ensured. The solution quality is explored through diagrams. The accuracy of the method is substantiated by comparing it with the Runge–Kutta method of order 1.5 (R–K 1.5). The absolute error caused by the proposed technique is comparably much less than R–K 1.5. A simulation analysis is carried out with MATLAB, and an algorithm is developed.
... Proof. The following result provides the large parameter asymptotic behaviour of a 1 F 2 of the same style as the first term in (31), it is adapted from Luke (1969) ...
... where m − b = 0, 1, 2 . . . . The next result is adapted from Luke (1969) 7.3(8) and provides the large parameter asymptotic behaviour of a 1 F 2 of the same style as the second term in (31) ...
... The following identity is taken from Luke (1969) ...
This work provides theoretical foundations for kernel methods in the hyperspherical context. Specifically, we characterise the native spaces (reproducing kernel Hilbert spaces) and the Sobolev spaces associated with kernels defined over hyperspheres. Our results have direct consequences for kernel cubature, determining the rate of convergence of the worst case error, and expanding the applicability of cubature algorithms based on Stein's method. We first introduce a suitable characterisation on Sobolev spaces on the $d$-dimensional hypersphere embedded in $(d+1)$-dimensional Euclidean spaces. Our characterisation is based on the Fourier--Schoenberg sequences associated with a given kernel. Such sequences are hard (if not impossible) to compute analytically on $d$-dimensional spheres, but often feasible over Hilbert spheres. We circumvent this problem by finding a projection operator that allows to Fourier mapping from Hilbert into finite dimensional hyperspheres. We illustrate our findings through some parametric families of kernels.
... In this paper, we use the shifted version of 2D Legendre polynomials called 2D shifted Legendre polynomials [22] (2DSLP) to find an approximate solution of our problem under consideration. The orthogonal property of this polynomial produces triangular, tri-diagonal and diagonal matrices at different instances. ...
... Basic definitions of stochastic calculus [9,22,24] which are required for our further study has been highlighted in this section. Also, Table 1 shows the nomenclatures used throughout this paper. ...
... The Legendre polynomials P n (z) are the solutions of Legendre's Differential Equation [22]. The orthogonal property of Legendre polynomials is defined as ...
This paper deals with the study of a class of stochastic heat equation using the operational matrix of integration and stochastic integration based on two-dimensional shifted Legendre polynomials. The characteristics of these operational matrices together with the properties of shifted Legendre polynomials convert the stochastic heat equation into a system of algebraic equations. A theoretical analysis is also carried out to ensure the convergence of the proposed approximation technique. It has also been proved that the error function reduces to zero for larger values of the parameters involved. The applicability, validity, accuracy and efficiency of the proposed technique is tested with some numerical examples and the solution quality can be realized through various figures. The error curve is plotted, which justifies that the fluctuations in error fall within the error bound discussed in the theoretical analysis.
... Definition 1. The beta function (also called the Euler's integral of the first kind) is defined as (see [11,13]): ...
... Remark 2. By setting x = y = 1, p = q = 0 and m = 1 in (9), we get the Euler Beta function (1) (see [11], [13]) ...
The primary object of this article is to introduce (p, q)-beta logarithmic function with extended beta function by using the logarithmic mean. We evaluate different properties and representations of beta logarithmic function. Further, it is evaluated logarithmic distribution, hypergeometric and confluent hypergeometric functions via logarithmic mean are evaluated and their essential properties are studied. Numerous formulas of (p, q)-beta logarithmic functions such as integral formula, derivative formula, transformation formula and generating function are analyzed.
... The JPs with the real parameters (ζ > −1, η > −1) (see, Luke [27] and Szegö [28]), are a sequence of polynomials J (ζ , η) n (z)(n = 0, 1, 2, ...), constitute an orthogonal system concerning the weight function w (ζ , η) (z) = (1 − z) ζ (z + 1) η , that is, ...
... where in this context, let λ = ζ + η + 1, and (·) i denotes the Pochhammer symbol. Refer to [27] for the definition of generalized hypergeometric functions, including the special 3 F 2 . ...
This study explores the spectral Galerkin approach to solving the space-time Schrödinger, wave, Airy, and beam equations. In order to facilitate the creation of a semi-analytical approximation solution, it uses polynomial bases that are formed from a linear combination of Jacobi polynomials (JPs) in both spatial and temporal dimensions. By using these polynomials to expand the exact solution, the paper hopes to satisfy the homogeneous starting and Dirichlet boundary requirements. Notably, the Jacobi Galerkin (JG) method exhibits exponential convergence rates if the solution is sufficiently smooth. This result emphasizes the JG approach's potential as an effective numerical solution method, which has promise for a variety of applications in other domains where these equations occur, such as quantum mechanics, acoustics, optics, and structural mechanics.
