Asymptotic expansion of n-dimensional Faxén-type integrals

The asymptotic expansion of n-dimensional extensions of Faxén’s integral In(z) are derived for large complex values of the variable z. The theory relies on the asymptotics of the generalised hypergeometric, orWright, function. The coefficients in the exponential expansion are obtained by means of an algorithm applicable for arbitrary n. Numerical examples are given to illustrate the accuracy of the expansions

Asymptotic approximations for n!

Several approximations for n! have recently appeared in the literature. We show here how these approximations can be derived by expansion of certain polynomials in inverse powers of n and comparison with Stirling’s asymptotic series. Some new approximations are also given. The same procedure is applied to generate approximations for the ratio of two gamma functions

The asymptotic expansion of a generalised Mathieu series

We obtain the asymptotic expansion of a generalised Mathieu series and its alternating variant for large complex values of the variable by means of a Mellin transform approach. Numerical examples are presented to demonstrate the accuracy of these expansions

Exponentially small expansions of the confluent hypergeometric functions

The asymptotic expansions of the confluent hypergeometric functions 1F1(a; b; z) and U(a, b, z) are examined as |z|→∞ on the Stokes lines arg z = ±π. Particular attention is given to the exponentially small contributions associated with these two functions. Numerical results demonstrating the accuracy of the expansions are given

Asymptotics of the Gauss hypergeometric function with large parameters, I

We obtain asymptotic expansions for the Gauss hypergeometric function F(a+ε1λ,b+ε2λ;c+ε3λ;z) as |λ| →∞ when the εj are finite by an application of the method of steepest descents, thereby extending previous results corresponding to εj = 0, ±1. By means of connection formulas satisfied by F it is possible to arrange the above hypergeometric function into three basic groups. In Part I we consider the cases (i) ε1 > 0, ε2 = 0, ε3 = 1 and (ii) ε1 > 0, ε2 = −1, ε3 = 0; the third case ε1, ε2 >0, ε3 =1 is deferred to Part II. The resulting expansions are of Poincar´e type and hold in restricted domains of the complex z-plane. Numerical results illustrating the accuracy of the different expansions are given

Asymptotics of integrals of Hermite polynomials

Integrals involving products of Hermite polynomials with the weight factor exp (−x2) over the interval (−∞,∞) are considered. A result of Azor, Gillis and Victor (SIAM J. Math. Anal. 13 (1982) 879–890] is derived by analytic arguments and extended to higher order products. An asymptotic expansion in the case of a product of four Hermite polynomials Hn(x) as n→∞is obtained by a discrete analogue of Laplace’s method applied to sums

Exponentially small expansions in the asymptotics of the Wright function

We consider exponentially small expansions present in the asymptotics of the generalised hypergeometric function, or Wright function, pΨq(z) for large |z| that have not been considered in the existing theory. Our interest is principally with those functions of this class that possess either a finite algebraic expansion or no such expansion and with parameter values that produce exponentially small expansions in the neighbourhood of the negative real z axis. Numerical examples are presented to demonstrate the presence of these exponentially small expansions

The asymptotic expansion of a generalisation of the Euler-Jacobi series

We consider the asymptotic expansion of the sum Sp(a; w) = ∞Ʃ n=1 e−anp/nw as a → 0 in | arg a| <1/2π for arbitrary finite p > 0 and w > 0. Our attention is concentrated mainly on the case when p and w are both even integers, where the expansion consists of a finite algebraic expansion together with a sequence of increasingly subdominant exponential expansions. This exponentially small component produces a transformation for Sp (a; w) analogous to the well-known Poisson-Jacobi transformation for the sum with p = 2 and w = 0. Numerical results are given to illustrate the accuracy of the expansion obtained

The asymptotics of the mittag-leffler polynomials

We investigate the asymptotic behaviour of the Mittag-Leffler polynomials Gn(z) for large n and z, where z is a complex variable satisfying 0 arg z 12 π . A summary of the asymptotic properties of Gn(ix) for real values of x and an approximation for its extreme zeros as n→∞ are given. When the variables are such that z/n is finite, an expansion is obtained using the method of steepest descents applied to a suitable integral representation. This expansion holds everywhere in the first quadrant of the z -plane except in the neighbourhood of the point z=in , where there is a coalescence of saddle points. Numerical results are presented to illustrate the accuracy of the various expansions