The asymptotics of the generalised Bessel function

2017/11/07 by Paris, R B
#30B10 #30E15 #33C20 #34E05 #41A60 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1711.03006

Abstract

We demonstrate how the asymptotics for large |z| of the generalised Bessel function ₀Ψ₁(z)=∑_n=0^∞(zⁿ)/(Γ(an+b) n!), where a>-1 and b is any number (real or complex), may be obtained by exploiting the well-established asymptotic theory of the generalised Wright function _pΨ_q(z). A summary of this theory is given and an algorithm for determining the coefficients in the associated exponential expansions is discussed in an appendix. We pay particular attention to the case a=-1/2, where the expansion for z→±∞ consists of an exponentially small contribution that undergoes a Stokes phenomenon. We also examine the different nature of the asymptotic expansions as a function of arg z when -1

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