... (see http://functions.wolfram.com/07.23.26.0007.01). Also, on combining equations 5.4 (1) and 5.4 (13) of [14], we obtain the indefinite integral formula ...
... In this appendix, we define the modified Bessel function of the second kind, the Gaussian hypergeometric function and the Meijer G-function, and state some basic properties that are needed in this paper. Unless otherwise stated, these and further properties can be found in the standard references [9,14,21]. ...
... With this assumption, using the following identities [40,41]: ...
... This property allows us to rearrange the terms in Eqs. (40) and (41) and use the following expressions for the wave functions: ...
In this paper, we present a theoretical perspective concerning the scattering of electrons on a twisted light (TL) driven graphene quantum dot (GQD). Relatively recently discovered, TL is a novel type of electromagnetic field which carries a finite orbital angular momentum oriented on the propagation direction, besides its spin. This striking property of TL is due to its spatial structure. It is well known that the localization of electrons in a GQD is forbidden by the Klein tunneling, an effect that manifests by the perfect transmission of electrons through a potential barrier, regardless of its magnitude. Here we demonstrate that, for a suitable choice of the scattering regimes, there emerge scattering resonances characterized by trapping states of the incident electron inside the GQD for finite periods of time. The most interesting result is the prediction regarding the possibility to control the trapping times using a TL irradiation. Also, we mention that the investigation was performed for a frequency of the TL within the infrared spectrum.
... where 2 F 2 denotes the generalized hypergeometric function p F q with p = q = 2, and the details can be seen in Luke [29] or Zörnig [30]. This result will be used to obtain the cdf of a SWL model. ...
... It can be seen in Luke [29] that, using the expression of the pdf of Z given in (4) and applying (3), the result proposed is obtained. ...
The slash-weighted Lindley model is introduced due to the need to obtain a model with more kurtosis than the weighted Lindley distribution. Several expressions for the pdf of this model are given. Its cumulative distribution function is expressed in terms of a generalized hypergeometric function and the incomplete gamma function. Moments and maximum likelihood estimation were studied. A simulation study was carried out to illustrate the good performance of the estimates. Finally, two real applications are included.
... a hypergeometric function that may also be used to recover asymptotics of u n , see [15,Sect. 5.11.2] for asymptotic expansions of generalized hypergeometric functions. Indeed, the leading terms of an asymptotic expansion of U (z), provided by Maple, together with Depoissonization via the saddle point method, yield an alternative proof of Theorem 2.1. ...
... where J 0 and J 1 are Bessel functions of the first kind. We may use D(z) to recover asymptotics of d n , see [15,Sect. 5.11.4] for asymptotic expansions of Bessel functions. We find ...
Consider Young diagrams of n boxes distributed according to the Plancherel measure. So those diagrams could be the output of the RSK algorithm, when applied to random permutations of the set $$\{1,\ldots ,n\}$$ { 1 , … , n } . Here we are interested in asymptotics, as $$n\rightarrow \infty $$ n → ∞ , of expectations of certain functions of random Young diagrams, such as the number of bumping steps of the RSK algorithm that leads to that diagram, the side length of its Durfee square, or the logarithm of its probability. We can express these functions in terms of hook lengths or contents of the boxes of the diagram, which opens the door for application of known polynomiality results for Plancherel averages. We thus obtain representations of expectations as binomial convolutions, that can be further analyzed with the help of Rice’s integral or Poisson generating functions. Among our results is a very explicit expression for the constant appearing in the almost equipartition property of the Plancherel measure.
... Before stating Theorem 4.12, we recall the definitions of the hypergeometric series and the Meijer Gfunction. For r, s ∈ N, the (generalised) hypergeometric series (see [1,12,22]) is defined by ...
... For r ∈ Z + , the Meijer G-function G r,0 0,r (see [12,22] for more details and for information on the general G m,n s,r ) is defined by the Mellin-Barnes type integral ...
