Browsing by Author "Paris, Richard B."
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The asymptotics of a new exponential sum
Paris, Richard B. (Elsevier, 20090101)The absolutely convergent exponential sum [formula missing] is studied for m→+∞ and fixed p when the parameter θ is allowed to become large such that θ/m remains finite. This situation corresponds, in general, to the trace ... 
Asymptotics of integrals of Hermite polynomials
Paris, Richard B. (Hikari Ltd, 2010)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 ... 
Asymptotics of the Gauss hypergeometric function with large parameters, I
Paris, Richard B. (Element d.o.o., 2013)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 ... 
Asymptotics of the Gauss hypergeometric function with large parameters, II
Paris, Richard B. (Element d.o.o., 2013)We obtain asymptotic expansions by application of the method of steepest descents for the Gauss hypergeometric function F(a+ε1λ,b+ε2λ;c+λ;z) as λ → ∞ when 0<ε1 <1 and ε1 >1 where, without loss of generality, it is supposed ... 
The asymptotics of the generalised Hermite–Bell polynomials
Paris, Richard B. (Elsevier, 20091015)The Hermite–Bell polynomials are defined by [formula missing] for n=0,1,2,… and integer r≥2 and generalise the classical Hermite polynomials corresponding to r=2. We obtain an asymptotic expansion for [formula missing] as ... 
The asymptotics of the mittagleffler polynomials
Paris, Richard B. (Element d.o.o., 2012)We investigate the asymptotic behaviour of the MittagLeffler 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 ... 
The asymptotics of the mittagleffler polynomials
Paris, Richard B. (Journal of classical analysis, 2012)We investigate the asymptotic behaviour of the MittagLeffler 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 ... 
Branching particle systems and compound Poisson processes related to PólyaAeppli distributions
Paris, Richard B.; Vinogradov, Vladimir (Serials Publications, 201503)We establish numerous new refined local limit theorems for a class of compound Poisson processes with PólyaAeppli marginals, and for a particular family of the branching particle systems which undergo critical binary ... 
Certain transformations and summations for generalized hypergeometric series with integral parameter differences
Miller, A. R.; Paris, Richard B. (Taylor & Francis, 201101)Certain transformation and summation formulas for generalized hypergeometric series with integral parameter differences are derived. 
Clausen's series 3F2(1) with integral parameter differences and transformations of the hypergeometric function 2F2(x)
Miller, A. R.; Paris, Richard B. (Taylor & Francis, 2012)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 ... 
Comments on "New hypergeometric identities arising from Gauss’s second summation theorem"
Rakha, Medhat A.; Rathie, Arjun K.; Chopra, Purnima; Paris, Richard B. (Miskolc University Press, 2012)In 1997, Exton [J. Comput. Appl. Math. 88 (1997) 269–274] obtained a general transfor mation involving hypergeometric functions by elementary manipulation of series. A number of hypergeometric identities not previously ... 
A derivation of two quadratic transformations contiguous to that of Gauss via a differential equation approach
Meethal, S.; Rathie, Arjun K.; Paris, Richard B. (Hikari, 20150129)The purpose of this note is to provide an alternative proof of two quadratic transformation formulas contiguous to that of Gauss using a differential equation approach. 
A derivation of two transformation formulas contiguous to that of Kummer’s second theorem via a differential equation approach
Kodavanji, S.; Rathie, Arjun K.; Paris, Richard B. (Hilaris, 2015)The purpose of this note is to provide an alternative proof of two transformation formulas contiguous to that of Kummer’s second transformation for the confluent hypergeometric function 1F1 using a differential equation approach. 
The discrete analogue of Laplace’s method
Paris, Richard B. (Elsevier, 201105)We give a justification of the discrete analogue of Laplace’s method applied to the asymptotic estimation of sums consisting of positive terms. The case considered is the series related to the hypergeometric function ... 
Eulertype transformations for the generalized hypergeometric function r+2Fr+1(x)
Miller, A. R.; Paris, Richard B. (Springer Verlag, 201102)We provide generalizations of two of Euler’s classical transformation formulas for the Gauss hypergeometric function extended to the case of the generalized hypergeometric function r+2 F r+1(x) when there are additional ... 
Exactification of the method of steepest descents: the Bessel functions of large order and argument
Paris, Richard B. (The Royal Society, 200409)The Hadamard expansion procedure applied to Laplace–type integrals taken along contours in the complex plane enables an exact description of the method of steepest descents. This mode of expansion is illustrated by the ... 
The expansion of a finite number of terms of the Gauss hypergeometric function of unit argument and the Landau constants
Paris, Richard B. (Hilaris, 2015)We obtain convergent inverse factorial expansions for the sum Sn(a, b; c) of the first n ≥ 1 terms of the Gauss hypergeometric function 2F1(a, b; c; 1) of unit argument. The form of these expansions depends on the location ... 
Exponential asymptotics of the Voigt functions
Paris, Richard B. (Springer, 20150601)We obtain the asymptotic expansion of the Voigt functionss K(x, y) and L(x, y) for large (real) values of the variables x and y, paying particular attention to the exponentially small contributions. A Stokes phenomenon is ... 
Exponentially small expansions in the asymptotics of the Wright function
Paris, Richard B. (Elsevier, 201005)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 ... 
Exponentially small expansions of the confluent hypergeometric functions
Paris, Richard B. (Hikari, 2013)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 ...