Recurrence Relations For Integrals Research Articles

Efficient implementation of time-dependent auxiliary density functional theory.

The random phase approximation of time-dependent auxiliary density functional theory (TDADFT) is rederived from auxiliary density perturbation theory. Our exhaustive validation of TDADFT reveals an upshift of the excitation energies by ∼0.1eV with respect to standard time-dependent density functional theory. For the computationally efficient implementation of TDADFT, floating point operation optimized three-center electron repulsion integral recurrence relations and their double asymptotic expansions are implemented into the Davidson solver. The computational efficiency of TDADFT is benchmarked with four sets of molecules comprising alkanes, fullerenes, DNA fragments, and zeolites. The results show that TDADFT has a computational scaling between 1.3 and 1.9 with respect to the number of basis functions, which is lower than the scaling of standard time-dependent density functional theory. Due to its computational simplifications, TDADFT is particularly well suited for Born-Oppenheimer molecular dynamics simulations. As illustrative examples, we present the temperature effects on the gas-phase absorption spectra of benzene, naphthalene, and anthracene.

Some generalised extended incomplete beta functions and applications

This paper introduces generalised incomplete beta functions defined by the generalised beta function. Firstly, we provide some of the generalised beta function's basic properties, such as integral representations, summation formulas, Mellin transform, and beta distribution. We then present several fundamental properties, such as integral representations, summation formulas, and recurrence relations with the help of the generalised incomplete beta functions.

On Moments of Folded and Doubly Truncated Multivariate Extended Skew-Normal Distributions

ABSTRACT This article develops recurrence relations for integrals that relate the density of multivariate extended skew-normal (ESN) distribution, including the well-known skew-normal (SN) distribution introduced by Azzalini and Dalla-Valle and the popular multivariate normal distribution. These recursions offer a fast computation of arbitrary order product moments of the multivariate truncated extended skew-normal and multivariate folded extended skew-normal distributions with the product moments as a byproduct. In addition to the recurrence approach, we realized that any arbitrary moment of the truncated multivariate extended skew-normal distribution can be computed using a corresponding moment of a truncated multivariate normal distribution, pointing the way to a faster algorithm since a less number of integrals is required for its computation which results much simpler to evaluate. Since there are several methods available to calculate the first two moments of a multivariate truncated normal distribution, we propose an optimized method that offers a better performance in terms of time and accuracy, in addition to consider extreme cases in which other methods fail. Finally, we present an application in finance where multivariate tail conditional expectation (MTCE) for SN distributed data is calculated using analytical expressions involving normal left-truncated moments. The R MomTrunc package provides these new efficient methods for practitioners. Supplementary files for this article are available online.

Extension of Exton’s hypergeometric function K16

The purpose of this article is to introduce an extension of Exton’s hypergeometric function K16 by using the extended beta function given by Özergin et al. [11]. Some integral representations, generating functions, recurrence relations, transformation formulas, derivative formula and summation formulas are obtained for this extended function. Some special cases of the main results of this paper are also considered.

Some Results on the Extended Hypergeometric Matrix Functions and Related Functions

In this article, we discuss certain properties for generalized gamma and Euler’s beta matrix functions and the generalized hypergeometric matrix functions. The current results for these functions include integral representations, transformation formula, recurrence relations, and integral transforms.

Bicomplex Mittag-Leffler function and associated properties

With the increasing importance of the Mittag-Leffler function in the physical applications, these days many researchers are studying various generalizations and extensions of the Mittag-Leffler function. In this paper efforts are made to define bicomplex extension of the Mittag-Leffler function and also its analyticity and region of convergence are discussed. Various properties of the bicomplex Mittag-Leffler function including integral representation, recurrence relations, duplication formula and differential relations are established.