In the two main results presented here we give representations as products of bidiagonal matrices for the production matrices of certain generating polynomials of lattice paths and we show that those production matrices are the Hessenberg matrices associated with the components of the decomposition of symmetric multiple orthogonal polynomials. As a consequence of these results, we can use the recently found connection between lattice paths, branched continued fractions, and multiple orthogonal polynomials to analyse symmetric multiple orthogonal polynomials. We revisit known results about the location of the zeros of symmetric multiple orthogonal polynomials, we find combinatorial interpretations for the moments of the dual sequence of a symmetric multiple orthogonal polynomial sequence, we give an explicit proof of the existence of orthogonality measures supported on a starlike set for symmetric multiple orthogonal polynomials with positive recurrence coefficients as well as orthogonality measures on the positive real line for the components of the decomposition of these symmetric multiple orthogonal polynomials, and we give explicit formulas as terminating hypergeometric series for Appell sequences of symmetric multiple orthogonal polynomials with respect to an arbitrary number of measures whose densities can be expressed via Meijer G-functions.
... The first-and nth-order derivative formulas of I p (ε) are, respectively [36]: ...
... where 2 F 3 (., ., .) is the hypergeometric function defined by [36]: ...
In this paper, we establish a new auxiliary identity of the Bullen type for twice-differentiable functions in terms of fractional integral operators. Based on this new identity, some generalized Bullen-type inequalities are obtained by employing convexity properties. Concrete examples are given to illustrate the results, and the correctness is confirmed by graphical analysis. An analysis is provided on the estimations of bounds. According to calculations, improved Hölder and power mean inequalities give better upper-bound results than classical inequalities. Lastly, some applications to quadrature rules, modified Bessel functions and digamma functions are provided as well.
... Here when 0 < β < 1, and 0 < α < 1, the equation has solutions for the initial value problem of free particle diffusion under Cauchy boundary conditions that are expressed in terms of the Fox H-functions [33]. Graphically, the domain of these solutions is displayed as an anomalous diffusion phase diagram {β, α}-see Figure 2-in which the Gaussian solution corresponds to the point, {β = 1, α = 2}, and the mean squared displacement falls on the line, {α = 2β}; hence x 2 = 2D β,α t 2β/α , where D β,α has the units m α /s β . ...
... As an example, in Figure 6 we have superimposed a portion of Figure 2 (e.g., the {β, α} plane for fractional time and space diffusion with a constant coefficient) onto the phase diagram of pure water, Figure 4 (e.g., from the triple point, tp, to critical point, cp,). Taking inspiration from the Clausius-Clapeyron equation, we normalize the pressure P and temperature T using: α = 2 log 10 (P/P tp )/log 10 (P cp /P tp ) and β = T − T tp / T cp − T tp (33) Log Pressure (atm) The overlap divides the liquid and vapor phases of water into regions of sub-and super-diffusion, respectively, and suggests that beyond the critical point the supercritical fluid phase corresponds with the domain of Brownian motion in the mathematical phase diagram. This provisional or inchoate connection also appears to be valid for the simple gases listed in Tables 1 and 3. ...
The application of fractional calculus in the field of kinetic theory begins with questions raised by Bernoulli, Clausius, and Maxwell about the motion of molecules in gases and liquids. Causality, locality, and determinism underly the early work, which led to the development of statistical mechanics by Boltzmann, Gibbs, Enskog, and Chapman. However, memory and nonlocality influence the future course of molecular interactions (e.g., persistence of velocity and inelastic collisions); hence, modifications to the thermodynamic equations of state, the non-equilibrium transport equations, and the dynamics of phase transitions are needed to explain experimental measurements. In these situations, the inclusion of space- and time-fractional derivatives within the context of the continuous time random walk (CTRW) model of diffusion encodes particle jumps and trapping. Thus, we anticipate using fractional calculus to extend the classical equations of diffusion. The solutions obtained illuminate the structure and dynamics of the materials (gases and liquids) at the molecular, mesoscopic, and macroscopic time/length scales. The development of these models requires building connections between kinetic theory, physical chemistry, and applied mathematics. In this paper, we focus on the kinetic theory of gases and liquids, with particular emphasis on descriptions of phase transitions, inter-phase mixing, and the transport of mass, momentum, and energy. As an example, we combine the pressure–temperature phase diagrams of simple molecules with the corresponding anomalous diffusion phase diagram of fractional calculus. The overlap suggests links between sub- and super-diffusion and molecular motion in the liquid and the vapor phases.
... Here, it is important to remember that the generalized hypergeometric function is defined as Luke [23] where b j ≠ 0 , for all 1 ≤ j ≤ q. ...
