An expansion of the hypergeometric function in Bessel functions

Nagel, Bengt

doi:10.1063/1.1415431

Cited by 3 publications

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“…Our expressions also provide an alternate way to find the AEs for some HGFs with non-integer parameters that have previously appeared in physics. Examples of such HGFs are F n/2+s + ,n/2+s − n+1 ; x , with s ± = b ± 1 + (a 2 − 1)n 2 /[2(a − 1)], for real a, b, x and n → ∞ in [1]; F (n+is)/2,(n−is)/2 n+1 ; c n 2/3 − b 2 a 2 , with s = √ a 2 n 2 − 1/b for real a, b, c and n → ∞ in [28]; F n−n,n+1 m+1 ; 1−z 2 and F (n−m)/2+1,(n−m+1)/2 n+3/2 ; 1 z 2 for z complex, n, m → ∞ and m/(n + 1/2) constant in [27]; F −an,−bn 1−an ; −x for real a, b, x and n → ∞ in [21]; and F a,b c+1 ; − z 4ab for complex z and |a|, |b| → ∞, in [7,46]. All of these can be mapped to the cases evaluated here by substitutions for the parameters and/or the variable.…”

Section: Discussionmentioning

confidence: 99%

“…; 1 z 2 for z complex, n, m → ∞ and m/(n + 1/2) constant in [27]; F −an,−bn 1−an ; −x for real a, b, x and n → ∞ in [21]; and F a,b c+1 ; − z 4ab for complex z and |a|, |b| → ∞, in [7,46]. All of these can be mapped to the cases evaluated here by substitutions for the parameters and/or the variable.…”

Section: Discussionmentioning

confidence: 99%

“…The AEs of the HGF with more general values of ε i have been studied only for particular HGFs appearing in different problems in physics. Some examples are the solution of the compressible gas flow dynamics problem [1], the persistence problem of the solutions of the sine-Gordon equation in [28], the Legendre functions P −m n (z) and Q −m n (z) for n, m → ∞ in [27], the plaquette expansion for lattice-Hamiltonian systems [21], and the quantal description of the scattering of charged particles in atomic systems in [7,46].…”

Section: Introduction and Outline Of The Problemmentioning

confidence: 99%

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Asymptotic expansions of the hypergeometric function with two large parameters—application to the partition function of a lattice gas in a field of traps

Cvitković

Smith

Pande

2017

J. Phys. A: Math. Theor.

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The canonical partition function of a two-dimensional lattice gas in a field of randomly placed traps, like many other problems in physics, evaluates to the Gauss hypergeometric function 2 F 1 (a, b; c; z) in the limit when one or more of its parameters become large. This limit is difficult to compute from first principles, and finding the asymptotic expansions of the hypergeometric function is therefore an important task. While some possible cases of the asymptotic expansions of 2 F 1 (a, b; c; z) have been provided in the literature, they are all limited by a narrow domain of validity, either in the complex plane of the variable or in the parameter space. Overcoming this restriction, we provide new asymptotic expansions for the hypergeometric function with two large parameters, which are valid for the entire complex plane of z except for a few specific points. We show that these expansions work well even when we approach the possible singularity of 2 F 1 (a, b; c; z), |z| = 1, where the current expansions typically fail. Using our results we determine asymptotically the partition function of a lattice gas in a field of traps in the different possible physical limits of few/many particles and few/many traps, illustrating the applicability of the derived asymptotic expansions of the HGF in physics.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introduction and Outline Of The Problemmentioning

confidence: 99%

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Asymptotic expansions of the hypergeometric function with two large parameters—application to the partition function of a lattice gas in a field of traps

Cvitković

Smith

Pande

2017

J. Phys. A: Math. Theor.

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“…5.7. (1)] for the original result and also [24,32] for further develoments. In the framework of the analysis of noncompact semisimple Lie groups, a similar result is also recalled in [20,Eq.…”

Section: By Settingmentioning

confidence: 99%

Scattering theory and an index theorem on the radial part of SL(2,R)

Inoue,

Richard

2022

Preprint

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We present the spectral and scattering theory of the Casimir operator acting on the radial part of SL(2, R). After a suitable decomposition, these investigations consist in studying a family of differential operators acting on the half-line. For these operators, explicit expressions can be found for the resolvent, for the spectral density, and for the Moeller wave operators, in terms of the Gauss hypergeometric function. An index theorem is also introduced and discussed. The resulting equality links various asymptotic behaviors of the hypergeometric function.

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Confluence expansions of the generalized hypergeometric function

Nagel

2003

Journal of Mathematical Physics

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By confluencing a subset of upper and lower parameters in the generalized hypergeometric function P F Q (a 1, ,...,a P ,c 1 ,...,c Q ;z) with the variable z one obtains a lower-order hypergeometric function in the limit when the confluence parameters go to infinity. It is shown that this function is the first term in a convergent expansion in terms of functions of the same type with parameters increasing stepwise by integers and coefficients which are polynomials in the reciprocals of the confluence parameters. These polynomials have nonvanishing lowest degree terms whose power increases with . The expansion can hence be used to derive asymptotic expansions for large but finite absolute values of the confluence parameters.

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An expansion of the hypergeometric function in Bessel functions

Abstract: An expansion of the hypergeometric function F12(a,b,c+1;−z2/4ab) in Bessel functions of argument z is derived. This expansion can be used to obtain an asymptotic expansion of the hypergeometric function for large absolute values of a and b.

Cited by 3 publications

References 1 publication

Asymptotic expansions of the hypergeometric function with two large parameters—application to the partition function of a lattice gas in a field of traps

Asymptotic expansions of the hypergeometric function with two large parameters—application to the partition function of a lattice gas in a field of traps

Scattering theory and an index theorem on the radial part of SL(2,R)

Confluence expansions of the generalized hypergeometric function

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