Recursion formulas for certain quadruple hypergeometric functions

A remarkably large number of hypergeometric (and generalized) functions and a variety of their extensions have been presented and investigated in the literature by many authors. In this paper, we introduce five new hypergeometric functions in four variables and then establish several recursion formulas for these new functions. Some interesting particular cases and consequences of the main results are also considered.

On the Kampé de Fériet Hypergeometric Matrix Function

In this study, we derive recursion formulas for the Kampé de Fériet hypergeometric matrix function. We also obtain some finite matrix and infinite matrix summation formulas for the Kampé de Fériet hypergeometric matrix function.

Crossings and Nestings Over Some Motzkin Objects and $q$-Motzkin Numbers

We examine the enumeration of certain Motzkin objects according to the numbers of crossings and nestings. With respect to continued fractions, we compute and express the distributions of the statistics of the numbers of crossings and nestings over three sets, namely the set of $4321$-avoiding involutions, the set of $3412$-avoiding involutions, and the set of $(321,3\bar{1}42)$-avoiding permutations. To get our results, we exploit the bijection of Biane restricted to the sets of $4321$- and $3412$-avoiding involutions which was characterized by Barnabei et al. and the bijection between $(321,3\bar{1}42)$-avoiding permutations and Motzkin paths, presented by Chen et al.. Furthermore, we manipulate the obtained continued fractions to get the recursion formulas for the polynomial distributions of crossings and nestings, and it follows that the results involve two new $q$-Motzkin numbers.

Computing the Permanent of the Laplacian Matrices of Nonbipartite Graphs

Let G be a graph with Laplacian matrix L G . Denote by per L G the permanent of L G . In this study, we investigate the problem of computing the permanent of the Laplacian matrix of nonbipartite graphs. We show that the permanent of the Laplacian matrix of some classes of nonbipartite graphs can be formulated as the composite of the determinants of two matrices related to those Laplacian matrices. In addition, some recursion formulas on per L G are deduced.

A (p,ν)-extension of Srivastava's triple hypergeometric function H_ B and its properties

In this paper, we obtain a ( p , ν ) {(p,\nu)} -extension of Srivastava’s triple hypergeometric function H B ⁢ ( ⋅ ) {H_{B}(\,\cdot\,)} , by using the extended beta function B p , ν ⁢ ( x , y ) {B_{p,\nu}(x,y)} introduced in [R. K. Parmar, P. Chopra and R. B. Paris, On an extension of extended beta and hypergeometric functions, J. Class. Anal. 11 2017, 2, 91–106]. We give some of the main properties of this extended function, which include several integral representations involving Exton’s hypergeometric function, the Mellin transform, a differential formula, recursion formulas and a bounded inequality.

Compound Binomial Model with Batch Markovian Arrival Process

A compound binomial model with batch Markovian arrival process was studied, and the specific definitions are introduced. We discussed the problem of ruin probabilities. Specially, the recursion formulas of the conditional finite-time ruin probability are obtained and the numerical algorithm of the conditional finite-time nonruin probability is proposed. We also discuss research on the compound binomial model with batch Markovian arrival process and threshold dividend. Recursion formulas of the Gerber–Shiu function and the first discounted dividend value are provided, and the expressions of the total discounted dividend value are obtained and proved. At the last part, some numerical illustrations were presented.

Recursion formulas for poly-Bernoulli numbers and their applications

Recurrence formulas for generalized poly-Bernoulli polynomials are given. The formula gives a positive answer to a question raised by Kaneko. Further, as applications, annihilation formulas for Arakawa-Kaneko type zeta-functions and a counting formula for lonesum matrices of a certain type are also discussed.

Heat Kernel Approximation on Kendall Shape Space

The heat kernel on Kendall shape subspaces is approximated by an expansion. The Kendall space is useful for representing the shapes associated to collections of landmarks’positions. The Minakshisundaram-Pleijel recursion formulas are used in order to calculate the closed-form approximations of the first and second coefficients of the heat kernel expansion. Prior to the exploitation of the recursion scheme, the expression of the Laplace-Beltrami operator is adapted to the targeted space using geodesic spherical and angular coordinates.

Some results on Srivastava’s triple hypergeometric matrix functions

Inspired by the recent work by Abd-Elmageed et al., who established recursion formulas satisfied by the first Appell matrix function, namely [Formula: see text], Sahai et al. presented various recursion formulas for the Gauss hypergeometric matrix function and all four Appell matrix functions. In this paper, we obtain recursion formulas and infinte summation formulas for Srivastava’s triple hypergeometric matrix functions [Formula: see text], [Formula: see text] and [Formula: see text].

Nonsingular recursion formulas for third-body perturbations in mean vectorial elements

The description of the long-term dynamics of highly elliptic orbits under third-body perturbations may require an expansion of the disturbing function in series of the semi-major axes ratio up to higher orders. To avoid dealing with long series in trigonometric functions, we refer the motion to the apsidal frame and efficiently remove the short-period effects of this expansion in vectorial form up to an arbitrary order. We then provide the variation equations of the two fundamental vectors of the Keplerian motion by analogous vectorial recurrences, which are free from singularities and take a compact form useful for the numerical propagation of the flow in mean elements.