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Several convergent expansions are available for most of the special functions of the mathematical physics, as well as some asymptotic expansions [NIST Handbook of Mathematical Functions, 2010]. Usually, both type of expansions are given in terms of elementary functions; the convergent expansions provide a good approximation for small values of a certain variable, whereas the asymptotic expansions provide a good approximation for large values of that variable. Also, quite often, those expansions are not uniform: the convergent expansions fail for large values of the variable and the asymptotic expansions fail for small values. In recent papers [Bujanda & all, 2018-2019] we have designed new expansions of certain special functions, given in terms of elementary functions, that are uniform in certain variables, providing good approximations of those special functions in large regions of the variables, in particular for large and small values of the variables. The technique used in [Bujanda & all, 2018-2019] is based on a suitable integral representation of the special function. In this paper we face the problem of designing a general theory of uniform approximations of special functions based on their integral representations. Then, we consider the following integral transform of a function g(t) with kernel h(t, z), F(z) := 1 0 h(t, z)g(t)dt. We require for the function h(t, z) to be uniformly bounded for z ∈D⊂ C by a function H(t) integrable in t ∈ [0, 1], and for the function g(t) to be analytic in an open region Ω that contains the open interval (0, 1). Then, we derive expansions of F(z) in terms of the moments of the function h, M[h(·, z), n] := 1 0 h(t, z)tndt, that are uniformly convergent for z ∈ D. The convergence of the expansion is of exponential order O(a−n), a > 1, when [0, 1] ∈ Ω and of power order O(n−b), b > 0, when [0, 1] ∈/ Ω. Most of the special functions F(z) having an integral representation may be cast in this form, possibly after an appropriate change of the integration variable. Then, special interest has the case when the moments M[h(·, z), n] are elementary functions of z, because in that case the uniformly convergent expansion derived for F(z) is given in terms of elementary functions. We illustrate the theory with several examples of special functions different from those considered in [Bujanda & all, 2018-2019]. This research was supported by the Spanish Ministry of Economía y Competitividad, project MTM2017-83490-P. The Universidad Pública de Navarra is acknowledged for its financial support.

We completely determine the set of s, t > 0 for which the function is a Bernstein function, that is Ls,t(x) is positive with completely monotonic derivative on (0, ∞). The complete monotonicity of several closely related functions is also established. © 2012 American Mathematical Society. 82 282 1097 1120 Cited By :2

Explicit formulas for general order multivariate Padé approximants of pseudo-multivariate functions are constructed on specific index sets. Examples include the multivariate forms of the exponential function E ( x _ ) = ∑ j 1 , j 2 , … , j m = 0 ∞ x 1 j 1 x 2 j 2 ⋯ x m j m ( j 1 + j 2 + ⋯ + j m ) ! , \begin{equation*} E\left (\underline {x}\right ) =\sum _{j_{1},j_{2},\ldots ,j_{m}=0}^{\infty } \frac {x_{1}^{j_{1}}x_{2}^{j_{2}}\cdots x_{m}^{j_{m}}}{\left ( j_{1}+j_{2}+\cdots +j_{m}\right ) !}, \end{equation*} the logarithm function L ( x _ ) = ∑ j 1 + j 2 + ⋯ + j m ≥ 1 x 1 j 1 x 2 j 2 ⋯ x m j m j 1 + j 2 + ⋯ + j m , \begin{equation*} L(\underline {x})=\sum _{j_{1}+j_{2}+\cdots +j_{m}\geq 1}\frac { x_{1}^{j_{1}}x_{2}^{j_{2}}\cdots x_{m}^{j_{m}}}{j_{1}+j_{2}+\cdots +j_{m}}, \end{equation*} the Lauricella function F D ( m ) ( a , 1 , … , 1 ; c ; x 1 , … , x m ) = ∑ j 1 , j 2 , … , j m = 0 ∞ ( a ) j 1 + ⋯ + j m ( c ) j 1 + ⋯ + j m x 1 j 1 ⋯ x m j m , \begin{equation*} F_{D}^{\left ( m\right ) }\left ( a,1,\ldots ,1;c;x_{1},\ldots ,x_{m}\right ) =\sum _{j_{1},j_{2},\ldots ,j_{m}=0}^{\infty }\frac {\left ( a\right ) _{j_{1}+\cdots +j_{m}}}{\left ( c\right ) _{j_{1}+\cdots +j_{m}}} x_{1}^{j_{1}}\cdots x_{m}^{j_{m}}, \end{equation*} and many more. We prove that the constructed approximants inherit the normality and consistency properties of their univariate relatives. These properties do not hold in general for multivariate Padé approximants. A truncation error upperbound is also given.

