Search Results (69)

We extend the classical Euler-type integral representations for the Appell functions $F_1$, $F_2$, and $F_3$, to the appropriate Kampé de Fériet functions by using integration against the Meijer–Nørlund G-function. In particular, these representations provide analytic continuation of the corresponding Kampé de Fériet functions. We further focus on the following two applications. First, we obtain sufficient conditions for complete monotonicity on the positive quadrant for three families of the Kampé de Fériet functions. These conditions can be expressed directly in terms of parameters and imply, among other things, joint log-convexity and related inequalities for partial derivatives of the Kampé de Fériet functions. Second, we show how known reduction and transformation formulas for the Appell and the generalized hypergeometric functions can be lifted to Kampé de Fériet functions by concatenating parameter arrays via the integral representations. This yields several reduction formulas, including extensions of some classical and new product identities. Further combining integration against the Meijer–Nørlund G-function with Slater’s double series transformation we obtain several exotic identities for infinite sums of the generalized hypergeometric functions. Full article

We study a generalized class of Jacobsthal–Lucas polynomials that depends on two parameters. First, we introduce essential formulas for these polynomials, involving their series representation, inverse formula, and moment formula. These formulas allow us to investigate this generalized class of polynomials further and to develop novel formulations. The essential standard linearization problem of these polynomials is solved, and the linearization coefficients are given in simple forms. In addition, some mixed linearization formulas with other classes of polynomials are presented. The derivative formulas of these polynomials, expressed as combinations of different polynomials, are given. By employing symbolic algebra methods—most notably Zeilberger’s algorithm and other well-known identities from the literature—many hypergeometric functions appearing in the coefficients can be reduced, resulting in simpler expressions. In addition, some definite integrals are evaluated using the newly introduced formulas. Full article

While many continuous distributions are known, the list of discrete ones (usually derived from counting) often reported is relatively short. This list becomes even shorter when dealing with dichotomous observables: binomial, hypergeometric, negative binomial, and uniform. Binomial distribution is important for medical studies, since a finite sample from a population included in a medical study with yes/no outcome resembles a series of independent Bernoulli trials. The problem of calculating the confidence interval (CI, with conventional risk of 5% or otherwise) is revealed to be a problem of combinatorics. Several algorithms dispute the exact calculation, each according to a formal definition of its exactness. For two algorithms, four previously proposed case studies are provided, for sample sizes of 30, 50, 100, 150, and 300. In these cases, at 1% significance level, ordered sequences defining the confidence bounds were generated for two formal definitions. Images of the error’s alternation are provided and discussed. Both algorithms propose symmetric solutions in terms of both CIs and actual coverage probabilities. The CIs are not symmetric relative to the observed variable, but are mirrored symmetric relative to the middle of the observed variable domain. When comparing the solutions proposed by the algorithms, with the increase in the sample size, the ratio of identical confidence levels is increased and the difference between actual and imposed coverage is shrunk. Full article

We investigate the influence of the higher-order curvature corrections on a static configuration with constant density in Einstein–Gauss–Bonnet (EGB) gravity. This analysis is applied to both neutral and charged fluid distributions in arbitrary spacetime dimensions. The EGB field equations are generated, and the condition of pressure isotropy is shown to generalise the general relativity equation. The gravitational potentials are unique in all spacetime dimensions for neutral gravitating spheres. Charged gravitating spheres are not unique and depend on the form of the electric field. Our treatment is extended to the particular case of a charged fluid distribution with a constant energy density and constant electric field intensity. The charged EGB field equations are integrated to give exact models in terms of hypergeometric functions which can also be written as a series. Full article

The squared Bessel process plays a central role in stochastic analysis, with broad applications in mathematical finance, physics, and probability theory. While explicit expressions for its transition probability density function (TPDF) under constant parameters are well known, analytical results in the case of time-dependent dimensions remain scarce. In this paper, we address a significantly challenging problem by deriving an analytical formula for the TPDF of a conic combination of independent squared Bessel processes with time-dependent dimensions. The result is expressed in terms of a Laguerre series expansion. Furthermore, we obtain closed-form expressions for the conditional moments of such conic combinations, represented via generalized hypergeometric functions. These results also yield new analytical formulas for the TPDF and conditional moments of both squared Bessel processes and Bessel processes with time-dependent dimensions. The proposed formulas provide a unified analytical framework for modeling and computation involving a broad class of time-inhomogeneous diffusion processes. The accuracy and computational efficiency of our formulas are verified through Monte Carlo simulations. As a practical application, we provide an analytical valuation of an interest rate swap, where the underlying short rate follows a conic combination of independent squared Bessel processes with time-dependent dimensions, thereby illustrating the theoretical and practical significance of our results in mathematical finance. Full article

