Related papers: A Remarkable Identity Involving Bessel Functions
In this work the authors use their contour integral method to derive a double integral connected to the modified Bessel function of the second kind and express it in terms of the Lerch function. There are some useful results relating double…
We present a new type of integral that is supposed to extend the usability of the Lebesgue integral in certain types of investigations. It is based on the Hausdorff dimension and measure. We examine the basic properties of the integral and…
We consider several families of binomial sum identities whose definition involves the absolute value function. In particular, we consider centered double sums of the form \[S_{\alpha,\beta}(n) :=…
In this paper, we continue our investigation of double sums where the inner sum is binomial but incomplete. We prove many new results for these types of double sums associated with binomial transform pairs. As applications we deduce new…
Often in mathematics it is useful to summarize a multivariate phenomenon with a single number and in fact, the determinant -- which is represented by det -- is one of the simplest cases. In fact, this number it is defined only for square…
In this short note we provide some algebraic identity with a proof exploiting its probabilistic interpretation. We show several consequences of the identity, in particular we obtain a new representation of a Stirling number of second kind,…
The present article is devoted to one example which related to the Salem function. The main attention is given to properties of one type of functions including items related to functional equations, graphs, the Lebesgue integral, etc.
Using probability theory we derive an expression for the sum of a series of definite integrals involving upper incomplete Gamma functions. In the proof, a normal variance mixture distribution with Beta mixing distributions plays a crucial…
We prove a new universal identity for umbral operators. This motivates the definition of a subclass satisfying a simplified identity, which we fully characterize. The results are illustrated with common examples of the theory of umbral…
By treating the multiple argument identity of the logarithm of the Gamma function as a functional equation, we obtain a curious infinite product representation of the $sinc$ function in terms of the cotangent function. This result is…
Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving…
A new formula is derived that generalises an earlier result for the infinite integral over three spherical Bessel functions. The analytical result involves a finite sum over associated Legendre functions, $P_l^m(x)$, of degree $l$ and order…
We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage…
We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting…
This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results…
In this paper, new integral representations for the Bessel $J$ and $I$ functions were presented and their results were used to derive an expression for the Modified Bessel $K$ function.
We examine convergent representations for the sum of a decaying exponential and a Bessel function in the form \[\sum_{n=1}^\infty \frac{e^{-an}}{(\frac{1}{2} bn)^\nu}\,J_\nu(bn),\] where $J_\nu(x)$ is the Bessel function of the first kind…
We obtain new trigonometric identities, which are product-to-sum type formulas for derivative of the cosecant and cotangent functions. Further, from specializations of our formulas, we derive new reciprocity laws of generalized Dedekind…
New index transforms are investigated, which contain as the kernel products of the Bessel and modified Bessel functions. Mapping properties and invertibility in Lebesgue spaces are studied for these operators. Relationships with the…
We consider a generalisation of a definite integral involving the Bessel function of the first kind. It is shown that this integral can be expressed in terms of the Fox-Wright function ${}_p\Psi_q(z)$ of one variable. Some consequences of…