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The use of operational methods of different nature is shown to be a fairly powerful tool to study different problems regarding the theory of Legendre and Legendre-like polynomials. We show how the use of the well known integral…

A new representation of solutions to the equation $-y"+q(x)y=\omega^2 y$ is obtained. For every $x$ the solution is represented as a Neumann series of Bessel functions depending on the spectral parameter $\omega$. Due to the fact that the…

We study the non homogeneous quadratic Bessel zeta function $\zeta_{RB}(s,\nu,a)$ defined as the sum of the square of the positive zeros of the Bessel function $J_\nu(z)$ plus a positive constant. In particular, we give explicit formulas…

This paper deals with the study of the zeros of the big $q$-Bessel functions. In particular, we prove a new orthogonality relations for this functions similar to the one for the classical Bessel functions. Also we give some applications…

Two representations of the extended gamma functions $\Gamma^{2,0}_{0,2}[(b,x)]$ are proved. These representations are exploited to find a transformation relation between two Fox's $H$-functions. These results are used to solve Fox's…

N. Kishore, Proc. Amer. Math. Soc. 14 (1963), 523, considered the Rayleigh functions sigma_n, sums of the negative even powers of the (non-zero) zeros of the Bessel function J_nu(z) and provided a convolution type sum formula for finding…

In this article we consider certain types of weighted generalized functions associated with nondegenerate quadratic forms. Such functions and their derivatives are used for constructing fundamental solutions of iterated ultra-hyperbolic…

We show connections between a special type of addition formulas and a theorem of Stieltjes and Rogers. We use different techniques to derive the desirable addition formulas. We apply our approach to derive special addition theorems for…

Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases, we show that we obtain the best possible constant or that our bounds are tight in certain limits. We…

A new generalization of the modified Bessel function of the second kind $K_{z}(x)$ is studied. Elegant series and integral representations, a differential-difference equation and asymptotic expansions are obtained for it thereby…

The two-dimensional Helmholtz equation separates in elliptic coordinates based on two distinct foci, a limit case of which includes polar coordinate systems when the two foci coalesce. This equation is invariant under the Euclidean group of…

A four-term recurrence relation for squared spherical Bessel functions is shown to yield closed-form expressions for several types of finite weighted sums of these functions. The resulting sum rules, which may contain an arbitrarily large…

Using Bauer's expansion and properties of spherical Bessel and Legender functions, we deduce a new transform and briefly indicate its use.

Most of the special functions of mathematical physics are connected with the representation of Lie groups. The action of elements $D$ of the associated Lie algebras as linear differential operators gives relations among the functions in a…

We introduce the fractal expansions, sequences of integers associated to a number. We show that these sequences characterize the O-sequences and encode some information on lex segment ideals. Moreover, we introduce a numerical functions…

We present (bi-)symmetric generating functions for the joint distributions of Euler-Stirling statistics on permutations, including the number of descents ($\mathsf{des}$), inverse descents ($\mathsf{ides}$), the number of left-to-right…

In this paper, we prove that the Bessel functions are locally integrable for all connected split reductive linear algebraic groups over a p-adic field $F$ and the Bessel distributions are given by integrals against these Bessel functions,…

Up-down permutations are counted by tangent resp. secant numbers. Considering words instead, where the letters are produced by independent geometric distributions, there are several ways of introducing this concept; in the limit they all…

We construct the existence theory for generalized fractional Bessel differential equations and find the solutions in the form of fractional or logarithmic fractional power series. We figure out the cases when the series solution is unique,…

We prove that every irreducible, admissible representation of GSp(4,F), where F is a non-archimedean local field of characteristic zero, admits a Bessel functional, provided the representation is not one-dimensional. Given such a…