New formulae for higher order derivatives and applications

We present new formulae (the Slevinsky–Safouhi formulae I and II) for the analytical development of higher order derivatives. These formulae, which are analytic and exact, represent the kth derivative as a discrete sum of only k+1 terms. Involved in the expression for the kth derivative are coefficients of the terms in the summation. These coefficients can be computed recursively and they are not subject to any computational instability. As examples of applications, we develop higher order derivatives of Legendre functions, Chebyshev polynomials of the first kind, Hermite functions and Bessel functions. We also show the general classes of functions to which our new formula is applicable and show how our formula can be applied to certain classes of differential equations. We also presented an application of the formulae of higher order derivatives combined with extrapolation methods in the numerical integration of spherical Bessel integral functions.     

Bessel Potentials in Ahlfors Regular Metric Spaces

In this paper we introduce Bessel potentials and the Sobolev potential spaces resulting from them in the context of Ahlfors regular metric spaces. The Bessel kernel is defined using a Coifman type approximation of the identity, and we show integration against it improves the regularity of Lipschitz, Besov and Sobolev-type functions. For potential spaces, we prove density of Lipschitz functions, and several embedding results, including Sobolev-type embedding theorems. Finally, using singular integrals techniques such as the T1 theorem, we find that for small orders of regularity Bessel potentials are inversible, its inverse in terms of the fractional derivative, and show a way to characterize potential spaces, concluding that a function belongs to the Sobolev potential space if and only if itself and its fractional derivative are in L^p. Moreover, this characterization allows us to prove these spaces in fact coincide with the classical potential Sobolev spaces in the Euclidean case.     

The Numerical Calculation of Odd-degree Polynomial Splines with Equi-spaced Knots

Constructive equations for polynomial splines of odd-degree2r+1 with knots x₁ = x₀+ih, i = 0(1)n are formulated in termsof even-order derivatives, odd-order derivatives being givenby explicit formulae which are shown to be identical with truncatedTaylor series expansions of the same form. The defining equationsare written in a manner which reveals a strong connection withthe well-known Numerov formula. Solution of the equations byblock iterative methods is considered for the case when evenderivatives are specified at x = x₀ and x_n. Block Jacobi andblock Gauss—Seidel iteration are shown to be convergentfor all positive r and optimum acceleration parameters for blockS.O.R. are given for r = 2(1)6. It is shown that distinct computationaladvantages result from relaxing the condition for a true splinefit and considering instead a truncated spline of higher order.     

Definiteness of the Peano Kernel Associated with the Polyharmonic Mean Value Property

The definiteness of the Peano kernel is proved for a functionalassociated with the mean-value property of Picone and Brambleand Payne for polyharmonic functions in the ball. An importantcorollary of this is that if a function f satisfying (–1)^ppf>0vanishes on p concentric spheres centered at 0, then f(0)>0.This generalizes a well-known property of subharmonic functions(which arise in the special case p = 1).     

S-adic L-Functions Attached to the Symmetric Square of a Newform

In order to apply the ideas of Iwasawa theory to the symmetricsquare of a newform, we need to be able to define non-archimedeananalogues of its complex L-series. The interpolated p-adic L-functionis closely connected via a "Main Conjecture" with certain Selmergroups over the cyclotomic Z_p-extension of Q. In the p-ordinarycase these functions are well understood. In this article we extend the interpolation to an arbitraryset S of good primes (not necessarily satisfying ordinarityconditions). The corresponding S-adic functions can be characterisedin terms of certain admissibility criteria. We also allow interpolationat particular primes dividing the level of the newform. One interesting application is to the symmetric square of amodular elliptic curve E defined over Q. Our constructions yieldp-adic L-functions at all primes of stable or semi-stable reduction.If p is ordinary or multiplicative the corresponding analyticfunction is bounded; if p is supersingular our function behaveslike log²(1 + T). 1991 Mathematics Subject Classification: 11F67,11F66, 11F33, 11F30     

Convergence of Magnus integral addition theorems for confluent hypergeometric functions

In 1946, Magnus presented an addition theorem for the confluent hypergeometric function of the second kind U with argument x+y expressed as an integral of a product of two U's, one with argument x and another with argument y. We take advantage of recently obtained asymptotics for U with large complex first parameter to determine a domain of convergence for Magnus' result. Using well-known specializations of U, we obtain corresponding integral addition theorems with precise domains of convergence for modified parabolic cylinder functions, and Hankel, Macdonald, and Bessel functions of the first and second kind with order zero and one.     