This paper's major goal is to provide a numerical approach for estimating solutions to a coupled system of convection-diffusion equations with Robin boundary conditions (RBCs). We devised a novel method that used four homogeneous RBCs to generate basis functions using generalized shifted Legendre polynomials (GSLPs) that satisfy these RBCs. We provide new operational matrices for the derivatives of the developed polynomials. The collocation approach and these operational matrices are utilized to find approximate solutions for the system under consideration. The given system subject to RBCs is turned into a set of algebraic equations that can be solved using any suitable numerical approach utilizing this technique. Theoretical convergence and error estimates are investigated. In conclusion, we provide three illustrative examples to demonstrate the practical implementation of the theoretical study we have just presented, highlighting the validity, usefulness, and applicability of the developed approach. The computed numerical results are compared to those obtained by other approaches. The methodology used in this study demonstrates a high level of concordance between approximate and exact solutions, as shown in the presented tables and figures.
... Now, we retrieve the classical beta function denoted by B(ζ, η) (see [8,14]) and it is defined by ...
Recently, a large number of beta type integral operators and their extensions have been developed and explored. This activity has been prompted by the significance of these operators as well as the possible uses they may have in a range of study domains. We start a new class of extended (p, q) type integral operators employing the generalized Mittag-Leffler function that was described by Khan et al. [6]. In addition, our findings are coherent in character and may be interpreted as fundamental equations, from which we have also derived a number of special cases.
... Alternatively, the power series expansions of the Γ-function (see [20] and [25] (Section 2.10)) are given by ...
By computing definite integrals, we shall examine binomial series of convergence rate ±1/2 and weighted by harmonic-like numbers. Several closed formulae in terms of the Riemann and Hurwitz zeta functions as well as logarithm and polylogarithm functions will be established, including a conjectured one made recently by Z.-W. Sun.
... The parameters a j (j = 1, · · · , n) and b k (k = 1, · · · , m) are such that no poles of ∏ n j=1 Γ(b k − s) coincide with poles of ∏ n j=1 Γ(1 − a j + s). Refer to the study by [10] for details about the contour L and about the convergence conditions of the integral. ...
An exact expression for R = P(X < Y) has been obtained when X and Y are independent and follow Birnbaum-Saunders (BS) distributions. Using some special functions, it was possible to express R analytically with minimal parameter restrictions. Monte Carlo simulations and two applications considering real datasets were carried out to show the performance of the BS models in reliability scenarios. The new expressions are accurate and easy to use, making the results of interest to practitioners using BS models.
... There is a relation [21]: ...
The object of this study is underground gas storage facilities (UGSF). The main problem being solved is to ensure effective management of the operation process of underground gas storage facilities (UGSF) at operational and forecast time intervals. One of the main factors that affect the operating modes of UGSF is significantly non-stationary filtration processes that take place in the bottomhole zones of wells. The complexity of assessing the multifactorial impact on depression/repression around the wells affects both the speed and accuracy of calculating the mode parameters of UGSF operation. Analysis of the results of well studies revealed a significant area of uncertainty in the calculation of the filtration resistance coefficients of their bottomhole zones. A satisfactory accuracy of the result in the expected time was achieved by building a model of integrated consideration of the influence of the parameters of all the bottomhole zones of the wells on the mode of UGSF operation. It turned out that the integrated consideration of the impact on the parameters of the bottomhole zones of the wells neutralized the effect of significant changes in the filtration resistance coefficients of the wells and ensured a sufficient speed of calculation of UGSF operation modes. Simultaneous simulation of ten operating UGSFs under the peak mode of withdrawal the entire available volume of active gas takes no more than six minutes. The speed of simulation of filtration processes in the bottomhole zones of wells ensured finding the best of them according to one or another criterion of operation mode quality. As a result of the research, a model was built and implemented by software, which was tested under real operating conditions and provides optimal planning of UGSF operating modes for given time intervals. Its use is an effective tool for the operational calculation of current modes and technical capacity of UGSF for a given pressure distribution in the system of main gas pipelines. The performance of the constructed mathematical methods has been confirmed by the results of numerical experiments
... b q and the argument z in terms of other special or elementary functions is of great interest. These reduction formulas are found in several compilations of the existing literature, such as those provided by Luke [3] (Sections 6.2 and 6.3) and Prudnikov et al. [4] (Chapter 7). A revision, as well as an extension of the tables presented by Prudnikov et al., was carried out by Krupnikov and Kölbig in [5]. ...
