Browsing by Author "Paris, Richard B."
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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 ... 
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 ... 
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 ... 
Exponentially small expansions of the Wright function on the Stokes lines
Paris, Richard B. (Springer, 201401)We investigate a particular aspect of the asymptotic expansion of the Wright function pΨq(z) for large z. In the case p = 1, q ⩾ 0, we establish the form of the exponentially small expansion of this function on certain ... 
An extension of Saalschütz's summation theorem for the series r+3Fr+2
Kim, Yong S.; Rathie, Arjun K.; Paris, Richard B. (Taylor & Francis, 2013)The aim in this research note is to provide an extension of Saalschütz's summation theorem for the series r+3Fr+2(1) when r pairs of numeratorial and denominatorial parameters differ by positive integers. The result is ... 
Fluctuation properties of compound PoissonErlang Lévy processes
Paris, Richard B.; Vinogradov, Vladimir (Serials Publications, 201306)We derive an expression in terms of the Wright function for the density of the ﬁrstpassage times (or FPT’s) for the PoissonErlang Levy processes. For Poissonexponential processes, we establish an analogue of Zolotarev ... 
A generalisation of an expansion for the Riemann zeta function involving incomplete gamma functions
Paris, Richard B. (Hikari, 2009)We derive an expansion for the Riemann zeta function ζ(s) involving incomplete gamma functions with their second argument proportional to n2p, where n is the summation index and p is a positive integer. The possibility is ... 
A generalised Kummertype transformation for the pFp(x) hypergeometric function
Miller, A. R.; Paris, Richard B. (University of Toronto Press, 2012)In a recent paper, Miller derived a Kummertype transformation for the generalised hypergeometric function pFp(x) when pairs of parameters differ by unity, by means of a reduction formula for a certain Kampé de Fériet ... 
Generalization of two theorems due to Ramanujan
Kim, Yong S.; Rathie, Arjun K.; Paris, Richard B. (Taylor & Francis, 2013)The aim in this paper is to provide generalizations of two interesting entries in Ramanujan's notebooks that relate sums involving the derivatives of a function (t) evaluated at 0 and 1. The generalizations obtained are ... 
Generalizations of two infinite product formulas
Chen, ChaoPing; Paris, Richard B. (Taylor & Francis, 2014)Two interesting infinite product formulas were presented by Choi, Lee and Srivastava [A generalization of Wilf's formula. Kodai Math J. 2003;26:44–48], who generalized an infinite product formula posed by Wilf as a problem ...