Search Results (81)

In this paper, we revisit the recent result of Luo, Xu, and Raina on an Erdélyi-type integral for Saran’s three-variable hypergeometric function $F_K$. We provide a new proof of this integral and derive an attractive new integral related to Appell’s function $F_2$. A further extension on the L-variable $F_K$ function, which appears in physics, is also discussed. Furthermore, we prove various q-Erdélyi-type integrals for the q-analogue of the $F_K$-function. An interesting discrete analogue is also included. We also provide a valuable compilation of the sources for known Erdélyi-type integrals of many different hypergeometric functions. Full article

We study one-dimensional diffusions reflected at a boundary and analyze their pathwise “episodes” away from the boundary through Itô’s excursion theory. Under a fixed height cap of $a>0$, each excursion is equipped with three natural marks: its lifetime $\zeta$, its maximum $M$, and an additive (area-type) functional $A_f={\int }_0^{\zeta }f\left(e_t\right)dt$. Our main object is the height-truncated Itô-excursion Laplace exponent ${\mathsf{\Psi }}_{\alpha ,\lambda ;af}:=n\left(1-e^{-\alpha \zeta -\lambda A_f}; M<a\right)$ which jointly characterizes episode duration and cumulative load while excluding barrier-crossing spikes. We establish a general boundary–flux representation: ${\mathsf{\Psi }}_{\alpha ,\lambda ;af}$ is obtained as a boundary flux (in scale) of the unique solution to a one-dimensional killed Feynman–Kac boundary-value problem on $(0, a)$. This transfer principle yields a unified and tractable route to explicit computation. We implement it in three solvable families—the reflected arithmetic Brownian motion, reflected Ornstein–Uhlenbeck diffusions, and squared Bessel/Bessel-type diffusions—obtaining closed forms in terms of Airy, parabolic-cylinder, and confluent hypergeometric/Whittaker functions. Using the Poisson point process structure of excursions indexed by local time, we derive explicit extreme-burst laws (maxima and order statistics) for the additive marks up to a local-time horizon, and connect tail intensities to Laplace exponents via numerical Laplace inversion. Finally, we identify the strictly truncated cumulative load in local time as a (typically infinite-activity) subordinator whose Lévy measure coincides with the excursion-mark intensity, linking cumulative-load and extreme-burst statistics through the same exponent. Full article

The paper considers the problem of the analytical extension of the ratios of generalized hypergeometric functions ${}_{3}F_2$ A new domain of analytic continuation for these ratios under certain conditions to parameters is established. In this case, the domain of analytic extension of the special function is the domain of convergence of its branched continued fraction expansion. This paper also provides an example of applying the obtained results to dilogarithm function. Full article

In this paper, we study a generalized Duffin–Kemmer equation for a spin-1 particle with two characteristics, anomalous magnetic moment and polarizability in the presence of external uniform magnetic and electric fields. After separating the variables, we obtained a system of 10 first-order partial differential equations for 10 functions $f_A(r,z)$. To resolve this complicated problem, we first took into account existing symmetry in the structure of the derived system. The main step consisted of applying a special method for fixing the r-dependence of ten functions $f_A(r,z),A=1,\dots ,10$. We used the approach of Fedorov–Gronskiy, according to which the complete 10-component wave function is decomposed into the sum of three projective constituents. The dependence of each component on the polar coordinate r is determined by only one corresponding function, $F_i\left(r\right),i=1,2,3$. These three basic functions are constructed in terms of confluent hypergeometric functions, and in this process a quantization rule arises due to the presence of a magnetic field.In fact, this approach is a step-by-step algebraization of the systems of equations in partial derivatives. After that, we derived a system of 10 ordinary differential equations for 10 functions $f_A\left(z\right)$. This system was solved using the elimination method and with the help of special linear combinined with the involved functions. As a result, we found three separated second-order differential equations, and their solutions were constructed in the terms of the confluent hypergeometric functions. Thus, in this paper, the three types of solutions for a vector particle with two additional electromagnetic characteristics in the presence of both external uniform magnetic and electric fields. Full article

In this study, a semi-analytical solution to the inhomogeneous Whittaker equation is developed for both initial and boundary value problems. A new class of special integral functions ${}^f\mathrm{Zi}_{\kappa ,\mu }\left(x\right)$, along with their derivatives, is introduced to facilitate the construction of the solution. The analytical properties of ${}^f\mathrm{Zi}_{\kappa ,\mu }\left(x\right)$ are rigorously investigated, and explicit closed-form expressions for ${}^f\mathrm{Zi}_{\kappa ,\mu }\left(x\right)$ and its derivatives are derived in terms of Whittaker functions ${\mathrm{M}}_{\kappa ,\mu }\left(z\right)$ and ${\mathrm{W}}_{\kappa ,\mu }\left(z\right)$, confluent hypergeometric functions, and other special functions including Bessel functions, modified Bessel functions, and the incomplete gamma functions, along with their respective derivatives. These expressions are obtained for specific parameter values using symbolic computation in Maple. The results contribute to the broader analytical framework for solving inhomogeneous linear differential equations with applications in engineering, mathematical physics, and biological modeling. Full article