Corrigendum to: “On the calculation of arbitrary multielectron molecular integrals over Slater‐Type orbitals using recurrence relations for overlap integrals I. Single‐Center expansion method” [Int. J. Quant. Chem., 78 (2000) 146–152

Corrigendum to: “On the calculation of arbitrary multielectron molecular integrals over Slater‐Type orbitals using recurrence relations for overlap integrals I. Single‐Center expansion method” [Int. J. Quant. Chem., 78 (2000) 146–152

On moments of folded and truncated multivariate Student-t distributions based on recurrence relations

The use of the first two moments of the truncated multivariate Student-t distribution has attracted increasing attention from a wide range of applications. This paper develops recurrence relations for integrals that involve the density of multivariate Student-t distributions. The proposed techniques allow for fast computation of arbitrary-order product moments of folded and truncated multivariate Student-t distributions and offer explicit expressions of low-order moments of folded and truncated multivariate Student-t distributions. A real data example containing positive censored responses is applied to illustrate the effectiveness and importance of the proposed methods. An R MomTrunc package is developed and publicly available on the CRAN repository.

Some analytical merits of Kummer-Type function associated with Mittag-Leffler parameters

As of late, the study of Fractional Calculus (FC) and Special Functions (SFs) has been interestingly prompted in various realms of mathematics, engineering and sciences. This is due to the considerable demonstrated potential of their applications. Among these SFs, Gamma function and Mittag-Leffler functions are the most renowned and distinguished. Numerous authors continue to study this line. The current analysis attempts to introduce and further examine new modifications of Gamma and Kummer function in terms of Mittag-Leffler functions, respectively. Several attributes and formulations of this new Kummer-type function that include integral representations, Beta transform, Laplace transform, derivative formulas, and recurrence relation are investigated. Furthermore, outcomes of Riemann-Liouville fractional integral and fractional derivative in relation to this newly established Kummer function are also investigated.

A new extension of Srivastava’s triple hypergeometric functions and their associated properties

AbstractIn this paper, we define a new extension of Srivastava’s triple hypergeometric functions by using a new extension of Pochhammer’s symbol that was recently proposed by Srivastava, Rahman and Nisar [H. M. Srivastava, G. Rahman and K. S. Nisar, Some extensions of the Pochhammer symbol and the associated hypergeometric functions, Iran. J. Sci. Technol. Trans. A Sci. 43 2019, 5, 2601–2606]. We present their certain basic properties such as integral representations, derivative formulas, and recurrence relations. Also, certain new special cases have been identified and some known results are recovered from main results.

Note on the Hurwitz–Lerch Zeta Function of Two Variables

A number of generalized Hurwitz–Lerch zeta functions have been presented and investigated. In this study, by choosing a known extended Hurwitz–Lerch zeta function of two variables, which has been very recently presented, in a systematic way, we propose to establish certain formulas and representations for this extended Hurwitz–Lerch zeta function such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to yield corresponding results for several less generalized Hurwitz–Lerch zeta functions than the extended Hurwitz–Lerch zeta function considered here. For further investigation, among possibly various more generalized Hurwitz–Lerch zeta functions than the one considered here, two more generalized settings are provided.

New extensions of associated Laguerre polynomials

Abstract The main object of this paper is to present new extensions of associated Laguerre polynomials. Some integral representations, recurrence relations, generating functions and summation formulae are obtained for these new extended Laguerre polynomials.

Extended Bessel-Maitland function and its properties pertaining to integral transforms and fractional calculus

The aim of this paper is to establish an (presumably new) extension of generalized Bessel-Maitland function by using the extension of extended beta function. In addition, investigate several important properties namely integral representation, derivatives, recurrence relation, Beta transform and Mellin transform. Further, certain properties of the Riemann-Liouville fractional calculus associated with extended generalized Bessel-Maitland function are also investigated.

A Cox-de Boor-type recurrence relation for C1 multi-degree splines

Multi-degree splines are piecewise functions comprised of polynomial segments of different degrees. A subclass of such splines, that we refer to as C1 MD-splines, is featured by arbitrary continuity between pieces of same degree and at most C1 continuity between pieces of different degrees. For these spline spaces a B-spline basis can be defined by means of an integral recurrence relation, as an instance of the more general construction in Beccari et al. (2017). In this paper, we provide efficient formulas for evaluating C1 MD-splines and their derivatives, akin to the classical B-spline recurrence relations. Furthermore we derive algorithms for geometric design, including knot insertion and local degree elevation. Finally we demonstrate the utility of these splines, not only for geometric modeling, but also for graphical applications, discussing in particular the advantages for modeling and storing vector images.