Some results are established giving conditions on f ( x ) f(x) so that its main diagonal Padé approximation R n ( x ) {R_n}(x) is of the form P n ( x ) / P n ( − x ) {P_n}(x)/{P_n}( - x) , where P n ( x ) {P_n}(x) is a polynomial in x of degree n. A number of applications to special functions are presented. Numerical computations are given for the gamma function using the "bignum" arithmetical facilities of formula manipulation languages REDUCE2, FORMAC.

The ellipsoidal wave equation, which is the most general equation derived by separation of the Helmholtz equation in confocal coordinates, presents unusual computational difficulties, and its solutions, despite their importance for physical applications, have not hitherto been effectively computed. This paper describes a successful technique, which involves the solution of a four-term recursion and the simultaneous handling of two eigenparameters. 1. Introduction. The ellipsoidal wave equation, or Lame wave equation, has been known in mathematical literature for a considerable time, certainly since 1926 when Ince (12) listed it among the equations derivable from the general second order linear differential equation with five regular singularities. More significantly, it is the equation which emerges when the technique of separation of variables is applied to the Helmholtz equation in ellipsoidal coordinates (3, 1.6). Physically, therefore, its solution is a matter of considerable interest; there are applications, among others, to problems of scalar and electromagnetic scattering by an ellipsoid, and diffraction by an elliptic plate or through an elliptic aperture. The equation has been the subject of some fairly extensive theoretical analysis, and quite a lot is known about the general properties of its solutions; e.g., (2). Remarkably, however, computed results are almost wholly lacking, and indeed the usual computational techniques which serve for most other special functions are simply not applicable; this is because of mathematical difficulties to be mentioned shortly. The only numerical results-or formulae which can give such results- hitherto published are in the form of perturbation series or asymptotic series, corresponding roughly to low frequency and high frequency approximations, respec- tively (1), (7), (10). The purpose of this paper is to report on a general method for computing the eigenvalues and the solutions. There is an interesting parallel between this problem and the corresponding problem posed by Mathieu's equation; the latter was first formulated in 1868 and used in connection with an astronomical problem (15), (11). Considerable progress was made regarding its theoretical solution, but it was not until some 60 years after Mathieu's original memoir that Ince accomplished the first systematic tabulation (13), (14). For ellipsoidal wave functions, the corresponding gap has been similar; the greater problems of tabulation have been compensated by the availability of modern computing facilities. In fact, the difficulties are not so much computational as analytic; the limitation has been imposed not by any lack of speed or storage

The Mellin transformation is used as a method for discovery of cases where the partial derivatives with respect to parameters for certain Whittaker and Bessel functions can be expressed in terms of finite sums involving these functions. These results are easily generalized to the G-function, from which, by specialization, formulas involving hypergeometric and other functions can be obtained. 1. Introduction. In a recent paper, Laurenzi (4) obtained finite sum ex- pansions for the partial derivative of W,,1/2(Z) with respect to K at a positive integral values of K expressed in terms of W1/2(z). He also obtained analogs for M,11/2 (z). Similar expressions are known for certain other functions; for example, Abramowitz and Stegun (1) list partial derivatives with respect to order for the Bessel functions J,, Y,, I,, and K, at nonnegative integral values of v and at v = ? I and for the Legendre functions aPl/Ov and aP7-1/Ov at v = 0 and aP7-1/av at v = 1. The purpose of our paper is to indicate a method of discovering such formulas and their generalizations by use of the properties of the Mellin transformation along with simple identities involving the r and i1 functions.

This paper exposes properties of the Whittaker cardinal function and illustrates the use of this function as a mathematical tool. The cardinal function is derived using the Paley-Wiener theorem. The cardinal function and the central-difference expansions are linked through their similarities. A bound is obtained on the difference between the cardinal function and the function which it interpolates. Several cardinal functions of a number of special functions are examined. It is shown how the cardinal function provides a link between Fourier series and Fourier transforms, and how the cardinal function may be used to solve integral equations.