The present review discusses the solution of the Heun, confluent, biconfluent, double confluent, and triconfluent equations in terms of polynomial expansions, and applies the results to generate exactly solvable Schrödinger potentials. Although there are more general approaches to solve these differential equations in terms of the expansions of certain special functions, the importance of polynomial solutions is unquestionable, as most of the known potentials are solvable in terms of the hypergeometric and confluent hypergeometric functions; i.e., Natanzon-class potentials possess bound-state solutions in terms of classical orthogonal polynomials, to which the (confluent) hypergeometric functions can be reduced. Since some of the Heun-type equations contain the hypergeometric and/or confluent hypergeometric differential equations as special limits, the potentials generated from them may also contain Natanzon-class potentials as special cases. A power series expansion is assumed around one of the singular points of each differential equation, and recurrence relations are obtained for the expansion coefficients. With the exception of the triconfluent Heun equations, these are three-term recurrence relations, the termination of which is achieved by prescribing certain conditions. In the case of the biconfluent and double confluent Heun equations, the expansion coefficients can be obtained in the standard way, i.e., after finding the roots of an (N + 1)th-order polynomial in one of the parameters, which, in turn, follows from requiring the vanishing of an (N + 1) × (N + 1) determinant. However, in the case of the Heun and confluent Heun equations, the recurrence relation can be solved directly, and the solutions are obtained in terms of rationally extended X₁-type Jacobi and Laguerre polynomials, respectively. Examples for solvable potentials are presented for the Heun, confluent, biconfluent, and double confluent Heun equations, and alternative methods for obtaining the same potentials are also discussed. These are the schemes based on the rational extension of Bochner-type differential equations (for the Heun and confluent Heun equation) and solutions based on quasi-exact solvability (QES) and on continued fractions (for the biconfluent and double confluent equation). Possible further lines of investigations are also outlined concerning physical problems that require the solution of second-order differential equations, i.e., the Schrödinger equation with position-dependent mass and relativistic wave equations. Full article

This paper shows that certain ${}_{3}F_4$ hypergeometric functions can be expanded in sums of pair products of ${}_{1}F_2$ functions. In special cases, the ${}_{3}F_4$ hypergeometric functions reduce to ${}_{2}F_3$ functions. Further special cases allow one to reduce the ${}_{2}F_3$ functions to ${}_{1}F_2$ functions, and the sums to products of ${}_{0}F_1$ (Bessel) and ${}_{1}F_2$ functions. The class of hypergeometric functions with summation theorems are thereby expanded beyond those expressible as pair-products of ${}_{2}F_1$ functions, ${}_{3}F_2$ functions, and generalized Whittaker functions, into the realm of ${}_{p}F_q$ functions where $p<q$ for both the summand and terms in the series. Full article

Summation of infinite series has played a significant role in a broad range of problems in the physical sciences and is of interest in a purely mathematical context. In a prior paper, we found that the Fourier–Legendre series of a Bessel function of the first kind $J_N\left(kx\right)$ and modified Bessel functions of the first kind $I_N\left(kx\right)$ lead to an infinite set of series involving ${}_{1}F_2$ hypergeometric functions (extracted therefrom) that could be summed, having values that are inverse powers of the eight primes $1/\left(2^i{3}^j{5}^k{7}^l{11}^m{13}^n{17}^o{19}^p\right)$ multiplying powers of the coefficient k, for the first 22 terms in each series. The present paper shows how to generate additional, doubly infinite summed series involving ${}_{1}F_2$ hypergeometric functions from Chebyshev polynomial expansions of Bessel functions, and trebly infinite sets of summed series involving ${}_{1}F_2$ hypergeometric functions from Gegenbauer polynomial expansions of Bessel functions. That the parameters in these new cases can be varied at will significantly expands the landscape of applications for which they could provide a solution. Full article

In this paper, a new class of kernels of integral transforms of the Laplace convolution type that we named symmetrical Sonin kernels is introduced and investigated. For a symmetrical Sonin kernel given in terms of elementary or special functions, its associated kernel has the same form with possibly different parameter values. In the paper, several new kernels of this type are derived by means of the Sonin method in the time domain and using the Laplace integral transform in the frequency domain. Moreover, for the first time in the literature, a class of Sonin kernels in terms of the convolution series, which are a far-reaching generalization of the power series, is constructed. The new symmetrical Sonin kernels derived in the paper are represented in terms of the Wright function and the new special functions of the hypergeometric type that are extensions of the corresponding Horn functions in two variables. Full article

The paper considers the generalized hypergeometric function ${}_{3}F_2,$ which is important in various fields of mathematics, physics, and economics. The method is used, according to which the domains of the analytical continuation of the special functions are the domains of convergence of their expansions into a special family of functions, namely branched continued fractions. These expansions have wide domains of convergence and better computational properties, particularly compared with series, making them effective tools for representing special functions. New domains of the analytical continuation of the generalized hypergeometric function ${}_{3}F_2$ with real and complex parameters have been established. The paper also includes examples of the presentation and extension of some special functions. Full article