Herein, we calculate reduction formulas for some generalized hypergeometric functions m+1Fmz in terms of elementary functions as well as incomplete beta functions. For this purpose, we calculate the n-th order derivative of the function zγBzα,β with respect to z. As corollaries, we obtain reduction formulas of these m+1Fmz functions for argument unity in terms of elementary functions, as well as beta functions.
... where J 1/2 (t) and J − 1/2 (t) refer to the ordinary Bessel functions of the rst kind of orders ½ and -½, respectively. The three pairs of functions in (1,2), (3,4) and (5,6) are related to each other by [29] (7), ...
We present numerical algorithms for efficient and highly accurate computations of Fresnel’s sine and cosine integrals \(S\left(z\right)\) and \(C\left(z\right)\)for real (\(z=x\)) and complex (\(z=x+iy\)) arguments. The algorithms are based on series expansion for small values of | z |, expansion in Chebyshev subinterval polynomial approximations for intermediate real values (\(y=0\)), toggled with sum of half-integer order Bessel functions for complex arguments, together with asymptotic series expansion for large values of | z |. The present algorithms, implemented in a Fortran elemental module, can be run using any of the single , double or quadruple precision arithmetic. Results from the present code have been benchmarked versus comprehensive data tables, generated using Maple , Mathematica and Matlab symbolic toolbox . Compared to a maximum of 16 significant digits in the literature, the present algorithms can calculate \(S\left(x\right)\) and C(x) up to 28 significant digits in the range \(xϵ[0, {10}^{6}]\) and \(S\left(z\right)\) and C(z) to the same accuracy in the domain where \(\left|S\right(z\left)\right|\) and \(\left|C\left(z\right)\right|\) each is less than the largest finite floating point number in the precision under consideration with \(\left|arg z\right|<\frac{\pi }{2}\) or more specifically \(3\times {10}^{-4}<\left|\frac{y}{x}\right|<{2\times 10}^{3}\) for \(S\left(z\right)\) and \({4\times 10}^{-4}<\left|\frac{y}{x}\right|<{2.2\times 10}^{3}\) for \(C\left(z\right)\).
... Here, it is important to remember that the generalized hypergeometric function is defined as [26] where b j ≠ 0 , for all 1 ≤ j ≤ q. ...
The main aim of the current paper is to construct a numerical algorithm for the numerical solutions of second-order linear and nonlinear differential equations subject to Robin boundary conditions. A basis function in terms of the shifted Chebyshev polynomials of the first kind that satisfy the homogeneous Robin boundary conditions is constructed. It has established operational matrices for derivatives of the constructed polynomials. The obtained solutions are spectral and are consequences of the application of collocation method. This method converts the problem governed by their boundary conditions into systems of linear or nonlinear algebraic equations, which can be solved by any convenient numerical solver. The theoretical convergence and error estimates are discussed. Finally, we support the presented theoretical study by presenting seven examples to ensure the accuracy, efficiency, and applicability of the constructed algorithm. The obtained numerical results are compared with the exact solutions and results from other methods. The method produces highly accurate agreement between the approximate and exact solutions, which are displayed in tables and figures.
... Proofs. To prove our results in Theorem 4.1, we require the following recurrence relations of the confluent function 1 1 F (Luke, Y. L., 1969): ...
The main object of this paper is to introduce an extension of Exton's quadruple
hypergeometric function K14 by using the extended Euler's beta function obtained earlier by Özergin, Özarslanand and Altin (2011). For this extended function, we investigate various properties such as integral representations, recurrence relations, generating functions, transformation formulas and summation formulas. Some special cases of the main results of this paper are also considered.
... in [1] and also as Exercise 9 in [4, Chapter 2]. The proof of the second formula appears in Section 4.7 of [16]. ...
The second virial coefficient for the Mie potential is evaluated using the method of brackets. This method converts a definite integral into a series in the parameters of the problem, in this case this is the temperature $T$. The results obtained here are consistent with some known special cases, such as the Lenard-Jones potential. The asymptotic properties of the second virial coefficient in molecular thermodynamic systems and complex fluid modeling are described in the limiting cases of $T \rightarrow 0$ and $T \rightarrow \infty$.
... By using ( [18,Eq. (14), p. 21]), and ...
... and the following standard estimate (see [8]): ...
We study sub-Hilbert spaces in Fock–Sobolev spaces of entire functions on n-dimensional complex space Cn. These sub-Hilbert spaces arise as the action of certain Toeplitz operators on the Fock–Sobolev space. Our approach is based on analyzing the asymptotic behavior of the reproducing kernel of the required sub-Hilbert space.
