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Motivated mainly by certain interesting recent extensions of the Gamma, Beta and hypergeometric functions, we introduce here new extensions of the Beta function, hypergeometric and confluent hypergeometric functions. We systematically…
Using some basic properties of the gamma function, we evaluate a simple class of infinite products involving Dirichlet characters as a finite product of gamma functions and, in the case of odd characters, as a finite product of sines. As a…
By applying projection operators to state vectors of coordinates we obtain subspaces in which these states are no longer normalized according to Dirac's delta function but normalized according to what we call "incomplete delta functions".…
We present explicit expressions for multi-fold logarithmic integrals that are equivalent to sums over polygamma functions at integer argument. Such relations find application in perturbative quantum field theory, quantum chemistry, analytic…
This paper is a tutorial that demonstrates various methods from the Colombeau theory of generalized functions in the context of semilinear wave equations. The Colombeau generalized functions constitute differential algebras that contain the…
The paper considers a universal approach that allows one to quite simply obtain nonlinear asymptotic estimates of various summation functions. It is shown the application of this approach to the asymptotic estimation of divergent Dirichlet…
We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions…
The problem of evaluation of higher derivatives of Airy functions in a closed form is investigated. General expressions for the polynomials which have arisen in explicit formulae for these derivatives are given in terms of particular values…
In this short article, we non-perturbatively derive a recursive formula for the Green's function associated with finitely many point Dirac delta potentials in one dimension. We also extend this formula to the case for the Dirac delta…
This note is to show that the position-space embedding in \cite{ESP2021embedding} in the position and occupation bases can be obtained by considering the dynamics of Dirac delta function $$\delta(\mathbf{x}- \mathbf{z}(t)) =…
We have shown that in some region where the Euler integral of the first kind diverges, the Euler formula defines a generalized function. The connected of this generalized function with the Dirac delta function is found.
In this paper, we give a two dimensional analogue of the Euler-MacLaurin summation formula. By using this formula, we obtain an integral representation of Weil's elliptic functions which was introduced in the book "Elliptic functions…
We introduce the regular product for Cullen-regular quaternionic functions in a manner that does not depend upon a representation in power series but upon another, weaker kind of representation. The special case when the functions are…
Dirac-Coulomb type differential equation and its solution relativistic exponential-type spinor orbitals are introduced. They provide a revised form for operator invariants, namely Dirac invariants, simplifying the treatment of the angular…
For $0\neq x>-1$ let $$\Delta(x)={{\ln \Gamma(x+1)} \over x}.$$ Recently Adell and Alzer proved the complete monotonicity of $\Delta'$ on $(-1,\infty)$ by giving an integral representation of $(-1)^n \Delta^{(n+1)}(x)$ in terms of the…
In this paper, by using the method of Contour Integral Representations and the Theorem of Residues and integral representations of series, we discuss the analytic representa- tions of parametric Euler sums that involve harmonic numbers…
Spherical Bessel functions appear commonly in many areas of physics wherein there is both translation and rotation invariance, and often integrals over products of several arise. Thus, analytic evaluation of such integrals with different…
A new algorithm to calculate Coulomb wave functions with all of its arguments complex is proposed. For that purpose, standard methods such as continued fractions and power/asymptotic series are combined with direct integrations of the…
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 series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…