In the theory of hypergeometric series and generalized hypergeometric series, classical summation theorems, such as the two of Gauss and those of Kummer and Bailey for the series ${}_{2}F_1$; those of Watson, Dixon, Whipple, and Saalschutz for the series ${}_{3}F_2$; and others, play a key role. Applications of these classical summation theorems are well known. Berndt pointed out that a large number of interesting summations (including Ramanujan’s summations and the Gregory–Leibniz pi summation) can be obtained very quickly by employing the above-mentioned classical summation theorems. Also, several interesting results involving products of generalized hypergeometric series have been obtained by Bailey by employing the above-mentioned classical summation theorems. Recently, the above-mentioned classical summation theorems have been generalized and extended. In our present investigations, our aim is to demonstrate the applications of the extended Kummer’s summation theorem in establishing (i) extensions of Gauss’s second summation theorem and Bailey’s summation theorem; (ii) extensions of several summations (including Ramanujan’s summations); (iii) extensions of several results involving products of generalized hypergeometric series; and (iv) an extension of classical Dixon’s summation theorem. As special cases, we recover several known summations (including several Ramanujan summations and the Gregory–Leibniz pi summation) and various results involving products of generalized hypergeometric series due to Bailey. Full article

A sequence of n trials from a finite population with no replacement is described by the hypergeometric distribution as the number of successes. Calculating the likelihood that factory-produced items would be defective is one of the most popular uses of the hypergeometric distribution in industrial quality control. Very recently, several researchers have applied this distribution on certain families of analytic functions. In this study, we provide certain adequate criteria for the generalized hypergeometric distribution series to be in two families of analytic functions defined in the open unit disk. Furthermore, we consider an integral operator for the hypergeometric distribution. Some corollaries will be implied from our main results. Full article

Utilizing spatial diversity, the distributed multiple-input multiple-output (MIMO) radar has the potential advantage of improving system detection performance. In this paper, the incoherent detection performance of distributed multiple-input multiple-output (MIMO) radars is investigated for Rice fluctuating targets. To calculate the incoherent detection probability, the moment generating function (MGF) of the Rice variable is expanded as the infinite series form. By inverting the product of MGFs of multiple independent Rice variables, new closed-form expressions for the probability density function (PDF) of the sum of independent and weighted squares of Rice variables are proposed. The proposed PDF expression for the sum of independent, non-identically distributed (i.n.i.d.) Rice variables involves an infinite series in terms of the confluent Lauricella function. Specially, the PDF for the sum of independent identically distributed (i.i.d.) Rice is expressed as the confluent hypergeometric function-based infinite series. In addition, the uniform convergence of the proposed PDF expression is also validated. Using this proposed expression, the closed-form and approximate expressions of the incoherent detection probability of MIMO radar are derived, respectively. Numerically evaluated results are illustrated and compared with Monte Carlo (MC) simulations to validate the accuracy of the derivations. Full article

By employing the coefficient extraction method, a class of binomial series involving harmonic numbers will be reviewed through three hypergeometric ${}_{2}F_1\left(y^2\right)$-series. Numerous closed-form generating functions for infinite series containing binomial coefficients and harmonic numbers will be established, including several conjectured ones. Full article

The principal aim of this paper is to introduce the extended Voigt-type function ${\mathbb{V}}_{\mu ,\nu }(x,y)$ and its counterpart extension ${\mathbb{W}}_{\mu ,\nu }(x,y)$, involving the Neumann function $Y_{\nu }$ in the kernel of the representing integral. The newly defined integral reduces to the classical Voigt functions $K(x,y)$ and $L(x,y)$, and to their generalizations by Srivastava and Miller, by the unification of Klusch. Following an approach by Srivastava and Pogány, we also present the multiparameter and multivariable versions ${\mathbb{V}}_{\mu ,\nu }^{\left(r\right)}(x,y),{\mathbb{W}}_{\mu ,\nu }^{\left(r\right)}(x,y)$ and the r positive integer of the initial extensions ${\mathbb{V}}_{\mu ,\nu }(x,y),{\mathbb{W}}_{\mu ,\nu }(x,y)$. Several computable series expansions are obtained for the discussed Voigt-type functions in terms of Humbert confluent hypergeometric functions ${\Psi }_2^{\left(r\right)}$. Furthermore, by transforming the input extended Voigt-type functions by the Grünwald–Letnikov fractional derivative, we establish representation formulae in terms of the associated Legendre functions of the second kind $Q_{\eta }^{-\nu }$ in the two-parameter and two-variable cases. Finally, functional bounding inequalities are given for ${\mathbb{V}}_{\mu ,\nu }(x,y)$ and ${\mathbb{W}}_{\mu ,\nu }(x,y)$. Particularly interesting results are presented for the Neumann function $Y_{\nu }$ and for the Struve ${\mathbf{H}}_{\nu }$ function in the form of several functional bounds. The article ends with a thorough discussion and closing remarks. Full article