Some Incomplete Hypergeometric Functions and Incomplete Riemann-Liouville Fractional Integral Operators

Very recently, the incomplete Pochhammer ratios were 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 relations. Furthermore, incomplete Riemann-Liouville fractional integral operators are introduced. This definition helps us to obtain linear and bilinear generating relations for the new incomplete Gauss hypergeometric functions.

Range-Separated Hybrid Functionals with Variational Fitted Exact Exchange

This work presents a variationally fitted long-range exact exchange algorithm that can be used for the computation of range-separated hybrid density functionals in the linear combination of Gaussian type orbital (LCGTO) approximation. The obtained LCGTO energy and gradient expressions are free of four-center integrals and employ modified three-center integral recurrence relations to obtain optimal computational performance. The accuracy and performance of selected range-separated hybrid functionals with variational fitted long-range exact exchange are analyzed and discussed. A parallel implementation is presented and benchmarked.

On multi-degree splines

Multi-degree splines are piecewise polynomial functions having sections of different degrees. For these splines, we discuss the construction of a B-spline basis by means of integral recurrence relations, extending the class of multi-degree splines that can be derived by existing approaches. We then propose a new alternative method for constructing and evaluating the B-spline basis, based on the use of so-called transition functions. Using the transition functions we develop general algorithms for knot-insertion, degree elevation and conversion to Bézier form, essential tools for applications in geometric modeling. We present numerical examples and briefly discuss how the same idea can be used in order to construct geometrically continuous multi-degree splines.

A probabilistic angle on one-loop scalar integrals

Recasting the N-point one-loop scalar integral as a probabilistic problem allows the derivation of integral recurrence relations, as well as exact analytical expressions in the most common cases. ϵ expansions are derived by writing a formula that relates an N-point function in decimal dimensions to an N-point function in integer dimensions. As an example, we give relations for the massive five-point function in and dimensions. The reduction of tensor integrals of rank two with N = 5 is achieved showing the method’s potential. No hypergeometric functions are involved. Results are expressed as integrals of arcsine functions, whose analytical continuation is well known.

A Generalized Spectral Collocation Method with Tunable Accuracy for Fractional Differential Equations with End-Point Singularities

We develop spectral collocation methods for fractional differential equations with variable order with two end-point singularities. Specifically, we derive three-term recurrence relations for both integrals and derivatives of the weighted Jacobi polynomials of the form $(1+x)^{\mu_1}(1-x)^{\mu_2}P_{j}^{a,b}(x) \,({a,b,\mu_1,\mu_2>-1})$, which leads to the desired differentiation matrices. We apply the new differentiation matrices to construct collocation methods to solve fractional boundary value problems and fractional partial differential equations with two end-point singularities. We demonstrate that the singular basis enhances greatly the accuracy of the numerical solutions by properly tuning the parameters $\mu_1$ and $\mu_2$, even for cases for which we do not know explicitly the form of singularities in the solution at the boundaries. Comparison with other existing methods shows the superior accuracy of the proposed spectral collocation method.

Numerical approximation of GB-splines by a convolutional approach

Generalized splines are smooth functions belonging piecewisely to spaces which are a natural generalization of algebraic polynomials. GB-splines are a B-spline-like basis for generalized splines, and they are usually defined by means of an integral recurrence relation which makes their evaluation quite cumbersome and computationally expensive. We present a simple strategy for approximating the values of a cardinal GB-spline of arbitrary degree p, with a particular focus on hyperbolic and trigonometric GB-splines due to their interest in applications. The proposed strategy is based on the Fourier properties of cardinal GB-splines. The approximant is expressed as a linear combination of scaled and dilated versions of (polynomial) cardinal B-splines of degree p, whose coefficients can be efficiently computed via discrete convolution. Sharp error estimates are provided and illustrated with some numerical examples.