We construct a testing function space, which is equipped with the topology that is generated by Lν,p - multinorm of the differential operatorAx = x2 - x d/dx [x d/dx],and its k-th iterates Akx, where k = 0, 1, ... , and A0xφ = φ. Comparing with other testing-function spaces, we introduce in its dual the Kontorovich-Lebedev transformation for distributions with respect to a complex index. The existence, uniqueness, imbedding and inversion properties are investigated. As an application we find a solution...

Mathematics Subject Classification: Primary 33E20, 44A10; Secondary 33C10, 33C20, 44A20By using integral representations for several Mathieu type series, a number of integral transforms of Hankel type are derived here for general families of Mathieu type series. These results generalize the corresponding ones on the Fourier transforms of Mathieu type series, obtained recently by Elezovic et al. [4], Tomovski [19] and Tomovski and Vu Kim Tuan [20].

Mathematics Subject Classification: 33D15, 44A10, 44A20The present paper deals with the evaluation of the q-Laplace transforms of a product of basic analogues of the Bessel functions. As applications, several useful special cases have been deduced.

The zeros $c_k\left(\nu \right)$ of the solution $z(t,\nu )$ of the differential equation $z^{\text{'}\text{'}}+q(t,\nu )z=0$ are investigated when $\underset{t\to \infty }{lim}q(t,\nu )=1$, ${\int }^{\infty }|q(t,\nu )-1|dt<\infty$ and $q(t,\nu )$ has some monotonicity properties as $t\to \infty$. The notion $c_{\kappa }\left(\nu \right)$ is introduced also for $\kappa$ real, too. We are particularly interested in solutions $z(t,\nu )$ which are “close" to the functions $sint$, $cost$ when $t$ is large. We derive a formula for $d{c}_{\kappa }\left(\nu \right)/d\nu$ and apply the result to Bessel differential equation, where we introduce new pair of linearly independent solutions replacing the usual pair $J_{\nu }\left(t\right)$, $Y_{\nu }\left(t\right)$. We show the concavity of $c_{\kappa }\left(\nu \right)$ for $\left|\nu \right|\ge \frac{1}{2}$ and also...

Si prova una nuova formula di rappresentazione per la famosa funzione di Airy. Ne viene data applicazione per la determinazione di certi bounds significativi per la funzione stessa.

The paper deals with the computation of Riccati-Bessel functions. A modification of Miller method is presented together with estimates of relative errors.

We deal with several classes of integral transformations of the form $$f\left(x\right)\to D{\int }_{{ℝ}_{+}^2}\frac{1}{u}({\mathrm{e}}^{-ucosh(x+v)}+{\mathrm{e}}^{-ucosh(x-v)})h\left(u\right)f\left(v\right)\mathrm{d}u\mathrm{d}v,$$ where $D$ is an operator. In case $D$ is the identity operator, we obtain several operator properties on $L_p\left({ℝ}_{+}\right)$ with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on $L_2\left({ℝ}_{+}\right)$ and define the inversion formula. Further, for an other class of differential operators of finite...