Incomplete Bessel K function/generalized incomplete gamma function

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Some mathematicians defined this type incomplete-version of Bessel function or this type generalized-version of incomplete gamma function:[1][2][3][4][5]

Kv(x,y)=1extyttv+1dt
γ(α,x;b)=0xtα1etbtdt
Γ(α,x;b)=xtα1etbtdt

Properties

Kv(x,y)=xvΓ(v,x;xy)
Kv(x,y)+Kv(y,x)=2xv2yv2Kv(2xy)
γ(α,x;0)=γ(α,x)
Γ(α,x;0)=Γ(α,x)
γ(α,x;b)+Γ(α,x;b)=2bα2Kα(2b)

One of the advantage of defining this type incomplete-version of Bessel function Kv(x,y) is that even for example the associated Anger–Weber function defined in Digital Library of Mathematical Functions[6] can related:

Aν(z)=1π0eνtzsinhtdt=1π0e(ν+1)tzet2+z2etd(et)=1π1ezt2+z2ttν+1dt=1πKν(z2,z2)

recurrence relations

Kv(x,y) satisfy this recurrence relation:

xKv1(x,y)+vKv(x,y)yKv+1(x,y)=exy

References

  1. "incompleteBesselK function | R Documentation". www.rdocumentation.org.
  2. "incompleteBesselK: The Incomplete Bessel K Function in DistributionUtils: Distribution Utilities". rdrr.io.
  3. Harris, Frank E. (2008). "Incomplete Bessel, generalized incomplete gamma, or leaky aquifer functions" (PDF). Journal of Computational and Applied Mathematics. 215: 260–269. doi:10.1016/j.cam.2007.04.008. Retrieved 2020-01-08.
  4. "Generalized incomplete gamma function and its application". 2018-01-14. Retrieved 2020-01-08.
  5. "Archived copy" (PDF). S2CID 126117454. Archived from the original (PDF) on 2019-12-23. Retrieved 2019-12-23. Cite journal requires |journal= (help)CS1 maint: archived copy as title (link)
  6. Paris, R. B. (2010), "Anger-Weber Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248