Browsing by Author "Paris, Richard B."
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An alternative proof of the extended Saalschütz summation theorem for the r + 3Fr + 2(1) series with applications
Kim, Yong S.; Rathie, Arjun K.; Paris, Richard B. (John Wiley & Sons, 20150306)A simple proof is given of a new summation formula recently added in the literature for a terminating r + 3Fr + 2(1) hypergeometric series for the case when r pairs of numeratorial and denominatorial parameters differ by ... 
Asymptotic approximations for n!
Paris, Richard B. (Hikari, 2011)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 ... 
Asymptotic evaluation of an integral arising in quantum harmonic oscillator tunnelling probabilities
Paris, Richard B. (Hikari, 20150427)We obtain an asymptotic evaluation of the integral ʃ ∞ √2n+1 e−x2 H2 n (x) dx for n → ∞, where Hn(x) is the Hermite polynomial. This integral is used to determine the probability for the quantum harmonic oscillator in the ... 
An asymptotic expansion for the Stieltjes constants
Paris, Richard B. (Hilaris, 2015)The Stieltjes constants γn appear in the coefficients in the Laurent expansion of the Riemann zeta function ζ(s) about the simple pole s = 1. We present an asymptotic expansion for γn as n → ∞ based on the approach described ... 
The asymptotic expansion of a generalisation of the EulerJacobi series
Paris, Richard B. (European Journal of Pure and Applied Mathematics (EJPAM), 2016)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 ... 
The asymptotic expansion of a generalised Mathieu series
Paris, Richard B. (Hikari, 2013)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 ... 
Asymptotic expansion of ndimensional Faxéntype integrals
Paris, Richard B. (European Journal of Pure and Applied Mathematics (EJPAM), 2010)The asymptotic expansion of ndimensional 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, ... 
Asymptotics and MellinBarnes integrals
Paris, Richard B.; Kaminski, D. (Cambridge University Press, 2001) 
The asymptotics of a generalised Beta function
Paris, Richard B. (Hilaris, 2015)We consider the generalised Beta function introduced by Chaudhry et al. [J. Comp. Appl. Math. 78 (1997) 19–32] defined by B(x, y; p) = ʃ 1 0 tx−1(1 − t)y−1 exp [−p / 4t(1 − t)] dt, where ℜ(p) > 0 and the parameters x and ... 
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 ... 
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.