This paper considers the problem of approximating some Appell’s hypergeometric functions $F_2$ by their branched continued fraction expansions. Using the formula for the difference of two approximants of a branched continued fraction, we established the truncation error bounds for such expansions. In addition, we provided another proof of the convergence of branched continued fraction expansions to the ratio of Appell’s hypergeometric functions $F_2$. Finally, we also provide examples to demonstrate the effectiveness of branched continued fractions as a tool for approximating special functions. Full article

Indefinite integrals of products of exponential functions, power functions and generalized hypergeometric functions of some types are considered. Necessary and sufficient conditions are established for the algebraic independence of large sets of such functions (for various parameters) and their derivatives, as well as their values. All the algebraic relations between these functions are written out explicitly. Full article

The rapid densification of wireless networks demands efficient evaluation of special functions underpinning system-level performance metrics. To facilitate research, we introduce a computational framework tailored for the zero-balanced Gauss hypergeometric function $\Psi (x,y)\triangleq {}_{2}F_1(1,x;1+x;-y)$, a fundamental mathematical kernel emerging in Signal-to-Interference-plus-Noise Ratio (SINR) coverage analysis of non-uniform cellular deployments. Specifically, we propose a novel Reciprocal-Argument Transformation Algorithm (RTA), derived rigorously from a Mellin–Barnes reciprocal-argument identity, achieving geometric convergence with $O\left(1/y\right)$. By integrating RTA with a Pfaff-series solver into a hybrid algorithm guided by a golden-ratio switching criterion, our approach ensures optimal efficiency and numerical stability. Comprehensive validation demonstrates that the hybrid algorithm reliably attains machine-precision accuracy ($\sim {10}^{-16}$) within 1 μs per evaluation, dramatically accelerating calculations in realistic scenarios from hours to fractions of a second. Consequently, our method significantly enhances the feasibility of tractable optimization in ultra-dense non-uniform cellular networks, bridging the computational gap in large-scale wireless performance modeling. Full article

We establish a theory whose structure is based on a fixed variable and an algebraic inequality and which improves the well-known least squares theory. The mentioned fixed variable plays a basic role in creating such a theory. In this direction, some new concepts, such as p-covariances with respect to a fixed variable, p-correlation coefficients with respect to a fixed variable, and p-uncorrelatedness with respect to a fixed variable, are defined in order to establish least p-variance approximations. We then obtain a specific system, called the p-covariances linear system, and apply the p-uncorrelatedness condition on its elements to find a general representation for p-uncorrelated variables. Afterwards, we apply the concept of p-uncorrelatedness for continuous functions, particularly for polynomial sequences, and we find some new sequences, such as a generic two-parameter hypergeometric polynomial of the ${}_{4}F_3$ type that satisfies a p-uncorrelatedness property. In the sequel, we obtain an upper bound for 1-covariances, an improvement to the approximate solutions of over-determined systems and an improvement to the Bessel inequality and Parseval identity. Finally, we generalize the concept of least p-variance approximations based on several fixed orthogonal variables. Full article

In the Geometric Theory of Analytic Functions, classes of functions with several normalizations are considered. We consider the symmetric idea for harmonic functions. Classes of harmonic functions f with normalization $f\left(0\right)=f_{\overline{z}}^{\prime }\left(0\right)=0,$$f_z^{\prime }\left(0\right)=1$ are usually considered in the geometric theory of harmonic functions. The normalization is called the classical normalization. We can obtain some interesting results by using Montel’s normalization $f\left(0\right)=f_{\overline{z}}^{\prime }\left(0\right)=0,$$f_z^{\prime }\left(\rho \right)-f_{\overline{z}}^{\prime }\left(\rho \right)=1$, where $\rho \in [0,1).$ In the paper, we consider the class of harmonic functions with Montel’s normalization associated with the generalized hypergeometric function. 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

The evaluation of low-degree hypergeometric polynomials to zero defines algebraic hypersurfaces in the affine space of the free parameters and the argument of the hypergeometric function. This article investigates the algebraic surfaces defined by the hypergeometric equation ${}_2\mathrm{F}_1(-N,b;c;z)=0$ with $N=3$ or $N=4$. As a captivating application, these surfaces parametrize certain families of genus 0 Belyi maps. Thereby, this article contributes to the systematic enumeration of Belyi maps. 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

The paper considers the problem of representation and extension of Appell’s hypergeometric functions by a special family of functions—branched continued fractions. Here, we establish new symmetric domains of the analytical continuation of Appell’s hypergeometric function $F_2$ with real and complex parameters, using their branched continued fraction expansions whose elements are polynomials in the space ${\mathbb{C}}^2$. To do this, we used a technique that extends the domain of convergence of the branched continued fraction, which is already known for a small domain, to a larger domain, as well as the PC method to prove that it is also the domain of analytical continuation. A few examples are provided at the end to illustrate this. 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