... Here and throughout the paper we will use the standard symbol p F q (a; b; z) for the generalized hypergeometric function with parameter vectors a ∈ C p , b ∈ C q \{0, −1, . . . }, see [2, Section 2.1], [24,Section 5.1], [32, for precise definitions and details. We will omit the argument z = 1 from the above notation throughout the paper. ...
... from Equations (2.19a) in [24], (9.31.3) in [25] and Equations (5.11.1), (6.2.11.1) and (6.2.11.2) in [26], and (6.36) in [27]. ...
Using a contour integral method, the de Montmort-Prudnikov sum is extended to derive a new series representation involving the incomplete gamma function. The series is uniformly convergent and completely analytical, which can be evaluated for general complex ranges of the parameters involved. Applications and evaluations of this formula are discussed.
... Hypergeometric functions and their multivariate analogs are well studied objects in mathematics. The classical references include Érdelyi [1], Luke [2], Bailey [3], Slater [4] just to mention few. A very nice survey article about multivariate hypergeometric function of "Appell's type" was written by M. Schlosser in [5]. ...
We are going to study properties of "hypergeometrization" -- an operator which act on analytic functions near the origin by inserting two Pochhammer symbols into their Taylor series. In essence, this operator maps elementary function into hypergeometric. The main goal is to produce number of "change of variable" formulas for this operator which, in turn, can be used to derive great number of transform for multivariate hypergeometric functions.
This paper shows that certain 3F4 hypergeometric functions can be expanded in sums of pair products of 2F3 functions, which reduce in special cases to 2F3 functions expanded in sums of pair products of 1F2 functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, 2F1 functions, and 3F2 functions into the realm of pFq functions where p<q for both the summand and terms in the series. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation.
In this research note, we have obtained some new classical summation relations of certain Appell's double hypergeometric functions associated with theory of approximation. This novel relation include, as special case, a set of well-known results. It studies the famous works of the authors [1], [2], [14], [15], [33]-[38].
A proposed method for finding an approximate solution of the nonlinear ordinary differential equations two-point boundary value problem is proposed. It simplifies the problem approximately to a problem of solving a set of nonlinear algebraic equations. The basic idea of the method is to utilize the properties of orthogonal polynomials and the approximate operational matrices of the nonlinear functionalf(x(t), u(t), t), and also the transformation matrix between the back vector and the current time vector for the general orthogonal polynomials. A method for solving the nonlinear two-point boundary value problems for descriptor systems is also given.
We present numerical algorithms for efficient and highly accurate computations of Fresnel’s sine and cosine integrals S ( z ) and C ( z ) for real ( z = x ) and complex ( z = x + iy ) arguments. The algorithms are based on series expansion for small values of | z |, expansion in Chebyshev subinterval polynomial approximations for intermediate real values ( y = 0), toggled with sum of half-integer order Bessel functions for complex arguments, together with asymptotic series expansion for large values of | z |. The present algorithms, implemented in a Fortran elemental module, can be run using any of the single , double , or quadruple precision arithmetic. Results from the present code have been benchmarked versus comprehensive data tables, generated using Maple , Mathematica , and Matlab symbolic toolbox . Compared to a maximum of 16 significant digits in the literature, the present algorithms can calculate S ( x ) and C ( x ) up to 28 significant digits in the range xϵ [0, 10 ⁶ ] and S ( z ) and C ( z ) to the same accuracy in the domain where | S ( z )| and | C ( z )| each is less than the largest finite floating point number in the precision under consideration with $$\mid \mathit{\arg}\ z\mid <\frac{\pi }{2}$$ ∣ arg z ∣ < π 2 or more specifically $$3\times {10}^{-4}<\left|\frac{y}{x}\right|<2\times {10}^3$$ 3 × 10 - 4 < y x < 2 × 10 3 for S ( z ) and $$4\times {10}^{-4}<\left|\frac{y}{x}\right|<2.2\times {10}^3$$ 4 × 10 - 4 < y x < 2.2 × 10 3 for C ( z ).
A bstract
We employ the axiomatic framework of Rychkov and Tan to investigate the critical O( N ) vector model with a line defect in (4 − ϵ ) dimensions. We assume the fixed point is described by defect conformal field theory and show that the critical value of the defect coupling to the bulk field is uniquely fixed without resorting to diagrammatic calculations. We also study various defect localized operators by the axiomatic method, where the analyticity of correlation functions plays a crucial role in determining the conformal dimensions of defect composite operators. In all cases, including operators with operator mixing, we reproduce the leading anomalous dimensions obtained by perturbative calculations.
