Asymptotic expansion of the generalized hypergeometric function \({}_p F_q(z)\) as \(z\to\infty\) for \(p<q\). (English) Zbl 1528.33008
Summary: The asymptotic expansion of the generalized hypergeometric function \({}_p F_q(z)\) as \(z\to+\infty\) involves a coefficient sequence \(\{ c_k\}\). Explicit formulas are given for this sequence when \(0\leq p<q\). The result is based on an integral representation of the generalized hypergeometric function that allows application of Watson’s lemma.
MSC:
| 33C20 | Generalized hypergeometric series, \({}_pF_q\) |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
Software:
DLMFReferences:
| [1] | Luke, Y., Integrals of Bessel Functions, Dover Reprint (Dover Publications, Mineola, New York, 2014). |
| [2] | Nörlund, N. E., Hypergeometric functions, Acta Math.94 (1955) 289-349. · Zbl 0067.29402 |
| [3] | Olver, F. W. J., Asymptotics and Special Functions (A K Peters, Natick, Massachusetts, 1997). · Zbl 0982.41018 |
| [4] | Olver, F. W. J., Lozier, D. W., Boisvert, F. F. and Clark, C. W. (eds.), NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010). · Zbl 1198.00002 |
| [5] | Volkmer, H. and Wood, J., A note on the asymptotic expansion of generalized hypergeometric functions, Anal. Appl.12 (2014) 1-9. · Zbl 1282.33012 |
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