Krall and Fink (1949) defined the Bessel polynomials as the function
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(1)
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(2)
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where 
is a modified Bessel function
of the second kind. They are very similar to the modified
spherical bessel function of the second kind 
. The first few are
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(3)
|
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(4)
|
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(5)
|
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(6)
|
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(7)
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(OEIS A001497). These functions satisfy the differential equation
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(8)
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Carlitz (1957) subsequently considered the related polynomials
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(9)
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This polynomial forms an associated Sheffer sequence with
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(10)
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This gives the generating function
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(11)
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The explicit formula is
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(12)
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(13)
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where 
is a double factorial and 
is a confluent
hypergeometric function of the first kind. The first few polynomials are
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(14)
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(15)
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(16)
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(17)
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(OEIS A104548).
The polynomials satisfy the recurrence formula
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(18)
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