The cumulative bivariate t t -distribution associated with random variables T 1 = X 1 / ( S / k ) 1 / 2 {T_1} = {X_1}/{(S/k)^{1/2}} , T 2 = X 2 / ( S / k ) 1 / 2 {T_2} = {X_2}/{(S/k)^{1/2}} is considered where X 1 {X_1} , X 2 {X_2} are bivariate normal with correlation coefficient ρ \rho and S S is an independent χ 2 {\chi ^2} random variable with k k degrees of freedom. Representations in terms of series and simple, one-dimensional quadratures are presented together with efficient computational procedures for the special functions used in numerical evaluation.

In this report, we discuss the representation of bivariate functions in double series of Chebyshev polynomials. For an application, we tabulate coefficients which are accurate to 20 decimals for the evaluation of ( 2 z / π ) 1 / 2 e z K v ( z ) {(2z/\pi )^{1/2}}{e^z}{K_v}(z) for all z ≧ 5 z \geqq 5 and all v , 0 ≦ v ≦ 1 v,0 \leqq v \leqq 1 . Since K v ( z ) {K_v}(z) is an even function in v and satisfies a three-term recurrence formula in v which is stable when used in the forward direction, we can readily evaluate K v ( z ) {K_v}(z) for all z ≧ 5 z \geqq 5 and all v ≧ 0 v \geqq 0 . Only 205 coefficients are required to achieve an accuracy of about 20 decimals for the z and v ranges described. Extension of these ideas for the evaluation of all Bessel functions and other important bivariate functions is under way.

In this paper, an integral analogue of Taylor’s series \[ f ( z ) = ∫ − ∞ ∞ D ω f ( z 0 ) ( z − z 0 ) ω / Γ ( ω + 1 ) d ω f(z) = \int _{ - \infty }^\infty {{D^\omega }f({z_0})} {(z - {z_0})^\omega }/\Gamma (\omega + 1)d\omega \] is discussed. D ω f ( z ) {D^\omega }f(z) is a fractional derivative of order ω \omega . Extensions of this integral are also given, one of which is an integral analogue of Lagrange’s expansion. These integrals are shown to be generalizations of the Fourier integral theorem. Several special cases of these integrals are computed, and a table of Fourier transforms emerges.

This paper presents, without derivation, several generalized density functions together with their characteristic functions. The densities are expressed variously in terms of special functions such as: I v ( x ) {I_v}(x) , the modified Bessel function of the first kind of order v; K v ( x ) {K_v}(x) , the modified Bessel function of the second kind of order v; 1 F 1 ( a ; b ; x ) _1{F_1}(a;b;x) , the confluent hypergeometric function; 2 F 1 ( a , b ; c ; x ) _2{F_1}(a,b;c;x) , the hypergeometric function; W a , b ( x ) {W_{a,b}}(x) , Whittaker’s function; Φ 3 ( β ; γ ; b x , c x ) {\Phi _3}(\beta ;\gamma ;bx,cx) , a generalized hypergeometric function (type I); \[ Φ 2 ( b , c , d ; γ ; λ x , τ x , β x ) , {\Phi _2}(b,c,d;\gamma ;\lambda x,\tau x,\beta x), \] a generalized hypergeometric function (type II); and ϕ λ μ ( b v μ ) \phi _\lambda ^\mu (b{v^\mu }) , a generalized Bessel type function. The first five cases are summarized from the work of Laha [7], Pearson [25] and Raj [26] while Cases 13 through 19 have not previously appeared in the literature of statistics or Fourier transforms. In what follows, the usual notation f ( x ) f(x) , for a density function, and φ ( t ) \varphi (t) , for a characteristic function, will be used with all parameters considered as real quantities: \[ φ ( t ) = ∫ − ∞ ∞ exp ⁡ ( i t x ) f ( x ) d x . \varphi (t) = \int _{ - \infty }^\infty {\exp (itx)f(x)\;dx.} \]

This paper is concerned with numerical procedures for the evaluation of the zeros, with respect to order, of Hankel functions and their derivatives in cases when the arguments of these functions are held fixed. Using Olver’s asymptotic expansions, two auxiliary tables have been computed, one appropriate for real and the other for purely imaginary argument. These tables, included herein, permit the calculation of rather accurate approximations to the desired ν \nu -zeros for wide ranges of argument and index. Moreover, from the given tabular entries, the errors attendant with any approximate ν \nu -zero so determined can be easily estimated.

I present corrected values for the Tables IV and V together with an extension of the Tables I-V. I also present an 80D value of my constant F.