Certain Rational Function Approximations to (1+x2) -1/2

With a view to numerical applications, particular rational functionapproximations to (1+x²) – are constructed by using continuedfraction expansions. It is found that the zeros of the denominatorpolynomials, which occur in the convergents of the continuedfractions, can be determined explicitly. This fact enables therational function approximations to be expressed convenientlyin partial fractions. The resulting expressions are then appliedto examples in which x is regarded either as a real variable,or as the Laplace operator, the latter case giving rise to someinteresting approximations to Bessel functions of integral orderand complex argument. The character of the errors of the variousapproximations is discussed, and expressions for the error boundsdeveloped. These are in good agreement with computed results.The method followed in this paper is capable of further development.In particular it can be extended to those functions which possesssuitable Taylor and asymptotic expansions.     

Application of pseudo‐trigonometry to Bessel series and integrals of Bessel functions

In this article, an extension of the Laplace transform of J_n (t) to pseudo‐trigonometric function is discussed. We are seeking elementary functions expressed by Bessel series. It is shown that the result is applicable to the solution of the first‐order differential equation. The expression of modified Bessel integral formulas in pseudo‐trigonometric function is also discussed.     

On the summation of Schlömilch's series

ABSTRACT

Schlömilch's series is named after the German mathematician Oscar Xavier Schlömilch, who derived it in 1857 as a Fourier series type expansion in terms of the Bessel function of the first kind. However, except for Bessel functions, here we consider an expansion in terms of Struve functions or Bessel and Struve integrals as well. The method for obtaining a sum of Schlömilch's series in terms of the Bessel or Struve functions is based on the summation of trigonometric series, which can be represented in terms of the Riemann zeta and related functions of reciprocal powers and in certain cases can be brought in the closed form, meaning that the infinite series are represented by finite sums. By using Krylov's method we obtain the convergence acceleration of the trigonometric series.     

Arithmetical Results on Certain Multivariate Power Series

Power series with non-zero convergence radius R(f) are considered, and the arithmeticalnature (that is, irrationality, or even transcendence) of thecorresponding multivariate series is studied if x₁, ..., x_m and the sequence (f(n)) satisfy appropriatearithmetical conditions. It follows that such arithmetical resultscan be written down easily if linear independence results onthe function F(x), defined in |x| < R(f) by the originalone-dimensional power series, and possibly its derivatives atthe points x_µ are known. Some typical applications areexplicitly stated. 2000 Mathematics Subject Classification 11J72.     

A transform method for linear evolution PDEs on a finite interval

We study initial boundary value problems for linear scalar evolutionpartial differential equations, with spatial derivatives ofarbitrary order, posed on the domain {t > 0, 0 < x <L}. We show that the solution can be expressed as an integralin the complex k-plane. This integral is defined in terms ofan x-transform of the initial condition and a t-transform ofthe boundary conditions. The derivation of this integral representationrelies on the analysis of the global relation, which is an algebraicrelation defined in the complex k-plane coupling all boundaryvalues of the solution. For particular cases, such as the case of periodic boundaryconditions, or the case of boundary value problems for even-orderPDEs, it is possible to obtain directly from the global relationan alternative representation for the solution, in the formof an infinite series. We stress, however, that there existinitial boundary value problems for which the only representationis an integral which cannot be written as an infinite series.An example of such a problem is provided by the linearized versionof the KdV equation. Similarly, in general the solution of odd-orderlinear initial boundary value problems on a finite intervalcannot be expressed in terms of an infinite series.     

The evaluation of Bessel functions via exp-arc integrals

A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the choice of the series changing at some appropriate argument magnitude, depending on the number of digits required. In a recent paper, D. Borwein, J. Borwein, and R. Crandall [D. Borwein, J.M. Borwein, R. Crandall, Effective Laguerre asymptotics, preprint at http://locutus.cs.dal.ca:8088/archive/00000334/] derived a series for an “exp-arc” integral which gave rise to an absolutely convergent series for the J and I Bessel functions with integral order. Such series can be rapidly evaluated via recursion and elementary operations, and provide a viable alternative to the conventional ascending-asymptotic switching. In the present work, we extend the method to deal with Bessel functions of general (non-integral) order, as well as to deal with the Y and K Bessel functions.     

Laplace's method on a computer algebra system with an application to the real valued modified Bessel functions

We examine a Maple implementation of two distinct approaches to Laplace's method used to obtain asymptotic expansions of Laplace-type integrals. One algorithm uses power series reversion, whereas the other expands all quantities in Taylor or Puiseux series. These algorithms are used to derive asymptotic expansions for the real valued modified Bessel functions of pure imaginary order and real argument that mimic the well-known corresponding expansions for the unmodified Bessel functions.     

