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I don't have experience with hypergeoemtric functions, but wish to compute the following limit:

$\lim_{x→\infty}{F([1],[a,b];-\frac{x^2}{4})}$, where $a,b$ are non-integer real parameters.

I tried to use Maple to calculate the limit and the result is 0. I tried to prove it or calculate it by hand and use an integral representation and some standard transformations but could not get the result.

Any help would be appreciated. Thanks!!

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The large-$x$ limit is only zero if $a+b>3/2$: $$_1{F}_2({1}; {a, b}; -x^2/4)=$$ $$=\sqrt{\tfrac{1}{\pi}}\Gamma (a) \Gamma (b) \sin \left(\tfrac{\pi}{2} (a+b-\tfrac{1}{2})-x\right)(2/x)^{a+b-3/2}+{\cal O}(1/x^2)$$

For example, when $a=b=1/2$ the amplitude of the oscillations increases as $\sqrt x$ (left plot), and when $a=b=3/4$ the amplitude does not decay (right plot):

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  • $\begingroup$ Yes! This is great! Thanks a lot for your help and nice pictures. May I ask where can I find this expression for 1F2? I went to the L.Luke's book. There are lots of transformations. Do I need to use the Bessel function? Thanks again! $\endgroup$
    – user431550
    Commented Feb 23, 2018 at 21:24
  • $\begingroup$ I used Mathematica for this, but I presume it should be in Lin & Wong --- although I can't check because it's behind a paywall :( $\endgroup$
    – Carlo Beenakker
    Commented Feb 23, 2018 at 21:43
  • $\begingroup$ Okay, no problem:). Thank you very much and your reference book! I will try to look for this book in my school library. $\endgroup$
    – user431550
    Commented Feb 24, 2018 at 0:18
  • $\begingroup$ @Carlo Beenakker: Can you kindly present the Mathematica code used by you to this end (I have in mind the asymptotics.)? BTW, the statement "I tried to use Maple to calculate the limit and the result is 0" in the question does not correspond to reality for Maple 2017.3 which fails with it $\endgroup$
    – user64494
    Commented Feb 24, 2018 at 8:04
  • $\begingroup$ Wolfram Alpha code $\endgroup$
    – Carlo Beenakker
    Commented Feb 24, 2018 at 9:09

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