On new approximations for the modified Bessel function of the second kind \(K_0(x)\)

A new series representation of the modified Bessel function of the second kind \(K_0(x)\) in terms of simple elementary functions (Kummer's function) is obtained. The accuracy of different orders in this expansion is analysed and has been shown not to be so good as those of different approximations found in the literature. In the sequel, new polynomial approximations for \(K_0(x)\), in the limits \(0< x\leq 2\) and \(2\leq x < \infty\), are obtained. They are shown to be much more accurate than the two best classical approximations given by the Abramowitz and Stegun's Handbook, for those intervals.

Certain Laplace transforms of convolution type integrals involving product of two special pFp functions

Recently the authors obtained several Laplace transforms of convolution type integrals involving Kummer’s function 1F1 [Appl. Anal. Discrete Math., 2018, 12(1), 257-272]. In this paper, the authors aim at presenting several new and interesting Laplace transforms of convolution type integrals involving product of two special generalized hypergeometric functions pFp by employing classical summation theorems for the series 2F1, 3F2, 4F3 and 5F4 available in the literature.

THE HEUN–SCHRÖDINGER RADIAL EQUATION FOR DH-ATOMS

This article deals with the connection between Schrödinger's multidimensional equation for DH-atoms (D≥1) and the confluent Heun equation with two auxiliary parameters ν and τ, where |1-ν| = o(1) and τ∈ℚ+, which influence the spectrum of eigenvalues, the Coulomb potential and the radial function. The case τ = ν = 1 and D = 3 corresponds to the "standard" form of Schrödinger's equation for a 3H-atom. With the help of parameter ν, e.g., some "quantum corrections" may be considered. The cases 0<τ<1 and τ>1, but â = (n-l-1)τ≥0 is an integer, change the "geometry" of the electron cloud in the atom, i.e. the so-called "exotic" 3H-like atoms arise, where Kummer's function 1F1(-â; c; z) has â zeros and the discrete spectrum depends only on Z/(νn) but not on l and τ. Diagrams of the radial functions [Formula: see text] as n≤3 are given.

ZEROS OF SCHRÖDINGER'S RADIAL FUNCTION Rnl(r) AND KUMMER'S FUNCTION 1F1(-a; c; z) AND THEIR "ANGLE" DISTRIBUTIONS

In the present paper exact formulae for the calculation of zeros of Rnl(r) and 1F1(-a; c; z), where z = 2λr, a = n - l - 1 ≥ 0 and c = 2l + 2 ≥ 2 are presented. For a ≤ 4 the method due to Tartallia and Cardono, and that due to L. Ferrai, L. Euler and J. L. Lagrange are used. In other cases (a > 4)numerical methods are employed to obtain the results (to within 10-15). For greater geometrical obviousness of the irregulary distribution (as a > 3) of zeros xk = zk - (c + a - 1) on the axis y = 0, the circular diagrams with the radius [Formula: see text] are presented for the first time. It is possible to notice some singularities of distribution of these zeros and their images — the points Tk — on the circle. For a = 3 and 4 their exact "angle" asymptotics (as c → ∞) are obtained. It is shown that in the basis of the L. Ferrari, L. Euler and J.-L. Lagrange methods, using for solving the equation 1F1(-4; c; z) = 0, one equation is obtained viz., the cubic resolvent equation of FEL-type. Calculating of zeros xk of the Rnl(r) and 1F1(z) functions enable us to show the "singular" cases (a, c) = (4, 6), (6, 4), (8, 14), …

Continued Fraction Analysis of the Duration of an Excursion in an M/M/∞ System

We show in this paper how the Laplace transform θ* of the duration θ of an excursion by the occupation process {Λ t } of an M/M/∞ system above a given threshold can be obtained by means of continued fraction analysis. The representation of θ* by a continued fraction is established and the [m−1/m] Padé approximants are computed by means of well known orthogonal polynomials, namely associated Charlier polynomials. It turns out that the continued fraction considered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion properties of hypergeometric functions, the representation of the Laplace transform θ* by means of Kummer's function is obtained. This allows us to recover an earlier result obtained via complex analysis and the use of the strong Markov property satisfied by the occupation process {Λ t }. The continued fraction representation enables us to further characterize the distribution of the random variable θ.

Continued Fraction Analysis of the Duration of an Excursion in an M/M/∞ System

We show in this paper how the Laplace transform θ* of the duration θ of an excursion by the occupation process {Λt} of an M/M/∞ system above a given threshold can be obtained by means of continued fraction analysis. The representation of θ* by a continued fraction is established and the [m−1/m] Padé approximants are computed by means of well known orthogonal polynomials, namely associated Charlier polynomials. It turns out that the continued fraction considered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion properties of hypergeometric functions, the representation of the Laplace transform θ* by means of Kummer's function is obtained. This allows us to recover an earlier result obtained via complex analysis and the use of the strong Markov property satisfied by the occupation process {Λt}. The continued fraction representation enables us to further characterize the distribution of the random variable θ.

On The Best Estimators of Coefficients of Variation, skewness and Kurtosis of Pareto Distribution and Their Variances

Here we develop the uniformly minimum variance unbiased (best) estimators of coefficient of variation, Pearson's coefficients of skewness and kurtosis of two-parameter Pareto distribution, the variance of these best estimators and the best estimators of those variances. The best estimators are in terms of Kummer's function 1 F1 and Humbert's function ø2 ; and their variances are in terms of F2 , the Appel function of second kind and K 12 , a quadruple hypergeornetric function invented by Exton during the early 1970's.

The Best Estimators of Quantiles and the Three Means of the Pareto Distribution

In this paper we develop the uniformly minimum variance unbiased (best) estimators of the quantiles, mean, geometric mean and harmonic mean of the Pareto distribuion in the case when both the shape parameter a and the scale parameter k are unknown and in cases when one of them alone is unknown. The best estimates are in terms of the Bessel Function o F1 and Kummer's function 1 F1. The variance of the best estimator are found out, which are in terms of F2 , the Appell function of second kind and ψ2 , a confluent hypergeometric function of two variables. Further we prove that every best estimator of this paper is strongly consistent.