Richard B. Paris

Extended hypergeometric and confluent hypergeometric functions

An extension of the beta function by introducing an extra parameter, which proved to be useful earlier, is applied here to extend the hypergeometric and confluent hypergeometric functions. Since the latter functions contain many of the familiar special functions as sub-cases, this extension is expected to prove to be useful. The object of the present paper is to study this extension and its relationship with the hypergeometric and confluent hypergeometric functions.

Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function

Abstract In this paper, we present some completely monotonic functions and asymptotic expansions related to the gamma function. Based on the obtained expansions, we provide new bounds for Γ ( x + 1 ) / Γ x + 1 2 and Γ x + 1 2 .

A uniform asymptotic expansion for the incomplete gamma function

We describe a new uniform asymptotic expansion for the incomplete gamma function Γ(a,z) valid for large values of z. This expansion contains a complementary error function of an argument measuring transition across the point z = a (which is different from that in the well-known uniform expansion for large a of Temme), with easily computable coefficients that do not involve a removable singularity at z = a. Our expansion is, however, valid in a smaller domain of the parameters than that of Temme. Numerical examples are given to illustrate the accuracy of the expansion.

New properties and representations for members of the power-variance family. II

This is the continuation of [V. Vinogradov, R.B. Paris, and O. Yanushkevichiene, New properties and representations for members of the power-variance family. I, Lith. Math. J., 52(4):444–461, 2012]. Members of the powervariance family of distributions became popular in stochastic modeling which necessitates a further investigation of their properties. Here, we establish Zolotarev duality of the refined saddlepoint-type approximations for all members of this family, thereby providing an interpretation of the Letac–Mora reciprocity of the corresponding NEFs. Several illustrative examples are given. Subtle properties of related special functions are established.

On the zeros of the Pearcey integral

Zeroes for real values of the arguments of the Pearcey integral are numerically evaluated and plotted. From this numerical examination, it is apparent that these zeroes display a high degree of structure, the character of which is revealed through asymptotic analysis. Refinements to the resulting approximations are supplied near the papers end.

Clausen's series 3F2(1) with integral parameter differences and transformations of the hypergeometric function 2F2(x)

We obtain summation formulas for the hypergeometric series 3 F 2(1) with at least one pair of numeratorial and denominatorial parameters differing by a negative integer. The results derived for the latter are used to obtain Kummer-type transformations for the generalized hypergeometric function 2 F 2(x) and reduction formulas for certain Kampé de Fériet functions. Certain summations for the partial sums of the Gauss hypergeometric series 2 F 1(1) are also obtained.

Certain transformations and summations for generalized hypergeometric series with integral parameter differences

Certain transformation and summation formulas for generalized hypergeometric series with integral parameter differences are derived.

Asymptotic expansions of Laplace -type integrals III: 409

Abstract We consider the asymptotic expansion for large λ of Laplace-type integrals of the form ∫ 0 ∞ ∫ 0 ∞ g(x,y) e −λf(x,y) d x d y for a wide class of amplitude functions g(x,y) and ‘polynomial’ (noninteger powers are permitted) phases f(x,y) possessing an isolated, though possibly degenerate, critical point at the origin. The resulting algebraic expansions valid in a certain sector of the complex λ plane are based on recent results obtained in Kaminski and Paris (Philos. Trans. Roy. Soc. London A 356 (1998) 583–623; 625–667) when g(x,y)≡1. The limitation of the validity of the algebraic expansion to this sector as certain coefficients in the phase function are allowed to take on complex values is due to the appearance of exponential contributions. This is examined in detail in the special case when the phase function corresponds to a single internal point in the associated Newton diagram. Numerical examples illustrating the accuracy of the expansions are discussed.

An extension of Saalschütz's summation theorem for the series r+3Fr+2

The aim in this research note is to provide an extension of Saalschützs summation theorem for the series r+3Fr+2(1) when r pairs of numeratorial and denominatorial parameters differ by positive integers. The result is obtained by exploiting a generalization of an Euler-type transformation recently derived by Miller and Paris [Transformation formulas for the generalized hypergeometric function with integral parameter differences. Rocky Mountain J Math. 2013;43, to appear].

On a new class of summation formulae involving the Laguerre polynomial

By elementary manipulation of series, a general transformation involving the generalized hypergeometric function is established. Kummer’s first theorem, the classical Gauss summation theorem and the generalized Kummer summation theorem due to Lavoie et al. [Generalizations of Whipple’s theorem on the sum of a 3 F 2, J. Comput. Appl. Math. 72 (1996), pp. 293–300] are then applied to obtain a new class of summation formulae involving the Laguerre polynomial, which have not previously appeared in the literature. Several related results due to Exton have also been given in a corrected form.