The Fox-Wright function admits a convergent expansion near its positive singularity with recursively computed coefficients, together with explicit expressions for the jump and average value on the banks of the branch cut.
A note on Fox's H function in the light of Braaksma's results
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abstract
In our previous works we found a power series expansion of a particular case of Fox's $H$ function $H^{q,0}_{p,q}$ in a neighborhood of its positive singularity. An inverse factorial series expansion of the integrand of $H^{q,0}_{p,q}$ served as our main tool. However, a necessary restriction on parameters is missing in those works. In this note we fill this gap and give a simpler and shorter proof of the expansion around the positive singular point. We further identify more precisely the abscissa of convergence of the underlying inverse factorial series. Our new proof hinges on a slight generalization of a particular case of Braaksma's theorem about analytic continuation of Fox's $H$ function.
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math.CA 1years
2019 1verdicts
UNVERDICTED 1representative citing papers
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The Fox-Wright function near the singularity and branch cut
The Fox-Wright function admits a convergent expansion near its positive singularity with recursively computed coefficients, together with explicit expressions for the jump and average value on the banks of the branch cut.