Sun, in 2022, introduced a conjectured evaluation for a series of convergence rate $\frac{1}{2}$ involving harmonic numbers. We prove both this conjecture and a stronger version of this conjecture, using a summation technique based on a beta-type integral we had previously introduced. Our full proof also requires applications of Bailey's ${}_{2}F_{1}\left( \frac{1}{2} \right)$-formula, Dixon's ${}_{3}F_{2}(1)$-formula, an almost-poised version of Dixon's formula due to Chu, Watson's formula for ${}_{3}F_{2}(1)$-series, the Gauss summation theorem, Euler's formula for ${}_{2}F_{1}$-series, elliptic integral singular values, and lemniscate-like constants recently introduced by Campbell and Chu. The techniques involved in our proof are useful, more broadly, in the reduction of difficult sums of convergence rate $\frac{1}{2}$ to previously evaluable expressions.
Named essentially after their close relationship with the modified Bessel function Kν(z) of the second kind, which is known also as the Macdonald function (or, with a slightly different definition, the Basset function), the so-called Bessel polynomials yn(x) and the generalized Bessel polynomials yn(x;α,β) stemmed naturally in some systematic investigations of the classical wave equation in spherical polar coordinates. Our main purpose in this invited survey-cum-expository review article is to present an introductory overview of the Bessel polynomials yn(x) and the generalized Bessel polynomials yn(x;α,β) involving the asymmetric parameters α and β. Each of these polynomial systems, as well as their reversed forms θn(x) and θn(x;α,β), has been widely and extensively investigated and applied in the existing literature on the subject. We also briefly consider some recent developments based upon the basic (or quantum or q-) extensions of the Bessel polynomials. Several general families of hypergeometric polynomials, which are actually the truncated or terminating forms of the series representing the generalized hypergeometric function rFs with r symmetric numerator parameters and s symmetric denominator parameters, are also investigated, together with the corresponding basic (or quantum or q-) hypergeometric functions and the basic (or quantum or q-) hypergeometric polynomials associated with rΦs which also involves r symmetric numerator parameters and s symmetric denominator parameters.
In this paper we use a contour integral method to derive a bilateral generating function in the form of a double series involving Chebyshev polynomials expressed in terms of the incomplete gamma function. Generating functions for the Chebyshev polynomial are also derived and summarized. Special cases are evaluated in terms of composite forms of both Chebyshev polynomials and the incomplete gamma function.
In this study, we propose an approximate solution based on two‐dimensional shifted Legendre polynomials to solve nonlinear stochastic partial differential equations with variable coefficients. For this purpose, we have considered a Fisher‐Kolmogorov‐Petrovsky‐Piskunov (Fisher–KPP) equation with space uniform white noise for the same. New stochastic operational matrix of integration based on shifted Legendre polynomials is generated. This operational matrix reduces the problem under study into solving a system of algebraic equations. The convergence analysis is discussed, and the error bound in L2$$ {L}^2 $$ norm is derived and obtained as eN(t,x)2≤6K(16N4)2(1−6S)$$ {\left\Vert {e}_N\left(t,x\right)\right\Vert}^2\le \frac{6K}{{\left(16{N}^4\right)}^2\left(1-6S\right)} $$. The theoretical analysis confirms that as the degree of the approximation polynomial is increased, the solution is on par with the exact solution. The numerical examples confirm the accuracy and the applicability of the proposed method. A comparative study is carried out with explicit 1.5 order Runge–Kutta method. The time complexity of the proposed technique is also studied.
Alternative Legendre polynomials (ALPs) are used for approximating the solution of nonlinear Volterra-Fredholm integral equations (VFIEs) of the rst kind. The problem is rst transformed into a second kind VFIE and then, using the ALP operational matrices of integration and the product, the solution of the later problem is reduced to the one of solving a system of nonlinear algebraic equations with unknown ALP coecients of the exact solution. An error analysis of the method is given. Results of numerical examples are presented, which illustrate that the proposed method achieve good accuracy even using a small number of ALPs.
