×

Asymptotics of the Gauss hypergeometric function with large parameters. I. (English) Zbl 1412.33008

Summary: We obtain asymptotic expansions for the Gauss hypergeometric function \[ F(a +\varepsilon_1\lambda,\,b +\varepsilon_2\lambda;\,c+\varepsilon_3\lambda;\,z) \] as \(|\lambda|\rightarrow\infty\) when the \(\varepsilon_j\) are finite by an application of the method of steepest descents, thereby extending previous results corresponding to \(\varepsilon_j= 0, \pm 1\). By means of connection formulas satisfied by \(F\), it is possible to arrange the above hypergeometric function into three basic groups. In Part I, we consider the cases (i) \(\varepsilon_1 > 0\), \(\varepsilon_2= 0\), \(\varepsilon_3= 1\) and (ii) \(\varepsilon_1 > 0\), \(\varepsilon_2= -1\), \(\varepsilon_3= 0\); the third case \(\varepsilon_1\), \(\varepsilon_2> 0\), \(\varepsilon_3= 1\) is deferred to Part II. The resulting expansions are of Poincaré type and hold in restricted domains of the complex \(z\)-plane. Numerical results illustrating the accuracy of the different expansions are given.

MSC:

33C05 Classical hypergeometric functions, \({}_2F_1\)
34E05 Asymptotic expansions of solutions to ordinary differential equations
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)

Software:

DLMF
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] T. M. CHERRY, Asymptotic expansions for the hypergeometric functions occurring in gas-flow theory, Proc. Roy. Soc. London A202 (1950) 507-522. · Zbl 0037.33102
[2] D. S. JONES, Rawlin’s method and the diaphonous cone, J. Mech. Appl. Math. 53 (2000) 91-109. · Zbl 1161.78316
[3] D. S. JONES, Asymptotics of the hypergeometric function, Math. Meth. Appl. Sci. 24 (2001) 369-389. · Zbl 0979.33002
[4] R. B. DINGLE, Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, London, 1973. · Zbl 0279.41030
[5] M. J. LIGHTHILL, The hodograph transformation in trans-sonic flow. II Auxiliary theorems on the hypergeometric functionsψn(τ), Proc. Roy. Soc. London A191 (1947) 341-351. · Zbl 0029.17801
[6] Y. L. LUKE, The Special Functions and Their Approximation, Vol. I, Academic Press, New York, 1969.
[7] A. B. OLDEDAALHUIS, Uniform asymptotic expansions for hypergeometric functions with large parameters, I, Anal. Appl. (Singap.) 1 (2003) 111-120.
[8] A. B. OLDEDAALHUIS, Uniform asymptotic expansions for hypergeometric functions with large parameters, II, Anal. Appl. (Singap.) 1 (2003) 121-128.
[9] A. B. OLDEDAALHUIS, Uniform asymptotic expansions for hypergeometric functions with large parameters, III, Anal. Appl. (Singap.) 8 (2010) 199-210.
[10] F. W. J. OLVER, Asymptotics and Special Functions, Academic Press, New York, 1974. Reprinted A. K. Peters, Massachussets, 1997. · Zbl 0303.41035
[11] F. W. J. OLVER, D. W. LOZIER, R. F. BOISVERT ANDC. W. CLARK(eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010. · Zbl 1198.00002
[12] R. B. PARIS, Hadamard Expansions and Hyperasymptotic Evaluation: An Extension of the Method of Steepest Descents, Encyclopedia of Mathematics and its Applications, Vol. 141, Cambridge University Press, Cambridge, 2011. · Zbl 1223.41021
[13] R. B. PARIS, Asymptotics of the Gauss hypergeometric function with large parameters, II, J. Classical Anal. 3, 1 (2013), 1-15. · Zbl 1412.33009
[14] R. B. PARIS ANDD. KAMINSKI, Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, Cambridge, 2001.
[15] A. D. RAWLINS, Diffraction by, or diffusion into, a penetrable wedge, Proc. Roy. Soc. London A455 (1999) 2655-2686. · Zbl 1062.78502
[16] B. RIEMANN, Sullo svolgimento del quoziente di due serie ipergeometriche in funzione continua infinita, in Collected Works of Bernhard Riemann (ed. H. Weber) 145-153, Dover, New York, 1953. the first logarithm is found to pass on to adjacent Riemann sheets of the log function.
[17] L. J. SLATER, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1960. · Zbl 0086.27502
[18] N. M. TEMME, Large parameter cases of the Gauss hypergeometric function, J. Comput. Appl. Math. 153 (2003) 441-462. · Zbl 1019.33003
[19] G. N. WATSON, Asymptotic expansions of hypergeometric functions, Trans. Cambridge Philos. Soc. 22 (1918) 277-308.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.