Search Results (258)

We study the half-integer family $F(n,p)={}_3{F}_2\left(-\frac{2n+1}{2},1,\frac{2p-1}{2};\frac{2n+5}{2},\frac{2p+1}{2};1\right).$ Although this family belongs to the Karlsson–Minton class of ${}_3{F}_2\left(1\right)$ series with negative integral parameter differences, the most directly relevant symmetric formula of Shpot and Srivastava becomes singular at the free numerator parameter $a=1$(here a denotes the first numerator parameter of the Shpot–Srivastava reduction formula, the value of which controls the Beta factors in their evaluation). We show that this apparent singularity is removable and identify its finite value with the independently derived closed form. First, Euler’s integral representation and a recurrence for $J_{m,n}\left(x\right)={\int }_0^1{v}^{2m}{(1-x{v}^2)}^{-n}dv$ yield a closed form for ${}_2{F}_1\left(-\frac{2n+1}{2},1;\frac{2n+5}{2};x\right)$ as a polynomial in $(1-x)$ plus $arctanh\left(\sqrt{x}\right)/\sqrt{x}$. Rainville’s integral formula then reduces $F(n,p)$ to a rational part and a finite rational linear combination of odd harmonic sums $H_r^{\mathrm{odd}}={\sum }_{j=1}^r{(2j-1)}^{-1}$. For all integers $n\ge 0$ and $p\ge n+3$, this proves $F(n,p)\in \mathbb{Q}$. The threshold is sharp: at the adjacent boundary $p=n+2$ we obtain $F(n,n+2)=R_n+C_n{\pi }^2$ with $R_n,C_n\in \mathbb{Q}$ and $C_n\ne 0$, so rationality fails. We also prove, by analyticity and the independently derived formula, that the $a\to 1$ limit of the Shpot–Srivastava representation equals the present closed form. For computation, the exact assembly uses $O(n+p)$ arithmetic operations under the unit-cost model, while a direct fixed-precision evaluation is ill-conditioned for large n. A stable series/closed-form hybrid for the ${}_2{F}_1$ factor combined with adaptive positive-kernel Gauss–Jacobi quadrature gives relative errors at the machine-precision scale over the tested range up to $n=64$. Full article

This paper investigates a class of Leonardo polynomials defined by a nonhomogeneous recurrence relation, whose algebraic structure differs significantly from that of classical Fibonacci-type polynomial families. First, we develop a new connection formula expressing Leonardo polynomials in terms of Fibonacci polynomials, and then we derive the corresponding inverse connection formula. These two identities are subsequently employed to develop an explicit power-form representation of the Leonardo polynomials together with its corresponding monomial inversion formula. Building upon these fundamental representations, the paper derives new derivative and connection formulas involving generalized Fibonacci and generalized Lucas polynomial families in terms of the Leonardo basis. Several consequences for the classical Fibonacci and Lucas polynomials are also established. Furthermore, new product identities involving Leonardo polynomials are developed. The derived representations also reveal parity-dependent symmetry patterns in the Leonardo polynomial coefficients and clarify how these patterns are inherited from the associated Fibonacci- and Lucas-type bases. Many of the resulting coefficients are expressed in terms of terminating hypergeometric functions, yielding compact representations that are expected to be useful in approximation theory and operational methods for differential equations. Full article

Classical summation theorems for generalized hypergeometric series constitute an important tool in the study of special functions. Motivated by earlier generalizations of these theorems, this paper derives several new summation formulas for generalized hypergeometric functions by employing the generalized classical summation theorems of Lavoie et al. within a well-known hypergeometric identity. The proposed formulas extend previously reported results, including those of Eslahchi and Masjed-Jamei, which arise as special cases of the present work. The obtained identities provide a unified framework for deriving summation formulas and may be useful in further investigations involving generalized hypergeometric functions and related identities. Full article

In this paper, we study the k-Kummer hypergeometric function $M_k(a,c;w)$, a generalization of the classical Kummer confluent hypergeometric function, arising as a solution of the k-confluent hypergeometric differential equation. We establish a Kummer-type transformation formula and derive derivative identities, contiguous relations, and addition and multiplication formulas. In addition, we obtain closed-form expressions for the Laplace, Mellin, Stieltjes, Sumudu, and Riemann–Liouville fractional integral transforms involving $M_k(a,c;w)$. The results presented here extend the classical theory of confluent hypergeometric functions to the k-generalized setting and provide analytic tools for further investigations of generalized differential equations and integral transforms. Full article

We consider a generalization of the modified Korteweg–de Vries–Burgers equation with a nonlinear source. Using the Painlevé test for partial differential equation with the Kruskal variable, we show that the corresponding Cauchy problem cannot be solved by the inverse scattering transform. However, the equation admits a two-wave solution, which is obtained by means of the Cole–Hopf transformation. Taking into account the traveling wave reduction, we derive the resulting nonlinear ordinary differential equation and determine the parameter conditions under which it passes the Painlevé test. This finding suggests the possible existence of the general solution for the ordinary differential equation, which can be reduced to the linear third-order equation. The general solution of the resulting linear equation is expressed in terms of the hypergeometric function. Full article

