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