Integral representations and summations of the modified Struve function

It is known that the Struve function H _ν and the modified Struve function L _ν are closely connected to the Bessel function of the first kind J _ν and to the modified Bessel function of the first kind I _ν and possess representations through higher transcendental functions like the generalized hypergeometric ₁ F ₂ and the Meijer G function. Also, the NIST project and Wolfram formula collection contain a set of Kapteyn type series expansions for L _ν (x). In this paper firstly, we obtain various another type integral representation formulae for L _ν (x) using the technique developed by D. Jankov and the authors. Secondly, we present some summation results for different kind of Neumann, Kapteyn and Schlömilch series built by I _ν (x) and L _ν (x) which are connected by a Sonin–Gubler formula, and by the associated modified Struve differential equation. Finally, solving a Fredholm type convolutional integral equation of the first kind, Bromwich–Wagner line integral expressions are derived for the Bessel function of first kind J _ν and for an associated generalized Schlömilch series.     

On the nth Quantum Derivative

The nth quantum derivative D_nf(x) of the real-valued functionf is defined for each real non-zero x as where is the q-binomial coefficient.If the nth Peano derivative exists at x, which is to say thatif f can be approximated by an nth degree polynomial at thepoint x, then it is not hard to see that D_nf(x) must also existat that point. Consideration of the function |1–x| atx = 1 shows that the second quantum derivative is more generalthan the second Peano derivative. However, it can be shown thatthe existence of the nth quantum derivative at each point ofa set necessarily implies the existence of the nth Peano derivativeat almost every point of that set.     

Mathieu functions and coulomb spheroidal functions in the electrostatic probe theory

A spherical probe placed in a slowly moving collisional plasma with a large Debye length λ_D → ∞ is considered. The partial differential equation describing the electron concentration around the probe is reduced to two ordinary differential equations, namely, to the equation for Coulomb spheroidal functions and Mathieu’s modified equation with the parameter a of the latter related to the eigenvalue λ of the former by the relation a = λ + 1/4. It is shown that the solutions of Mathieu’s equation are Mathieu functions of half-integer order, which are expressed as series in terms of spherical Bessel functions and series of products of Bessel functions. These Mathieu functions are numerically constructed for Mathieu’s modified and usual equations.     

Some Eigenfunction Methods for Computing a Numerical Fourier Transform

A number of methods for calculating the Fourier transform ofa function given numerically are studied. These methods exploitthe fact that the Hermite functions are eigen-functions of theFourier transform. The transforms of four types of functionsare considered: (i) functions of the form p(x) exp (–x²/2),where p(x) is a polynomial, (ii) functions with bounded support.(iii) rapidly decreasing functions, and (iv) functions whosetransform has bounded support. In each case algorithms for calculatingthe transformed function are derived. Error estimates are madein two of the cases and results of numerical experiments presentedin an appendix.     

Approximate Monotonicity: Theory and Applications

The ideas of value distribution for measurable functions fromR to R are applied to functions which are approximately monotonicon sets of positive measure. (For definitions see 1.) A functionp(x) is introduced, describing the local relative value distributionin the neighbourhood of a point x, and it is shown that almosteverywhere p(x) = 0 or wherever p(x) exists, implying approximate differentiability, with thefunction approximately oscillatory elsewhere. These resultsare applied to the analysis of angular boundary behaviour forHerglotz functions, where they have implications for the spectralanalysis of differential and other operators.     

The Computation of all the Derivatives of a B-spline Basis

Splines are currently much used in the field of interpolationto functions and their derivatives. In this context for a givenargument two relationships between derivatives of B-spline basesof consecutive orders are derived. Using these relationshipsit is shown there are (K—1)!((K—m1)!m!) schemesfor the evaluation of the mth derivative of a B-spline basisof order k. Analyses of error growth in terms of a matrix notationare carried out in order to see which of the schemes is themost numerically stable, for uniform or highly non-uniform knotsets. The computation of the B-spline basis of order ^K and its(K—1)th derivative are shown to have small a priori relativeerror bounds.     

Convolution of Rayleigh functions with respect to the Bessel index

A convolution of Rayleigh functions with respect to the Bessel index can be treated as a special function in its own right. It appears in constructing global-in-time solutions for some semilinear evolution equations in circular domains and may control the smoothing effect due to nonlinearity. An explicit representation for it is derived which involves the special function ψ(x) (the logarithmic derivative of the Γ-function). The properties of the convolution in question are established. Asymptotic expansions for small and large values of the argument are obtained and the graph is presented.