A mathematical analysis of poroacoustic traveling wave phenomena is presented. Assuming that the fluid phase satisfies the perfect gas law and that the drag offered by the porous matrix is described by Darcy's law, exact traveling wave solutions (TWS)s, as well as asymptotic/approximate expressions, are derived and examined. In particular, stability issues are addressed, shock and acceleration waves are shown to arise, and special/limiting cases are noted. Lastly, connections to other fields are...

We study the action of elementary shift operators on spherical functions on ordered symmetric spaces ${ℳ}_{m,n}$ of Cayley type, where $m\in \mathbb{N}$ denotes the multiplicity of the short roots and $n\in \mathbb{N}$ the rank of the symmetric space. For $m$ even we apply this to prove a Paley-Wiener theorem for the spherical Laplace transform defined on ${ℳ}_{m,n}$ by a reduction to the rank 1 case. Finally we generalize our notions and results to $B{C}_n$ type root systems.

2000 Math. Subject Classification: 26A33; 33E12, 33E30, 44A15, 45J05The Caputo fractional derivative is one of the most used definitions of a fractional derivative along with the Riemann-Liouville and the Grünwald- Letnikov ones. Whereas the Riemann-Liouville definition of a fractional derivative is usually employed in mathematical texts and not so frequently in applications, and the Grünwald-Letnikov definition – for numerical approximation of both Caputo and Riemann-Liouville fractional derivatives,...

We study extension of $p$-trigonometric functions ${sin}_p$ and ${cos}_p$ to complex domain. For $p=4,6,8,\cdots$, the function ${sin}_p$ satisfies the initial value problem which is equivalent to (*) $$-{\left(u^{\text{'}}\right)}^{p-2}{u}^{\text{'}\text{'}}-u^{p-1}=0,u\left(0\right)=0,u^{\text{'}}\left(0\right)=1$$ in $ℝ$. In our recent paper, Girg, Kotrla (2014), we showed that ${sin}_p\left(x\right)$ is a real analytic function for $p=4,6,8,\cdots$ on $(-{\pi }_p/2,{\pi }_p/2)$, where ${\pi }_p/2={\int }_0^1{(1-s^p)}^{-1/p}$. This allows us to extend ${sin}_p$ to complex domain by its Maclaurin series convergent on the disc $\{z\in ℂ:|z|<{\pi }_p/2\}$. The question is whether this extensions ${sin}_p\left(z\right)$ satisfies (*) in the sense of differential equations in complex domain. This interesting...

In this work we consider the Dunkl operator on the complex plane, defined by $${𝒟}_{k}f\left(z\right)=\frac{d}{dz}f\left(z\right)+k\frac{f\left(z\right)-f(-z)}{z},k\ge 0.$$ We define a convolution product associated with ${𝒟}_k$ denoted ${*}_k$ and we study the integro-differential-difference equations of the type $\mu {*}_{k}f={\sum }_{n=0}^{\infty }{a}_{n,k}{𝒟}_k^{n}f$, where $\left(a_{n,k}\right)$ is a sequence of complex numbers and $\mu$ is a measure over the real line. We show that many of these equations provide representations for particular classes of entire functions of exponential type.

A procedure is proposed in order to expand $w={\prod }_{j=1}^N{P}_{i_j}\left(x\right)={\sum }_{k=0}^M{L}_k{P}_k\left(x\right)$ where $P_i\left(x\right)$ belongs to aclassical orthogonal polynomial sequence (Jacobi, Bessel, Laguerre and Hermite) ($M={\sum }_{j=1}^N{i}_j$). We first derive a linear differential equation of order $2^N$ satisfied by w, fromwhich we deduce a recurrence relation in k for the linearizationcoefficients $L_k$. We develop in detail the two cases ${\left[P_i\left(x\right)\right]}^N$, $P_i\left(x\right)P_j\left(x\right)P_k\left(x\right)$ and give the recurrencerelation in some cases (N=3,4), when the polynomials $P_i\left(x\right)$are monic Hermite orthogonal polynomials.

Let $P_k$ be any sequence of classical orthogonal polynomials of a discrete variable. We give explicitly a recurrence relation (in k) for the coefficients in $P_i{P}_j={\sum }_{k}c(i,j,k)P_k$, in terms of the coefficients σ and τ of the Pearson equation satisfied by the weight function ϱ, and the coefficients of the three-term recurrence relation and of two structure relations obeyed by $P_k$.

This work is devoted to the study of a Cauchy problem for a certain family of q-difference-differential equations having Fuchsian and irregular singularities. For given formal initial conditions, we first prove the existence of a unique formal power series X̂(t,z) solving the problem. Under appropriate conditions, q-Borel and q-Laplace techniques (firstly developed by J.-P. Ramis and C. Zhang) help us in order to construct actual holomorphic solutions of the Cauchy problem whose q-asymptotic expansion...

We prove identities involving sums of Legendre and Jacobi polynomials. The identities are related to Green’s functions for powers of the invariant Laplacian and to the Minakshisundaram-Pleijel zeta function.