In this paper, the Tricomi problem for a second-kind mixed-type equation with a lower-order term is studied in an unbounded domain. The elliptic part of the domain is a vertical half-strip, while the hyperbolic part is bounded by characteristics. Homogeneous Dirichlet conditions are imposed on the walls of the half-strip, gluing conditions are given on the parabolic degeneracy line, and the trace of the desired solution is prescribed on one of the characteristics. The uniqueness of the solution is proved using the extremum principle and the Zaremba–Giraud principle. The existence of the solution is established by Green’s function method: in the elliptic part, Green’s function of the mixed problem is constructed in the form of a rapidly convergent series; in the hyperbolic part, a generalized solution of the Cauchy problem of a special class is used. The functional relations on the degeneracy line lead to a singular integral equation, which is regularized by the Carleman–Vekua method into a Fredholm integral equation of the second kind with a weak singularity. Explicit formulas for the trace of the solution and its normal derivative are obtained. For a specific set of parameters, a numerical visualization of the solution is performed, the gluing conditions are verified, and a physical interpretation of the obtained graphs is given in the context of transonic gas dynamics. The results can be useful for mathematical modeling of flows in Laval nozzles and other problems of mechanics. Full article

In this paper, we introduce a new difference operator, called the $\left(p,q\right)$-Hahn difference operator, which extends both the classical Hahn operator and the $\left(p,q\right)$-difference operator by incorporating an additional shift parameter, $\omega$. This extension allows the operator to reflect translation effects together with scaling behavior within a common framework. We investigate the basic algebraic properties of the operator, including linearity, product and quotient rules, and explicit formulas for power functions. A generalized Leibniz rule is established using $\left(p,q\right)$-binomial coefficients. In addition, a corresponding $\left(p,q\right)$-Hahn integral is defined, and a fundamental relation between the operator and the integral is obtained under suitable assumptions. Furthermore, several special and limiting cases are analyzed in order to clarify the connection between the proposed operator and existing difference operators. In particular, it is shown that the operator reduces to the classical derivative, the $q$-difference operator, and the standard $\left(p,q\right)$-difference operator under appropriate parameter choices. Finally, applications to $\left(p,q\right)$-Hahn Sturm–Liouville-type problems and hypergeometric-type difference equations are discussed. These results suggest that the proposed operator provides a consistent extension of existing difference-calculus structures. Full article

We calculate several finite integrals involving trigonometric and hyperbolic functions by applying the Laplace convolution theorem and known inverse Laplace transform formulas. As a consequence, we obtain a new integral representation of the Kelvin function $\mathrm{bei}\left(z\right)$, and a new reduction formula for a particular generalized hypergeometric function. In addition, we present several new inverse Laplace transform formulas that do not appear to have been reported in the existing literature. Full article

This study presents the development of a comprehensive framework wherein probabilistic and analytical techniques collaboratively produce identities pertinent to special functions. The fundamental premise is that ratios of random variables, particularly within the family of Gamma and Beta distributions, inherently generate integral representations and hypergeometric structures that can be harnessed to derive non-trivial summation formulas. A pivotal outcome of this investigation is that, in the context of independent and identically distributed random variables, symmetry plays a crucial role. A key property of ratios of i.i.d. random variables translates into straightforward probabilistic assertions that directly lead to precise analytic identities, thereby enabling the recovery of classical results such as Kummer’s and Watson’s summation theorems as consequences of distributional symmetry. When the assumption of identical distribution is relaxed, hypergeometric representations continue to exist; however, the absence of symmetry impedes closed-form evaluations of the same nature. This contrast underscores symmetry as the fundamental mechanism underlying exact summation formulas and elucidates why such identities are contingent upon specific parameter configurations. More generally, this methodology offers a probabilistic interpretation of hypergeometric identities and establishes a conceptual connection between probability theory and the theory of special functions. Furthermore, it suggests that more expansive constructions, based on non-identically distributed variables or iterated processes, could be fruitfully explored. In particular, examining ratios may lead to new identities or alternative derivations of classical results. Full article

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

This paper studies an extension of the bilateral gamma process assuming that the drift coefficient may jump at an exponentially distributed random time. The drift switching can reflect the symmetry between major economic events and moves of financial market indexes. The bilateral gamma distribution has an asymmetric form and fits well with different financial data when there are not external shocks. As the main results, we provide exact formulas for the probability density and incomplete moment-generating functions of the stated process. The expressions found are used for risk measurement and European option pricing. The new formulas are determined in particular by values of the incomplete gamma, Whittaker and confluent hypergeometric functions. Numerical examples of the computations are also afforded. The computation time for the formulas is under 4 s in a compiler compatible with MatLab. Full article

This paper employs Jacobians of matrix transformations to derive the density function of a matrix-variate generalized gamma distribution, together with its normalizing constant. By applying the inverse Mellin transform, explicit expressions for the density functions of the determinant and the trace are obtained in terms of generalized hypergeometric functions. The characteristic function and the first two moments follow from an associated density generator. Both the real and complex cases are treated, and several important special cases are identified. A simulation study reveals that the proposed model provides a more accurate fit than other distributions that are also defined on the cone of positive definite matrices. Moreover, it is shown to exhibit superior performance when applied to two empirical data sets. Applications involving the modeling of scatter matrices arising in financial studies, biostatistics, and reliability analysis are also discussed. 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

Closed-form analytic inverses allow explicit tracking of parameter effects, facilitate interpretation of experimental signals, and support solving inverse problems. Here, we derive a rigorous closed-form expression for the inverse Laplace transform of a class of shifted quasi-rational spectral functions with a square-root radical and a power-law decaying factor. These functions appear in coupled diffusion processes in physics and in the analysis of electromagnetic signal propagation through electrically cascaded networks, signal processing, and related areas. The transform is expressed as a finite sum of three generalized hypergeometric functions—two Kummer functions and one five-parameter Kampé de Fériet function—each multiplied by a monomial depending on the decay parameter. The validity of the result is confirmed by direct Laplace transformation, which recovers the original spectral function. Several known inverse transforms appear as limiting cases, illustrating the generality of the solution. Additionally, reduction formulas for a subclass of Kampé de Fériet functions demonstrate how the general solution encompasses previously known results and highlight the generality of the method. Full article