Related papers: Addition Theorems Via Continued Fractions
The goal of this paper is to extend the classical and multiplicative fractional derivatives. For this purpose, it is introduced the new extended modified Bessel function and also given an important relation between this new function…
Addition theorems can be constructed by doing three-dimensional Taylor expansions according to $f (\mathbf{r} + \mathbf{r}') = \exp (\mathbf{r}' \cdot \mathbf{\nabla}) f (\mathbf{r})$. Since, however, one is normally interested in addition…
We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial…
In this paper, we aim to present new extensions of incomplete gamma, beta, Gauss hypergeometric, confluent hypergeometric function and Appell-Lauricella hypergeometric functions, by using the extended Bessel function due to Boudjelkha [4].…
It is proved by the method of partial fraction expansions and Sturm's oscillation theory that the zeros of certain Hankel transforms are all real and distributed regularly between consecutive zeros of Bessel functions. As an application,…
A contiguous relation for complementry pairs of very well poised balanced ${}_{10}\phi_9$ basic hypergeometric functions is used to derive an explict expression for the associated continued fraction. This generalizes the continued fraction…
We provide a version of the celebrated theorem of Koml\'os in which, rather then random quantities, a sequence of finitely additive measures is considered. We obtain a form of the subsequence principle and some applications.
The established technique of eliminating upper or lower parameters in a general hypergeometric series is profitably exploited to create pathways among confluent hypergeometric functions, binomial functions, Bessel functions, and exponential…
The theoretical computing of special values assumed by the hypergeometric functions has a high interest not only on its own, but also in sight of the remarkable implications to both pure Mathematics and Mathematical Physics. Accordingly, in…
Elementary transformations of equations $A\psi=\lambda\psi$ are considered. The invertibility condition (Theorem 1) is established and similar transformations of Riccati equations in the case of second order differential operator $A$ are…
Fractional vector calculus is discussed in the spherical coordinate framework. A variation of the Legendre equation and fractional Bessel equation are solved by series expansion and numerically. Finally, we generalize the hypergeometric…
Series involving hypergeometric functions are used to derive, extend and evaluate integrals involving the product of two Bessel functions of the first kind $J_{u}(a z)$ $J_{v}(b z)$ with order $u,v$, studied by Landau et al. The method used…
We build a bridge from density combinatorics to dimension theory of continued fractions. We establish a fractal transference principle that transfers common properties of subsets of $\mathbb N$ with positive upper density to properties of…
A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized…
The paper is a survey of recent results in analysis of additive functions over function fields motivated by applications to various classes of special functions including Thakur's hypergeometric function. We consider basic notions and…
Motivated by applications in noncommutative geometry we prove several value range estimates for even convergents and tails, and odd reverse sequences of Stieltjes type continued fractions with bounded ratio of consecutive elements, and show…
Following van der Poorten, we consider a family of nonlinear maps which are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus $\mathrm{g}$. Using the connection with the classical theory of…
The notion of a duality between two derived functors as well as an extension theorem for derived functors to larger categories in which they need not be defined is introduced. These ideas are then applied to extend and study the coext…
Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death,…
In this paper we introduce a link between geometry of ordinary continued fractions and trajectories of points that moves according to the second Kepler law. We expand geometric interpretation of ordinary continued fractions to the case of…