./files/site1/files/%D8%A8%D8%B2%D9%85.pdf
We give an overview of the recursive characterisations of random matrix ensembles that are currently at the forefront of random matrix theory by way of studying two classes of ensembles using two different types of recursive schemes: Established theory on Selberg correlation integrals is used to derive linear differential equations on the eigenvalue densities and resolvents of the classical matrix ensembles, which lead to $1$-point recursions, understood to be analogues of the Harer-Zagier recursion, for the expansion coefficients of the associated $1$-point cumulants, while loop equation analysis is used to recursively compute some leading order correlator expansion coefficients pertaining to certain products of random matrices that have recently come into interest due to their connections to Muttalib-Borodin ensembles and integrals of Harish-Chandra-Itzykson-Zuber type. We also show how the aforementioned differential equations can be used to characterise the large $N$ limiting statistics of the classical matrix ensembles' eigenvalue densities in the global and edge scaling regimes and moreover give a comprehensive review of how the Isserlis-Wick theorem implies ribbon graph interpretations for mixed moments and cumulants of some of the ensembles at hand. It is expected that this thesis will serve as a valuable resource for readers wanting a well-rounded introduction to classical matrix ensembles, matrix product ensembles, $1$-point recursions, loop equations, Selberg correlation integrals, and ribbon graphs.
The O$(N)$ vector model in the presence of a boundary has a non-trivial fixed point in $(4-\epsilon)$ dimensions and exhibits critical behaviors described by boundary conformal field theory. The spectrum of boundary operators is investigated at the leading order in the $\epsilon$-expansion by diagrammatic and axiomatic approaches. In the latter, we extend the framework of Rychkov and Tan for the bulk theory to the case with a boundary and calculate the conformal dimensions of boundary composite operators with attention to the analyticity of correlation functions. In both approaches, we obtain consistent results.
We employ the axiomatic framework of Rychkov and Tan to investigate the critical O$(N)$ vector model with a line defect in $(4-\epsilon)$ dimensions. We assume the fixed point is described by defect conformal field theory and show that the critical value of the defect coupling to the bulk field is uniquely fixed without resorting to diagrammatic calculations. We also study various defect localized operators by the axiomatic method, where the analyticity of correlation functions plays a crucial role in determining the conformal dimensions of defect composite operators. In all cases, including operators with operator mixing, we reproduce the leading anomalous dimensions obtained by perturbative calculations.
In this study, the generalized Caputo-type fractional derivative is applied to construct a family of fractional advection-reaction-diffusion equations. The orthonormal discrete Chebyshev polynomials are considered to construct a computational method for these equations. The proposed method uses the approximation of the unknown solution by these polynomials and it employs the fractional derivative of these polynomials (which is obtained in this study) along with the collocation method to derive an algebraic system of equations. A smooth solution in terms of the mentioned polynomials is obtained for the introduced fractional problem by solving this system. The implementation and numerical simulations of the method are very simple. Three test problems have been studied to investigate the numerical accuracy of the method.
Remainders of various forms of series expressions of a function
$F(x)$
, which arises frequently in the high-frequency solutions of electromagnetic wave radiation and scattering problems, are transformed into some integrals. Then, the saddle-point method is applied to those integrals to obtain accurate approximations for terminating the series expressions. The accuracy improvements are illustrated by numerical examples and graphs.
In this paper, we address the integration problem of the isomonodromic system of quantum differential equations ($qDE$s) associated with the quantum cohomology of $\mathbb P^1$-bundles on Fano varieties. It is shown that bases of solutions of the $qDE$ of the total space of the $\mathbb P^1$-bundle can be reconstructed from the datum of bases of solutions of the corresponding $qDE$ associated with the base space. This represents a quantum analog of the classical Leray-Hirsch theorem in the context of the isomonodromic approach to quantum cohomology. The reconstruction procedure of the solutions can be performed in terms of some integral transforms, introduced in arXiv:2005.08262, called $Borel$ $(\alpha,\beta)$-$multitransf\!orms$. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the B\"ohmer-Tricomi incomplete Gamma function) as integral kernels. Remarkably, these integral kernels have a universal feature, being independent of the specifically chosen $\mathbb P^1$-bundle. When applied to projective bundles on products of projective spaces, our results give Mellin-Barnes integral representations of solutions of $qDE$s. As an example, we show how to integrate the $qDE$ of blow-up of $\mathbb P^2$ at one point via Borel multitransforms of solutions of the $qDE$ of $\mathbb P^1$.
In this invited survey-cum-expository review article, we present a brief and comprehensive account of some general families of linear and bilinear generating functions which are associated with orthogonal polynomials and such other higher transcendental functions as (for example) hypergeometric functions and hypergeometric polynomials in one, two and more variables. Many of the results as well as the methods and techniques used for their derivations, which are presented here, are intended to provide incentive and motivation for further research on the subject investigated in this article.
ResearchGate has not been able to resolve any references for this publication.