Bessel Polynomial
Krall and Fink (1949) defined the Bessel polynomials as the function
where K_n(x) is a modified Bessel function of the second kind. They are very similar to the modified spherical bessel function of the second kind k_n(x). The first few are
(OEIS A001497). These functions satisfy the differential equation
| x^2y^('')+(2x+2)y^'-n(n+1)y=0. |
(8)
|
Carlitz (1957) subsequently considered the related polynomials
| [画像: p_n(x)=x^ny_(n-1)(1/x). ] |
(9)
|
This polynomial forms an associated Sheffer sequence with
| f(t)=t-1/2t^2. |
(10)
|
This gives the generating function
The explicit formula is
where x!! is a double factorial and _1F_1(a;b;z) is a confluent hypergeometric function of the first kind. The first few polynomials are
(OEIS A104548).
The polynomials satisfy the recurrence formula
| p_n^('')(x)-2p_n^'(x)+2np_(n-1)(x)=0. |
(18)
|
See also
Bessel Function, Modified Spherical Bessel Function of the Second Kind, Sheffer SequenceExplore with Wolfram|Alpha
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References
Carlitz, L. "A Note on the Bessel Polynomials." Duke Math. J. 24, 151-162, 1957.Grosswald, E. Bessel Polynomials. New York: Springer-Verlag, 1978.Krall, H. L. and Fink, O. "A New Class of Orthogonal Polynomials: The Bessel Polynomials." Trans. Amer. Math. Soc. 65, 100-115, 1949.Roman, S. "The Bessel Polynomials." §4.1.7 in The Umbral Calculus. New York: Academic Press, pp. 78-82, 1984.Sloane, N. J. A. Sequences A001497, A001498, and A104548 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Bessel PolynomialCite this as:
Weisstein, Eric W. "Bessel Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BesselPolynomial.html