It is proved that generalized excursion measures can be constructed via time change of Itô’s Brownian excursion measure. A tightness-like condition on strings is introduced to prove a convergence theorem of generalized excursion measures. The convergence theorem is applied to obtain a conditional limit theorem, a kind of invariance principle where the limit is the Bessel meander.

Let up denote the normalized, generalized Bessel function of order p which depends on two parameters b and c and let λp(x) = up(x), x ≥ 0. It is proven that under some conditions imposed on p, b, and c the Askey inequality holds true for the function λp , i.e., that λp(x) +λp(y) ≤ 1 +λp(z), where x, y ≥ 0 and z = x + y. The lower and upper bounds for the function λp are also established.

The technique of Bakhvalov and Vasil'eva for evaluating Fourier integrals is generalized to integrals involving exponential and Bessel functions.

Named essentially after their close relationship with the modified Bessel function K?(z) of second kind, which is known also as Macdonald (or, a slightly different definition, Basset function), so-called polynomials yn(x) and generalized yn(x;?,?) stemmed naturally in some systematic investigations classical wave equation spherical polar coordinates. Our main purpose this invited survey-cum-exp...

In memoriam Csörgő Sándor, dear friend and great scholar Keywords: Bessel function of first kind J ν (z) Bounds for J ν (z) Positive zeros j ν,s of J ν (z) Generalized Kapteyn expansion Kapteyn series a b s t r a c t A new inequality is derived for the Bessel function of the first kind J ν (z), ν ∈ R, and a discussion of the consequent convergence is given for the generalized Kapteyn-type expan...

Virginia Kiryakova Dedicated to the 75th anniversary of Professor Gary Roach Abstract A generalization of the Poisson-type integral transformation proposed by Dimovski is applied to hyper-Bessel differential equations of arbitrary order, as a transmutation operator. By this method, their solutions are written in an explicit form, by evaluating operators of the generalized fractional calculus of...